Plenary, invited and confirmed contributed talks
Seventh and final update. September 10th, 16:00.

Avron, Joseph Plenary Swimming lessons for microbots
Bach, Volker Plenary Nonrelativistic QED The spectral analysis for Hamiltonians of nonrelativistic matter coupled to the quantized radiation field QED has been an active field of research in mathematical physics. A review of the progress during the last decade will be given. The problems which were addressed range from stability (of matter) properties, existence and nonexistence of a ground state, resonances and dynamical properties to scattering theory. Ultraviolet and infrared divergencies play a special role for these problems. The mathematical tools developed for the analysis includes by now several implementations of a renormalization group acting on operators.
De Bièvre, Stephan Plenary The semi-classical behaviour of quantum map eigenfunctions Quantum maps are a theoretical laboratory for questions in quantum chaos. I will discuss here some recent results in the understanding of the eigenfunction behaviour of such systems in the semi-classical limit.
Eliasson, Hakan Plenary Floquet solutions for the quasi-periodic Schroedinger equation in two dimensions
Erdos, Laszlo Plenary Towards the quantum Brownian motion Einstein's kinetic theory of the Brownian motion, based upon light water molecules continuously bombarding the heavy pollen, provided an explanation of diffusion from the Newtonian mechanics. Since the discovery of quantum mechanics it has been a challenge to verify the emergence of diffusion from the Schrodinger equation. In this talk I will report on a mathematically rigorous derivation of a diffusion equation as a long time scaling limit of a random Schrodinger equation in a weak, uncorrelated disorder potential This is a joint work with M. Salmhofer and H.T. Yau.
Exner, Pavel Plenary Quantum waveguides: mathematical problems In this talk we discuss three mathematical problems related to quantum waveguide systems; we present some recent results and review possible extensions and open questions. The first one, based on a common work with D.~Borisov, concerns asymptotic behaviour of discrete spectrum due to distant local perturbations. The second problem deals with approximation of Schr\"odinger operators on graphs by families of operators on appropriate manifolds; it relies on results obtained in collaboration with O.~Post. Finally, the third topic is an isoperimetric problem for point interaction Hamiltonians.
Helffer, Bernard Plenary Small eigenvalues of Witten Laplacians and metastability In this talk, our main object is the analysis of the small eigenvalues (as $h\ar 0$) of the Laplacian attached to the quadratic form $$ C_0^\infty(M)\ni v \mapsto h^2 \,\int_{\Omega}\left|\nabla v (x)\right|^2\;e^{-2f(x)/h}~dx\;, $$ where $f$ is a Morse function. Our aim is to extend and improve the previous results of Bovier-Eckhoff-Gayrard-Klein concerned with the case of $\rz^n$. In particular, we would like to analyze the compact case and the case with boundary. We will show how the the introduction of a Witten cohomology complex permits to obtain a very accurate asymptotics for the exponentially small eigenvalues. In particular, we analyze the effect of the boundary in the asymptotics. These results have been obtained in collaboration with M. Klein and (or) F. Nier.
Jitomirskaya, Svetlana Plenary The Ten Martini Problem We will present the solution of the Ten Martini Problem (Cantor spectrum for the Almost Mathieu operator) for all irrational frequencies and nonzero couplings. This is joint work with Artur Avila.
Klopp, Frédéric Plenary Strong resonant tunneling, level repulsion and spectral type for one-dimensional adiabatic quasi-periodic Schrödinger operator We consider one dimensional adiabatic quasi-periodic Schrödinger operators in the regime of strong resonant tunneling. We show the emergence of a level repulsion phenomenon which is seen to be very naturally related to the local spectral type of the operator: the more singular the spectrum, the weaker the repulsion.
Rivasseau, Vincent Plenary The Hubbard Model at half filling, a two dimensional non-Fermi liquid
Solovej, Jan Philip Plenary The energy asymptotics of a low density Fermi gas. In this talk I will discuss the thermodynamic ground state energy of a gas of fermionic particles with spin 1/2 interacting through a repulsive short range two-body potential. The thermodynamic limit of the ground state energy per particle will be studied as a function of the density. The aim of the talk is to present a proof of the two term asymptotic formula for this energy at low density. The first term is simply the same as for the free Fermi gas. The second term depends on the potential but only through its scattering length. This is joint work with E. Lieb and R. Seiringer.
Spohn, Herbert Plenary Phonon Boltzmann equation and its microscopic derivation We consider a simplified version of a dielectric crystal with either a weak nonlinear on-site potential or isotope disorder. We explain the kinetic scaling under which the phonon Boltzmann equation is obtained. In the case of isotope disorder a complete proof, jointly with J. Lukkarinen, is under construction.
Yngvason, Jakob Plenary Bose Einstein Condensation as a Quantum Phase Transition in an Optical Lattice One of the most remarkable recent developments in the study of ultracold Bose gases is the observation of a reversible transition from a Bose Einstein condensate to a state composed of localized atoms as the strength of a periodic, optical trapping potential is varied. In joint work with M. Aizenman, E.H. Lieb, R. Seiringer and J.P. Solovej a model of this phenomenon has been analyzed rigorously. The gas is a hard core lattice gas and the optical lattice is modelled by a periodic potential of strength $\lambda$. For small $\lambda$ and temperature BEC is proved to occur, while at large $\lambda$ BEC disappears, even in the ground state, which is a Mott-insulator state with a characteristic gap. The inter-particle interaction is essential for this effect.
Zelditch, Steve Plenary Random critical points and the vacuum selection problem of string/M theory According to string/M theory, our universe has small extra dimensions in the form a Calabi-Yau 3 fold. The vacuum selection problem is to select the physical CY 3-fold out of the moduli space of all CY 3-folds. The candidates CY manifolds are critical points of a functional on the moduli space, which (in the supersymmetric theory) is the norm-squared of a holomorphic section (the superpotential) of a line bundle over moduli space. (The same kind of critical point equation is also the attractor equation of black holes). The problem is that there are many candidate superpotentials and each has many critical points. M.R. Douglas has proposed a program to study the statistics of critical points: to count the number of critical points satisfying physical contraints, to determine how they are distributed in moduli space. My talk will report on work with M. R. Douglas and B. Shiffman giving rigorous results on these problems. Mathematically, the problem is to study the distribution of critical points of random holomorphic functions in a discrete and in a Gaussian ensemble.
Avila, Artur Invited The Zorich conjecture The motion of charged particles on Fermi surfaces in crystals leads one to consider the dynamics of translation flows on surfaces. It was discovered numerically by Zorich that the asymptotic behavior in homology of the trajectories of a typical translation flow in a surface of genus at least two differs signicantly from the genus one case. His quite precise description of this phenomenon depended on conjectured simplicity of the Lyapunov exponents of a certain cocycle over the relevant renormalization dynamics. I will discuss the proof of this conjecture. This is joint work with Marcelo Viana.
Buttiker, Markus Invited Scattering Theory of Dynamic Electrical Transport Quantum scattering theory is widely used to calculate conductance and noise of conductors which are so small that the electron wave nature is important. We emphasize a range of additional conduction problems which can be formulated with the help of scattering theory such as the dynamic conductance and in particular parametric quantum pumping. For dynamical problems the concept of partial density of states, injectivities and emissivities is central [1]. These are density of states for which the injecting or receiving contact or both are specified. We illustrate the role of these densities for quantum pumps and give the expression for the instantaneous current [2]. [1] Current partition in multiprobe conductors in the presence of slowly oscillating potentials M. Buttiker, H. Thomas, A. Pretre, Zeitschrift fur Physik B-Condensed Matter 94, 133 (1994) [2] Adiabatic quantum pump in the presence of external ac voltages M. Moskalets and M. Büttiker Phys. Rev. B 69, 205316 (2004)
Derezinski, Jan Invited Massless bosons in 1+1 dimension It is usually said that bosonic massless quantum fields in 1+1 dimension do not exist unless one introduces indefinite metric. I will describe a construction of such fields and their conformal invariance, which uses only positive definite Fock spaces.
Elgart, Alexander Invited Equality of the bulk and edge Hall conductances The integral quantum Hall effect can be explained either as resulting from bulk or edge currents (or, as it occurs in real samples, as a combination of both). The equality of two correponding conductances has been established by the number of authors for the case that the Fermi energy lies in the spectral gap of the bulk Hamiltonian. We prove this equality, by quite different means, in the more general setting that the bulk Hamiltonian exhibits dynamical localization in the vicinity of the Fermi energy. This is a joint work with G.-M. Graf and J.H. Schenker
Fröhlich, Jürg Invited Nonequilibrum statistical mechanics and thermodynamics I review some recent progress in deriving the fundamental laws of thermodynamics from nonequilibrium statistical mechanics. I focus on some special aspects of this general program related to the "isothermal theorem", which describes what happens when a thermodynamic system undergoes a process in thermal contact with a heat bath. The role of relative entropy will be emphasized.
Germinet, Francois Invited Localization and delocalization for some random Landau Hamiltonian We shall present recent results for some class of random Schrodinger operators with a constant magnetic field in two dimensions. Properties such as dynamical localization and transport (dynamical delocalization, current) will be shown to exist for different models with a random potential located either in the full physical space or in a half-space.
Glazek, Stanislaw Invited Limit cycles in quantum mechanics We discuss the simplest model Hamiltonian identified so far that exhibits a renormalization group limit cycle. The model is exactly soluble and suggests that limit cycles may be a commonplace for quantum Hamiltonians requiring renormalization, in contrast to experience to date with classical models of critical points, where fixed points are far more common.
Klich, Israel Invited Counting Statistics of Quantum Transport
Krikorian, Raphael Invited Lyapunov exponents or reducibility of quasi-periodic systems I will describe the following result we obtain with Artur Avila. Typical $SL(2,R)$-valued quasi-periodic cocycles either have positive (fibered) Lyapunov exponents or are reducible (in a sense to be specified). This result is also true for quasi-periodic Schrodinger operators and allows us to give a proof of the Aubry-Andre conjecture (concerning the Lebesgue measure of the spectrum for the almost Mathieu operator). This is a non perturbative result.
Kuchment, Peter Invited On quantum graph models The talk will provide a quick tour of quantum graph models of thin structures (quantum wires, thin photonic crystals, etc.) and of some their spectral structures.
Mantoiu, Marius Invited Functional Calculus in a Variable Magnetic Field We develop a pseudodifferential calculus of Weyl type in the presence of a variable magnetic field and put into evidence its connection with twisted crossed product C*-algebras, containing the resolvent family of magnetic Schroedinger operators.
Marklof, Jens Invited Weyl's law and quantum ergodicity for maps with divided phase space For a general class of unitary quantum maps, whose underlying classical phase space is divided into ergodic and non-ergodic components, we prove analogues of Weyl's law for the distribution of eigenphases, and the Schnirelman-Zelditch-Colin de Verdiere Theorem on the equidistribution of eigenfunctions with respect to the ergodic components of the classical map (quantum ergodicity). We apply our main theorems to quantised linked twist maps on the torus.
Merkli, Marco Invited Stability of Equilibria with a Condensate We consider a quantum system composed of a spatially infinitely extended free Bose gas with a condensate, interacting with a small system (quantum dot) which can trap finitely many Bosons. Due to spontaneous symmetry breaking in the presence of the condensate the system has many equilibrium states, for each fixed temperature. We extend the notion of Return to Equilibrium to systems possessing a multitude of equilibrium states and show in particular that a condensate coupled to a quantum dot has the property of Return to Equilibrium in a weak coupling sense: any local perturbation of an equilibrium state of the coupled system, evolving under the interacting dynamics, converges in the long time limit to some asymptotic state. The latter is, modulo an error term, an equilibrium state which depends in an explicit way on the local perturbation characterizing the initial condition (an effect due to long-range correlations). The error term vanishes in the small coupling limit.
Puig, Joaquim Invited Cantor Spectrum for Quasi-Periodic Schrödinger Operators. We will present some results concerning the Cantor structure of the spectrum of quasi-periodic Schrödinger operators.These are obtained studying the dynamics of the corresponding eigenvalue equations, specially the notion of reducibility and Floquet theory. We will deal with the Almost Mathieu case, and the solution of the ``Ten Martini Problem'' for Diophantine frequencies, as well as other models.
Rudnick, Zeev Invited Fluctuations of matrix elements for quantum chaotic systems For many classically chaotic systems it is believed that the quantum wave functions become uniformly distributed, that is the matrix elements of smooth observables tend to the phase space average of the observable. In this talk I will describe the FLUCTUATIONS of the matrix elements around the phase space average. For generic systems, it has been conjectured by Feingold and Peres that the matrix coefficients have a limiting variance which is given by the classical variance of the observable, and that the value distribution is Gaussian. I will check this predictions for two well-studied systems: Quantized toral automorphisms (Cat maps) in arbitrary dimensions, and the modular domain. In both cases interesting deviations from the generic predictions appear. In particular for the modular domain the higher moments of the normalized matrix coefficients are shown to blow up.
Schlein, Benjamin Invited Derivation of the Gross-Pitaevski Equation for the Dynamics of a Bose-Einstein Condensate We consider a system of N bosons in a volume of order one, interacting through a pair potential with scattering length of order 1/N (dilute gas). We prove that in the limit of large N the dynamics of the system is correctly described by the Gross Pitaevski Equation. Our analysis requires a modification of the original Hamiltonian, which cuts off the interaction whenever three or more particles come into a region much smaller than the typical interparticle distance.
Schulz-Baldes, Hermann Invited Perturbative results for products of random matrices Random products of symplectic matrices naturally appear in the study of quasi one-dimensional random Schroedinger operators. In many physically interesting situations, there is a coupling constant controlling the degree of non-commutativity. Perturbative techniques then allow to calculate the associated persistant quantities, namely the Lyapunov exponents, the variance in the associated central limit theorem and moments of the invariant measure. Applications concern the Anderson model on a strip, a test of single parameter scaling in one dimension as well as the random polymer model.
Seba, Petr Invited spin polarized electron filtering We theoretically analyze the possibility to use a ferromagnetic gate as a spin-polarization filter for one-dimensional electron systems formed in semiconductor heterostructures showing strong Rashba spin-orbit interaction. The proposed device is based on the effect of the breaking time-reversal symmetry due to the presence of weak magnetic fields. For a proper strength and magnetic field orientation there appears an energy interval in the electron energy spectrum at which the orientation of spin states is controlled by the direction of the electron velocity. It leads to the natural spin polarization of the electron current if the Fermi energy falls into this energy interval.
Soler, Juan Invited Dispersive properties and long-time dynamics of the Schrödinger-Poisson-X^\alpha system In this talk we analyze the asymptotic behaviour of solutions to the Schr\"odinger-Poisson-X^\alpha and, in particular, the Schr\"odinger-Poisson-Slater (SPS) system in the frame of semiconductor modeling. Depending on the potential energy and on the physical constants associated with the model, the repulsive SPS system develops stationary or periodic solutions. These solutions preserve the $L^p$ norm or exhibit dispersion properties. In comparison with the Schr\"odinger-Poisson (SP) system, only the last kind of solutions appear.
Stollmann, Peter Invited Generic singular continuous spectrum for geometric disorder We show that geometric disorder leads to purely singular continuous spectrum generically. The main input is a result of Simon known as the ``Wonderland theorem''. Here, we provide an alternative approach and actually a slight strengthening by showing that various sets of measures defined by regularity properties are generic in the set of all measures on a locally compact metric space. As a byproduct we obtain that a generic measure on euclidean space is singular continuous.
Teufel, Stefan Invited Precise coupling terms in adiabatic quantum evolution It is known that for multi-level time-dependent quantum systems one can construct superadiabatic representations in which the coupling between separated levels is exponentially small in the adiabatic limit. For two-state systems with analyitic real-symmetric Hamiltonian we construct such a superadiabatic representation and explicitly determine the asymptotic behavior of the exponentially small coupling term. First order perturbation theory in the superadiabatic representation then allows us to describe the time-development the of exponentially small adiabatic transitions, which, as predicted by M. Berry, have the universal form of an error function. This is joint work with V. Betz.
Adami, Riccardo Contributed The NLS in dimension one as the limit of a many-body problem We consider the problem of rigorously deriving the cubic nonlinearity arising in the one-body NLS from a N-body, linear dynamics. Such a problem is relevant in the study of Bose-Einstein condensates in the Gross-Pitaevskii (GP) regime, in which an effective equation with the same nonlinearity is prescribed. Lieb, Seiringer and Yngvason recently proved that, if the condensate lies in a very elongated trap, then the system becomes genuinely one-dimensional, and its behaviour is well described by the dynamics of a Lieb-Liniger gas: a one-dimensional gas of particles interacting by a two-body Dirac's delta potential. We show that in a suitable weak coupling limit such a system reproduces the GP regime at the level of BBGKY hierarchies, whereas the problem of deriving the one-particle GP equation is still unsolved. The result was obtained in collaboration with C.Bardos, F.Golse and A.Teta.
Benguria, Rafael Contributed H_2^+ in a strong magnetic field described via a solvable model We consider a hydrogen molecular ion (H_2^+) in the presence of a strong homogeneous magnetic field. In this regime, the effective Hamiltonian is almost one dimensional with a potential energy which looks like a sum of two Dirac delta functions. This model is solvable, but not close enough to our exact Hamiltonian for relevant strengths of the magnetic field. However, we show that the correct values of the equilibrium distance as well as the binding energy of the ground state of the ion can be obtained when incorporating perturbative corrections up to second order.
Bentosela, François Contributed Dynamics of Bloch electrons in constant electric field We study in one dimension the dynamics of an electron subjected to the sum of a periodic potential less singular than the Dirac comb and a linear potential. For a large class of initial conditions we prove the increase of the kinetic energy as time increases. As a by-product we get for the corresponding hamiltonian the existence of continuous spectrum.
Berglund, Nils Contributed On the stochastic exit problem - irreversible case
Bolte, Jens Contributed Semiclassical propagation of coherent states with spin-orbit interaction We study semiclassical approximations to the time evolution of coherent states for general spin-orbit coupling problems in two different semiclassical scenarios: The limit $\hbar\to 0$ is first taken with fixed spin quantum number $s$ and then with $\hbar s$ held constant. In these two cases different classical spin-orbit dynamics emerge. We prove that a coherent state propagated with a suitable classical dynamics approximates the quantum time evolution up to an error of size $\sqrt{\hbar}$ and identify an Ehrenfest time scale. Subsequently an improvement of the semiclassical error to an arbitray order $\hbar^{N/2}$ is achieved by a suitable deformation of the state that is propagated classically.
Borisov, Denis Contributed Spectrum of the Magnetic Schrodinger Operator in a Waveguide with Combined Boundary Conditions We consider the magnetic Schrodinger operator in a two-dimensional strip. On the boundary of the strip the Dirichlet boundary condition is imposed except for a fixed segment (window), where it switches to magnetic Neumann. We deal with a smooth compactly supported field as well as with the Aharonov-Bohm field. We give an estimate on the maximal length of the window, for which the discrete spectrum of the considered operator will be empty. In the case of a compactly supported field we also give a sufficient condition for the presence of eigenvalues below the essential spectrum.
Bruneau, Laurent Contributed A Hamiltonian model for linear friction We introduce and study a Hamiltonian model of a particle coupled to a dissipative environment in such a way that this particle experiences a friction force proportional to its velocity.
Cassanas, Roch Contributed Reduced Gutwiller formula with symmetry For a classical Hamiltonian with a finite group of symmetries, we give semi-classical asymptotics in a neighbourhood of an energy $E$ of the regularized spectral density of the quantum Hamiltonian restricted to symmetry subspaces of Peter-Weyl defined by characters of the group. If we suppose that the energy level $\Sigma_E$ is compact non-critical, and that its periodic orbits are non degenerate, we get a Gutzwiller type formula for the reduced Hamiltonian, whose oscillating part involves the symmetry properties of closed trajectories of $\Sigma_E$. The proof makes great use of the work of Combescure and Robert on coherent states.
Catto, Isabelle Contributed Enhanced binding revisited in nonrelativistic QED
Cheon, Taksu Contributed Quantum Abacus - Locational Qubit realization with Quantum Point Interactions We show that the $U(2)$ family of point interactions on a line can be utilized to provide the $U(2)$ family of qubit operations for quantum information processing. Qubits are realized as localized states in either side of the point interaction which represents a controllable gate. The manipulation of qubits proceeds in a manner analogous to the operation of an abacus. Ref: T.Cheon, I.Tsutsui and T.Fulop, "Quantum Abacus", quant-ph/0404039
Cornean, Horia Contributed A rigorous proof for the Landauer-Buttiker formula This is joint work with Arne Jensen and Valeriu Moldoveanu. We study quantum transport in finite systems coupled to particle reservoirs by semi-infinite leads, complementing some recent results obtained for quantum pumps by Avron and co-workers. Using a tight-binding framework, we rigorously prove the Landauer-Buttiker formula. As it is well known in physics, this formula states that the conductance coefficients for such systems (derived from a Kubo-type linear-response theory), are closely related to the S-matrix of the associated stationary scattering problem. As an application, the resonant transport through a quantum dot is discussed. The single charge tunneling processes are related to the spectral properties of an effective Hamiltonian via the Feshbach map.
De Oliveira, Cesar R. Contributed Dynamical delocalization for a 1D random model A discrete Hamiltonian with a random two-valued potential is considered. Although its spectrum is pure point, dynamical delocalization can be proven (for specific parameter values).
Dittrich, Jaroslav Contributed Quantum mechanics with piecewise constant mass The Hamiltonian of a particle moving with different effective masses in two different space domains is defined by the method of self-adjoint extensions. The additional interaction supported by the domains boundary is distiguished from the effects of different masses only. Mainly the one-dimensional case will be discussed.
Ekholm, Tomas Contributed Stability of the magnetic Schrödinger operator in a waveguide The spectrum of the Schrödinger operator in a quantum waveguide is known to be unstable in two and three dimensions. Any enlargement of the waveguide produces eigenvalues beneath the continuous spectrum. Also if the waveguide is bent eigenvalues will arise below the continuous spectrum. We consider the case when a magnetic field is added into the system. The spectrum of the magnetic Schrödinger operator is proved to be stable under small local deformations and also under small bending of the waveguide. The proof includes a magnetic Hardy-type inequality in the waveguide, which is interesting in its own.
Fournais, Soeren Contributed On the ground state of magnetic hamiltonians Joint with B. Helffer. In this talk I will present recent results on the asymptotics of the ground state of the Neumann realisation of the usual magnetic Schrodinger operator in a two-dimensional domain. By a procedure of reduction to the boundary we obtain a complete asymptotics, in the semiclassical regime, of the lowest lying eigenvalues. Due to applications to super-conductivity this problem has been studied a lot recently in the litterature. We mention the works by Bauman-Phillips-Tang, Lu-Pan, del Pino-Felmer- Sternberg and Helffer-Morame. Our results are a sharpening of these previous results.
Grebert, benoit Contributed Birkhoff normal form for pdes with tame modulus I will present an abstract Birkhoff normal form theorem for Hamiltonian partial differenrial equations. The theorem applies to semilinear equations with nonlinearity satisfying a property that will be called of tama modulus. The normal form can be used to deduce informations on the dynamics of the system. In particular, in the nonresonant case, one gets that any small solution remains close to a torus for very long times. Moreover one gets a long time estimate of the solution in higher sobolev norms. The result applies to several concrete examples ranging from the nonlinear wave equation with Dirichlet or periodic boundary conditions in one space dimension to some particular NLS and plate equations in d-space dimension. This is a joint work with Dario Bambusi
Gruber, Michael J. Contributed Zero field Hall effect for particles with spin 1/2 The integral QHE for Schrödinger electrons (particles described by a Schrödinger operator) in strong magnetic fields results in the quantisation of conductance. It is well-known by now that the remarkable stability of this effect can be traced back to its topological nature: the conductance is given by the Chern number of an adiabatic connection. We describe a QHE for Dirac fermions (particles described by a Dirac operator with spin 1/2) which is present even without a magnetic field and whose sign depends on the mass of the particles. The motivation for this is twofold: as a continuous limit of lattice models used in physics; and as a case study of the effect of the spectral gap of Dirac operators with mass on transport properties. We analyze the edge case (on the halfplane H²) and show that the conductance turns out to be given as a spectral flow through the gap. As such it is stable with respect to perturbations. We show furthermore how this spectral flow depends on the interplay between the sign of the mass and the boundary condition along the edge. This is joint work with Marianne Leitner.
Guillot, jean-claude Contributed The dressed non-relativistic electron in a magnetic field.( joint with L.Amour and B.Grébert ) We consider a non-relativistic electron interacting with a classical magnetic field pointing along the x_3-axis and with the quantized electromagnetic field.Because of the translation invariance along the x_3-axis we consider the reduced hamiltonian associated to the total momentum along the x_3-axis regularized by infrared and ultraviolet cutoffs.We give conditions under which the reduced hamiltonian has a ground state. The same approach can be applied to any free atom or ion interacting with the quantized electromagnetic field.
Hainzl, Christian Contributed Self-consistent no-photon QED in a Hartree-Fock type approximation
Harrell, Evans Contributed Universal spectral bounds for Schr\"odinger operators on surfaces Commutator relations are used to investigate the spectra of Schr\"odinger Hamiltonians, $H = -\Delta + V\left({x}\right),$ acting on functions of a smooth, compact $d$-dimensional manifold $M$ immersed in $\bbr^{\nu}, \nu \geq d+1$. Here $\Delta$ denotes the Laplace-Beltrami operator, and the real-valued potential--energy function $V(x)$ acts by multiplication. The manifold $M$ may be complete or it may have a boundary, in which case Dirichlet boundary conditions are imposed. It is found that the mean curvature of a manifold poses tight constraints on the spectrum of $H$. Further, a special algebraic r\^ole is found to be played by a Schr\"odinger operator with potential proportional to the square of the mean curvature: $$H_{g} := -\Delta + g h^2,$$ where $\nu = d+1$, $g$ is a real parameter, and $$h := \sum\limits_{j = 1}^{d} {\kappa_j},$$ with $\{\kappa_j\}$, $j = 1, \dots, d$ denoting the principal curvatures of $M$. For instance, by Theorem~\ref{thm3.1} and Corollary~\ref{cor4.5}, each eigenvalue gap of an arbitrary Schr\"odinger operator is bounded above by an expression using $H_{1/4}$. The ``isoperimetric" parts of these theorems state that these bounds are sharp for the fundamental eigenvalue gap and for infinitely many other eigenvalue gaps.
Hempel, Rainer Contributed Spectral properties of the Laplacian with Neumann boundary condition at a small obstacle. %%% AMS TeX %%%%%%%%%%%%%%%%%%%%%%%% We study the lowest eigenvalue $\lambda_1(\varepsilon)$ of the Laplacian $-\Delta$ in a bounded domain $\Omega \subset {\Bbb R}^d$, $d \ge 2$, from which a small compact set $\varepsilon K$ has been deleted, imposing Dirichlet boundary conditions along $\partial\Omega$ and Neumann boundary conditions on $\varepsilon\partial K$. We are mainly interested in results that require minimal regularity of $\partial K$; here our basic assumption is that the essential spectrum of the Laplacian should not reach down to $0$. We then show that $\lambda_1(\varepsilon)$ converges to $\Lambda_1$, the first Dirichlet eigenvalue of $\Omega$, as $\varepsilon \to 0$. Assuming some more regularity we also obtain upper and lower asymptotic bounds on $\lambda_1(\varepsilon) - \Lambda_1$, for $\varepsilon$ small, where we follow an idea of Burenkov and Davies. %% [J.\ Differential Equations {\bf 186} (2002)]. Finally, we discuss some applications in Mathematical Physics.
Hislop, Peter Contributed Integrated Density of States for Random Operators We discuss recent results on the integrated density of states for various families of random operators. This is joint work with J. M. Combes, F. Klopp, S. Nakamura, and G. Raikov.
Hradecky, Ivo Contributed Adiabatic analysis of a quantum model with time-dependent Aharonov-Bohm flux
Iantchenko, Alexei Contributed Scattering poles for two strictly convex obstacles To study the location of poles for the acoustic scattering matrix for two strictly convex obstacles with smooth boundaries, one uses an approximation of the quantized billiard operator M along the trapped ray between the two obstacles. Assuming that the boundaries are analytic and the eigenvalues of the Poincarй map are non-resonant we use the Birkhoff normal form for M to get the complete asymptotic expansions for the poles in any logarithmic neighborhood of real axis.
Jakubassa-Amundsen, Doris Contributed Spectral properties of a pseudorelativistic operator By means of a unitary transformation scheme borrowed from the study of quantum lattice systems, the Coulomb-Dirac operator of a few-electron ion is transformed into a pseudorelativistic operator which is block-diagonal up to any given order n in the potential strength with respect to projections onto the positive and negative spectral subspaces of the free Dirac operator. For n=2 (the Jansen-Hess operator), positivity is shown up to a critical potential strength, as well as the relative boundedness of the second-order potential terms with respect to the first-order terms. For the two-electron case, the essential spectrum is shown to contain the subset [2m,infty] (with m the electron mass), and its infimum agrees with the ground-state energy of the corresponding one-electron ion, increased by m (the relativistic HVZ Theorem).
Jecko, Thierry Contributed Non-trapping condition for Schrödinger matrix operators In this talk, I would like to discuss the generalization to matrix operators of the following fact: for semiclassical (w.r.t. h) Schrödinger operators, the boundary values of its resolvent at positive energy E is of size 1/h if and only if E is a non-trapping energy for the classical dynamic defined by the symbol of the operator. The main motivation here is the study of the Born-Oppenheimer approximation in the stationary scattering theory of molecules.
Kellendonk, Johannes Contributed Bulk versus boundary topological invariants We consider quantum systems described by covariant families of one-particle Schr\"odinger operators on half-spaces. Their behaviour far away from the boundary and that near the boundary or edge are not independent. In fact, we will explain in the framework of operator algebras that topologically quantised observables in the bulk (bulk topological invariants) are related to topologically quantised observables at the edge (boundary topological invariants). A famous example of this type is the quantum Hall effect in which the Hall conductivity can either be related to a current current correlation in the bulk or is simply the conductance of the edge current. We will present another example relating the value of the integrated density of states on a gap of the bulk spectrum to a force the boundary exhibits on the edge states. In two dimensions with perpendicular magnetic field this yields an integrated version of Streda's formula for the edge conductivity.
Keppeler, Stefan Contributed Semiclassical quantisation rules for the Dirac and Pauli equations We derive explicit semiclassical quantisation conditions for the Dirac and Pauli equations [1,2]. We show that the spin degree of freedom yields a contribution, deriving from a non-Abelian geometric phase, which is of the same order of magnitude as the Maslov correction in Einstein-Brillouin-Keller quantisation. In order to obtain this result a generalisation of the notion of integrability for a certain skew product flow of classical translational dynamics and classical spin precession has to be derived. As an example we discuss the relativistic Kepler problem with Thomas precession, whose treatment sheds some light on the amazing success of Sommerfeld's theory of fine structure [3]. The approach generalises to other multi-component wave equations. [1] S. Keppeler, Phys. Rev. Lett. 89 (2002) 210405 and Ann. Phys. (NY) 304 (2003) 40-71 [2] S. Keppeler, Spinning Particles: Semiclassics and Spectral Statistics, Spinger Tracts in Modern Physics, vol. 193, Springer-Verlag Berlin Heidelberg, 2003 [3] A. Sommerfeld, Ann. Phys. (Leipzig) 51 (1916) 1-94 125-167
Kondej, Sylwia Contributed Locally deformed leaky quantum wires We discuss two dimensional model of leaky quantum wire which is assumed to be asympotically straight. The compact deformation of quantum wire can be manifested, for example, as the finite number of dots or local perturbation of the straight line. For the former we show that the resonace problem can be explicitly solved. We also discuss the scattering problem for these types of models. The results are common work with P. Exner.
Korotyaev, Evgeni Contributed Spectral estimates for Schrodinger operator with periodic We consider the Schrodinger operator on the real line with a matrix valued 1-periodic potential (dimension N). The spectrum of this operator is absolutely continuous and consists of intervals separated by the gaps. We define the Lyapunov function, which is analytic on the N sheeted Riemann surface. On each sheet the Lyapunov function has the standard properties of the Lyapunov function for the scalar case. The Lyapunov function has branch points (resonances). We prove the existence of real and non-real resonances for some potentials. We determine the asymptotics of the periodic, anti-periodic spectrum and the resonances at high energy. We show that there exist two type of gaps: 1) a stable gap, the endpoints are periodic and anti-periodic eigenvalues, 2) a resonance (unstable) gap, the endpoints are resonances (real branch points). Moreover, the following results are obtained: 1) we define the quasimomentum as an analytic function on the Riemann surface; various properties and estimates of the quasimomentum are obtained, 2) we construct the conformal mapping, the real part is integrated density of states and the imaginary part is the Lyapunov exponent. We obtain various properties of this conformal mapping, which are similar to the case N=1. 3) we determine various new trace formulae for potentials and the integrated density of states, the Lyapunov exponent, 4) a priori estimates of gap lengths in terms of potentials are obtained.
Kovarik, Hynek Contributed Resonance Width in Crossed Electric and Magnetic Field We study the spectral properties of a charged particle confined to a two-dimensional plane and submitted to homogeneous magnetic and electric fields and an impurity potential $V$. We use the method of complex translations to prove that the life-times of resonances induced by the presence of electric field are at least Gaussian long as the electric field tends to zero.
Krejcirik, David Contributed A lower bound to the spectral threshold in curved tubes We consider the Laplacian in curved tubes of arbitrary cross-section rotating together with the Frenet frame along curves in Euclidean spaces of arbitrary dimension, subject to Dirichlet boundary conditions on the cylindrical surface and Neumann conditions at the ends of the tube. We prove that the spectral threshold of the Laplacian is estimated from below by the lowest eigenvalue of the Dirichlet Laplacian in a torus determined by the geometry of the tube. In two dimensions, we also investigate the case of combined Dirichlet and Robin conditions on the boundary curves.
Kriz, Jan Contributed Spectral properties of planar waveguides with combined boundary conditions The spectrum of the Laplace operator in a planar strip subject to the nontrivial combination of Dirichlet and Neumann boundary conditions is studied. The existence of discrete spectrum depends on the specific geometrical configuration of the waveguide.
Lauscher, Oliver Contributed Evidence for the nonperturbative renormalizability of Quantum Einstein Gravity In this talk I will summarize recent evidence supporting the conjecture that four-dimensional Quantum Einstein Gravity (QEG) is nonperturbatively renormalizable along the lines of Weinberg's asymptotic safety scenario. This would mean that QEG is mathematically consistent and predictive even at arbitrarily small length scales below the Planck length. Using a truncated version of the exact renormalization group (RG) equation of the effective average action the existence of a UV attractive non-Gaussian RG fixed point is established within two different truncations. It is precisely this fixed point which allows for the construction of a nonperturbative infinite cutoff-limit. For both truncations the numerical predictions of the RG flow in the fixed point regime agree with very high precision. Due to the consistency of the results it appears unlikely that the non-Gaussian fixed point is an artifact of the truncation.
Lenz, Daniel Contributed Cantor spectrum of Lebesgue measure zero for one-dimensional quasicrystals Aperiodically ordered solids, so called quasicrystals, have been studied quite intensely since they were discovered in 1984 by Shechtman/Blech/Gratias/Cahn. The associated one-dimensional Schroedinger operators tend to exhibit features as Cantor spectrum of Lebesgue measure zero, purely singularly continuous spectrum and anomalous transport. We present a new approach to Cantor spectrum of Lebesgue measure zero. This approach relies on ergodic theory. It does not need explicit knowledge of the construction of the potentials. It recovers all earlier results of this type and gives new ones. Recently, this approach has been extended in joint work with David Damanik. In particular, we can show Cantor spectrum of Lebesgue measure zero for almost all circle maps.
Lev, Ondrej Contributed Matrix elements near diagonal in the semiclassical limit, general oscillator case Using WKB approximation we compute semiclassical limit of matrix elements close to the diagonal for wide class of Schr\H odinger Hamiltonians on line.
Lobanov, Igor Contributed Spectral analysis on periodic decorated graphs The spectral properties of periodic Schr\"odinger operators on periodic graphs decorated by manifolds are studied by means of the reduced group $C^*$-algebras affiliated to the priodicity group. In terms of these algebras, conditions for the band structure and the Cantor structure for the Schr\"odinger operators are given. In the abelian case, the Bloch analysis is used and the absence of the singular continuous spectrum is proven. For a relatively wide class of the operators under consideration appearance of an infinite number of spectral gaps is shown. The presence of embedded eigenvalues is discussed.
Lukkarinen, Jani Contributed Energy transport in a harmonic crystal with small random mass perturbations: derivation of a phonon Boltzmann equation We consider the large-scale behavior of energy in an infinite classical harmonic crystal with a 3D cubic lattice structure and small perturbations in the masses of the particles. The mass perturbations are determined by independent, identically distributed random variables with zero mean and a variance v. We first show how the classical system can be transformed into a quantum dynamical system whose Wigner function provides a natural definition for the phase-space energy distribution of the classical system. As the quantum system is similar to the Anderson model, we are then able to study the behavior of the expectation value of the energy distribution by the time-dependent perturbation techniques of Erdös and Yau. We show that when v goes to zero, the kinetic scaling limit of the averaged energy distribution exists, and its time-evolution is governed by a phonon Boltzmann equation.
Melgaard, Michael Contributed Negative Discrete Spectrum of Perturbed Multivortex Aharonov-Bohm Hamiltonians The diamagnetic inequality is established for the Schr\"{o}dinger operator $H_{0}^{(d)}$ in $L^{2}(\Rl^{d})$, $d=2,3$, describing a particle moving in a magnetic field generated by finitely or infinitely many Aharonov-Bohm solenoids located at the points of a discrete set in $\Rl^{2}$, e.g., a lattice. This fact is used to prove the Lieb-Thirring inequality as well as CLR-type eigenvalue estimates for the perturbed Schr\"{o}dinger operator $H_{0}^{(d)}-V$, using new Hardy-type inequalities. Large coupling constant eigenvalue asymptotic formulas for the perturbed operators are also discussed.
Mityagin, Boris Contributed Asymptotics of instability zones of a periodic Schroedinger operator with a two term potential We give the asymptotics of instability zones of Hill-Schroedinger operators L, Ly = y'' + v(x)y, in the case of two term potentials v(x) = a cos2x + b cos4x, with any real nonzero $a$ and $b$. Presentation is based on the joint paper P.Djakov, B. Mityagin, Asymptotics of instability zones of Hill operators with a two term potentials, C. R. Math. Paris, v. 339 (2004), placed on their website on August 10, 2004.
Moller, Jacob Schach Contributed On a Pauli-Fierz model with translation invariance In this talk the structure of the bottom of the joint energy-momentum spectrum for a translation invariant linearly coupled Pauli-Fierz Hamiltonian is presented. Topics to be discussed include: An HVZ theorem, existence and non-existence of groundstates, and regularity properties of the bottom of the essential spectrum, as a function of total momentum.
Nachtergaele, Bruno Contributed Ordering of energy levels in Heisenberg models For a class of ferromagnetic Heisenberg models we prove that the smallest eigenvalues in the invariant subspaces of fixed total spin are monotone decreasing as a function of the total spin. This result can be regarded as a ferromagnetic companion to the famous theorem by Lieb and Mattis for the Heisenberg antiferromagnet on bipartite lattices.
Neidhardt, Hagen Contributed Hybrid models for semiconductors The paper analyzes an one dimensional current coupled hybrid model for semiconductors consisting of a drift-diffusion model without generation and recombination in which is embedded a quantum transmitting Schr\"odinger-Poisson system (QTSP-system). With respect to a further numerical treatment the embedded QTSP-system is replaced by a discretized system which leads to a dissipative hybrid model. It is shown that the dissipative hybrid model admits always a solution with constant current densities. All solutions are uniformly bounded where the bound depends only on the data of the hybrid model. The current densities are different from zero iff the boundary values of the electro-chemical potentials are different.
Nemcova, Katerina Contributed Approximation by point-interaction Hamiltonians in dimension two We show how operators with an attractive delta-potential supported by a graph can be modeled in the strong resolvent sense by point-interaction Hamiltonians. The result is illustrated on finding the spectral properties for two simple examples with the graph being a circle and a star, respectively. Furthermore, we use this method to search for resonances due to quantum tunneling or repeated reflections.
Nicoleau, François Contributed {Inverse scattering for a Schr\" odinger operator with a repulsive potential We We consider a pair of Hamiltonians $(H, H_0 )$ on $L^2 (\r^n )$ where $H_0= p^2 -x^2$ is a Schr\" odinger operator with a repulsive potential, and $H= H_0 +V(x)$. We show that, under suitable assumptions on the decay of the electric potential, $V$ is uniquely determined by the high energy limit of the scattering operator.
Nonnenmacher, Stéphane Contributed Fractal Weyl law for the quantized open Baker's map (coll. with M.Zworski) The "open Baker's map" is a canonical relation on the 2-torus, where a portion of the phase space escapes to infinity at each time step. The points which never escape form a simple Cantor set of dimension D<1. The quantization of this open map into a NxN subunitary matrix (N is the inverse of Planck's constant) provides a toy model for more realistic quantum scattering systems with chaotic classical limit. We analyze the spectrum of this (non-normal) matrix, in the semiclassical limit. We show that the number of eigenvalues ("resonances") in an annulus {0<r1<|E|<r2<1} scales like N^D, while the majority (~N) of the eigenvalues accumulate near 0. Such a map therefore provides a simple example of "Fractal Weyl law", as already exhibited for several classes of chaotic Hamiltonian systems.
Oktay, Veliev Contributed On the Multidimensional Schrodinger Operator with Periodic Potential It is obtained asymptotic formulas for the eigenvalues,Bloch functions and Fermi survaces of the d dimensional periodic Schrodinger operator for arbitrary dimension d.In particular, we prove the validity of the Bethe –Sommerfeld conjecture for arbitrary dimention and arbitrary lattices. Using it we constructively determine the spectral invariants of the multidimensional Schrodinger operator with periodic potential by the Floquet spectrum.Then we give an algorithm for finding the potential by the spectral invariants.
Otte, Peter Contributed An adiabatic theorem for section determinants of spectral projections. Computing transition probabilities in many-fermion systems leads to section determinants of linear operators. We derive an adiabatic-type theorem that expresses the section determinants of spectral projections of a self-adjoint operator through the solution to a Wiener-Hopf equation. The solution theory of this equation is discussed and for a special case a concrete criterion that ensures uniqueness of the solution is presented.
Panati, Gianluca Contributed Dynamics in perturbed periodic structures One of the central questions of solid state physics is to understand the motion of electrons in the periodic potential which is generated by the ionic cores, under the additional influence of external macroscopic electromagnetic potentials. While this problem is quantum mechanical, many electronic properties of solids can be understood already in the semiclassical approximation, by using the celebrated semiclassical model of solid state physics (Bloch and Peierls). A rigorous mathematical connection between the original quantum mechanical problem and the semiclassical model, for general electromagnetic potentials and general wavefunctions, has been recently established, together with the so-called Peierls substitution. In the seminar, I wish to discuss some recent developments and open problems in this area: 1) extension of the result to the case in which a strong constant magnetic field is present and semiclassical explanation of the Quantum Hall effect 2) triviality of the Bloch eigenbundle in the non-magnetic case (the only non-technical assumption in the cited result) 3) a rigorous derivation of the Hofstadter model and of the Harper model by using Peierls substitution.
Pankrashkin, Konstantin Contributed Continuity of integral kernels related to Schroedinger operators on manifolds For Schroedinger operators with singular (locally integrable) potentials on manifolds of bounded geometry we study continuity properties of some related integral kernels: the heat kernel, the Green function and kernels of some other functions of the operator. In particular, the conituity of the Green function outside the diagonal is proved and estimates for the resolvent norm are given. The results are obtained jointly with Jochen Bruening and Vladimir Geyler.
Paul, Thierry Contributed Long time semiclassical asymptotics We present some results on the long time semiclassical evolution for wave packets and observables-both geometrical and microlocal aspects are discussed. Some related results concerning quantum hydrodynamics and regularization of hyperbolic type non linear partial differential equations in the presence of shocks are also presented. Moreover semiclassical methods are used to study the quantum evolution for complex hamiltonians and the dispersionless (large number of particles) limit of (classical) Toda system.
Pedra, Walter Contributed Flot de renormalisation avec des surfaces de Fermi adaptatives We construct a renormalization group sequence for many-fermion systems. Its novel feature is that the Fermi surface is adjusted after every integration step so that no counterterms are needed to treat the infrared problem and the Fermi surface of the interacting system is obtained in a natural way. We construct this flow using convergent expansions for a class of weakly coupled two--dimensional fermion systems with a positively curved Fermi surface and use it to prove Fermi liquid behaviour above an exponentially small temperature. Our class of models includes the two-dimensional Hubbard model.
Pizzo, Alessandro Contributed Construction of the ground state in nonrelativistic Q.E.D. and its expansion in the coupling constant We consider a nonrelativistic electron moving in the Coulomb field of a single nucleus of unit charge and interacting with the soft modes of the quantized electromagnetic field. We analyze the corrections to the ground state of the Coulomb system in the case in which no infrared regularization is adopted. For this purpose we use a scaling analysis by which we can also provide an asymptotic expansion of the ground state (and the ground energy) up to any order in the coupling constant.
Popov, Igor Yu. Contributed Coupled dielectric waveguides with photonic crystal property Two 2D dielectric waveguides separated by thin high-contrast dielectric strip with small windows are considered. It is proved that there exists an eigenvalue below the threshold in the case of one coupling window for sufficiently small size of the window. Variational estimate of the distanse from the threshold is obtained. The existence of a band below the threshold is proved in the case of periodic system of coupling windows. A possibility of gap existence is shown.
Post, Olaf Contributed Residually finite groups, covering spaces and spectral gaps We consider Riemannian coverings with compact quotient and a residually finite covering transformation group G. We prove that for a given number N there is a metric on the covering space such that the corresponding Laplacian on the covering has at least N spectral gaps. Under a further assumption on the reduced group C*-algebra of G (fulfilled e.g. for free non-abelian groups) we can show in addition that the spectrum has band structure, i.e., that the number of gaps is at most finite in a given bounded spectral interval. (joint work with F. Lledó, RWTH Aachen)
Raganelli, Valerio Contributed Solutions for the Klein-Gordon equation In this work a method is put forward to find exactly the analytical solution of the time dependent Klein-Gordon equation for no matter which potential function that is smooth enough and for any smooth enough initial condition. A method to find in an analytical manner, under the same conditions, the eigenfunctions of the corresponding so called reduced form Klein-Gordon equation, that is, the time independent version thereof, is in turn presented and it is proposed a very small outline of some computational advantages for that method.
Raikov, Georgi Contributed On the singularities of the magnetic spectral shift function I will consider the 3D Schroedinger operator with constant magnetic field, perturbed by an electric potential which decays fast enough at infinity. I will describe the singularities at the Landau levels of the corresponding spectral shift function.
Ricaud, Benjamin Contributed A model of exciton in carbon nanotubes The exciton phenomena in low dimensional systems like carbon nanotubes modifies the optical response of the material. In order to study this particular effect, we propose a model of exciton in a carbon nanotube and study its spectrum. We exhibit general features of the spectrum and study in detail its ground state with the help of perturbative theory. We compare with variational approaches.
Romanov, Roman Contributed On instability of absolutely continuous spectrum of non-self-adjoint ordinary differential operators under slowly decaying perturbations
Rouleux, Michel Contributed Instantons and tunnel cycles We present some geometrical technics to compute instantons and semiclassical tunneling between multidimensional tori in phase space ; these apply for double wells in case of the Schr\"odinger operator $P=-h^2\Delta+V(x)$, or the magnetic Schr\"odinger operator $P=(hD-A(x))^2+V(x)$, or the Laplace-Beltrami operator on a 2-d Liouville surface.
Rousse, Vidian Contributed Avoided crossings in the Born-Oppenheimer approximation In the Born-Oppenheimer approximation, we use the smallness of the mass ratio between electrons and nuclei of a molecular system as a semi-classical parameter and the fact that a wave function associated to an electronic energy level remains adiabatically in that state. Unfortunately, when the energy levels present avoided crossings, the second approximation breaks down but we can compute the correction at leading order in term of a Landau-Zener transition between the two levels involved in the avoided crossings whatever the size.
Schubert, Roman Contributed Time evolution of wave packets for large times We study the semiclassical propagation of wave packets in quantum mechanics for large times. The results depend strongly on the ergodic properties of the underlying classical system. If it is Anosov we obtain universal equidistribution for the class of wavepackets we consider, whereas for classically integrable the behavior depends strongly on the initial conditions.
Siedentop, Heinz Contributed The Structure of the Relativistic Electron-Positron Field We will review some recent results on the properties of the electron-positron field. We will concentrate in particular on the existence of electron-positron pairs in the ground state.
Sims, Robert Contributed Localization for Discrete Random Word Models
Sobieczky, Florian Contributed Number of connected components of stationary randomly generated graphs By the spectral theory of a reducible Markov process on a randomly generated graph, tight bounds of the number of connected components are derived. Assuming stationarity and ergodicity,the number of connected components per unit number of vertices shows to exist in the infinite volume limit, independently of the realization of the graph. Using the theory of pertubations of finite rank, bounds can be derived which improve results from the literature by means of their numerical accessibility. (The results are derived from pertubation theory of finite rank, as formulated by Lifshits, and Krain.)
Sorensen, Thomas Ostergaard Contributed Properties of atomic electron densities. We study $\tilde\rho$, the spherically averaged electron density associated to an eigenfunction $\psi$ of the non-relativistic Schr\"odinger operator for an atom. We prove the existence of the third (one-sided) derivative at the nucleus, a lower exponential bound at infinity, and, when $\psi$ is the groundstate in some symmetry-subspace, strict positivity. This is joint work with S. Fournais, M. and T. Hoffmann-Ostenhof.
Soshnikov, Alexander Contributed Poisson Statistics for the Largest Eigenvalues of Random Matrices with Heavy Tails We study large Wigner random matrices and sample covariance random matrices in the case when the matrix entries have distributions with heavy tails. We prove that the largest eigenvalues of such random matrices have Poisson statistics in the limit.
Stovicek, Pavel Contributed On the stability of periodic quantum systems with pure point quasi-spectrum We consider a periodically time dependent Hamiltonian $H(t)$ whose monodromy operator has a pure point spectrum. We give sufficient conditions that guarantee the energy of the system to remain bounded in the course of time evolution. We also deduce bound, uniform in time, on transition probabilities for such systems.
Szpak, Nikodem Contributed Spontaneous particle creation within the external field approximation of QED Strong electrodynamic fields in QED may lead to destabilization of the vacuum. Physical intuition supported by non-rigorous calculations predicts spontaneous particle creation from vacuum if the strength of the electric field exceeds some threshold. Rigorously, the question about the existence of the effect is still open, even at the level of model fields. We treat the problem within an external field approximation of QED, which is in common use for strong fields, and discuss conditions for its validity. The spontaneous particle creation, defined by the adiabatic limit, is studied analytically and numerically, first for a model electric field. The aim of the study is to provide a conclusive model, which applied to future experimental results – like collision of two heavy nuclei (like U 92+ ) – will help to decide the question about the existence of the effect. We review the present state of the theory concerning spontaneous particle creation and report on the open theoretical problems.
Tcheremchantsev, Serguei Contributed Spectral and dynamical properties of Schrodinger operators with growing sparse potentials
Tchoulaevski, Victor Contributed Spectra of multi-particle quantum lattice systems We study spectral properties of N>1 quantum particles on a lattice of dimension d>=1, with a short-range two-body interaction, in an external random field gV(x,w) with i.i.d. values. We prove that if the common probability density of random variables V(x,.) is analytic and |g| is large enough, then, with probability one, the spectrum of the lattice Schroedinger operator is pure point, and all eigen-functions decay exponentially.
Tenuta, Lucattilio Contributed Semiclassical Analysis of Constrained Quantum Systems We study the dynamics of a quantum particle in $R^{n+m}$ constrained by a strong potential force to stay in the vicinity of a submanifold $M$. This approach might represent another way, besides imposing Dirichlet or Neumann conditions on the boundary of a narrow neighbourhood of $M$, to get a confined motion and to model electronic nanostructures, like quantum waveguides, which rely on the formation of a low-dimensional electron gas. We link the length scale of the confining potential to $\hbar$, so that the motion transverse to $M$ retains its quantum nature even in the semiclassical limit. We apply then a mixed adiabatic-semiclassical treatment (following a technique developed by G. Hagedorn) to get the leading order motion on $M$, which turns to be equal to that of the corresponding classical constrained system. Finally we examine an example of a singular constraining potential which, in the classical case, leads to a non deterministic motion on $M$ (Takens chaos). We show that semiclassical limit offers a natural way to reduce this degeneracy. (Joint work with Gianfausto Dell'Antonio)
Teta, Alessandro Contributed A model for decoherence We consider a quantum system consisting of a heavy particle interacting with a light particle in dimension three. For an initial state given in a product form, we characterize the asymptotic dynamics of the system in the limit of small mass ratio. The extension to the case of $n$ light particles is also analysed. The asymptotics is then used to give a rigorous analysis of the mechanism of decoherence, i.e. the suppression of superposition states due to the interaction with the environment, along the line of the description given by Joos and Zeh (1985). In particular it is explicitely computed the decoherence effect induced on the heavy particle by multiple scattering of the light ones. The work is in collaboration with R. Adami, R. Figari and D. Finco.
Tiedra de Aldecoa, Rafael Contributed Dirac operators with variable magnetic field of constant direction We carry out the spectral analysis of matrix valued perturbations of 3-dimensional Dirac operators with variable magnetic field of constant direction. Under suitable assumptions on the magnetic field and on the pertubations, we obtain a limiting absorption principle, we prove the absence of singular continuous spectrum in certain intervals and state properties of the point spectrum. Various situations, for example when the magnetic field is constant, periodic or diverging at infinity, are covered. The importance of an internal-type operator (a 2-dimensional Dirac operator) is also revealed in our study. The proofs rely on commutator methods.
Tip, Adriaan Contributed Perturbation theory of complex photonic band structure Photonic crystals are dielectrics characterized by an electric permeability (permittivity) that has a spatial periodic structure. Then, as in solid state physics, a band spectrum is present, the underlying equations being Maxwell's equations rather than the Schroedinger equation. As in the Schroedinger case a Bloch decomposition can be made, resulting in real point spectrum, depending on the Bloch vector k, thus giving rise to band structure, with the possibility that band gaps occur. The situation changes if absorption is present, a fairly common situation. Then the eigenvalues turn into resonances, here complex eigenvalues in the lower complex half plane. In earlier work we dicussed the mechanism behind this and we also obtained numerical resuts. The latter indicated that the imaginary part of the complex eigenvalues is rather small, suggesting a perturbation treatment starting from the non-absorptive case. Although the perturbation procedure is akin to Kato's perturbation theory, there are differences due to the circumstance that we are not dealing with the resolvent of a perturbed operator and that there is an infinite-dimensional null space which depends on the value of the absorption parameter.
Toloza, Julio Hugo Contributed Exponentially Accurate Quasimodes for the Time-Independent Born-Oppenheimer Approximation This is a joint work with George Hagedorn. We consider a simple molecular--type quantum system in which the nuclei have one degree of freedom. The Hamiltonian has the form \[ H(\epsilon)\ =\ -\,\frac{\epsilon^4}2\, \frac{\partial^2\phantom{i}}{\partial y^2}\ +\ h(y), \] where $h(y)$ is a family of self--adjoint operators that has an isolated, nondegenerate electron level ${\cal E}(y)$ in some open set of ${\mathbb R}$. Near a local minimum of the electron level ${\cal E}(y)$ that is not at a level crossing, we construct quasimodes that are exponentially accurate in the square of the Born--Oppenheimer parameter $\epsilon$ by optimal truncation of the Rayleigh--Schr\"odinger series. That is, we construct $E_\epsilon$ and $\Xi_\epsilon$, such that the $L^2$-norm of $\Xi_\epsilon$ is ${\cal O}(1)$ and the $L^2$-norm of $(H(\epsilon)\,-\,E_\epsilon)\,\Xi_\epsilon$ is upper bounded by $\exp\left(-\,{\Gamma}/{\epsilon^2}\,\right)$ for some $\Gamma>0$ independent of $\epsilon$.
Veselic, Ivan Contributed Spectral Analysis of Percolation Hamiltonians We study the family of Hamiltonians which corresponds to the adjacency operators on a percolation graph. We characterise the set of energies which are almost surely eigenvalues with finitely supported eigenfunctions. This set of energies is a dense subset of the algebraic integers. The integrated density of states has discontinuities precisely at this set of energies. We show that the convergence of the integrated densities of states of finite box Hamiltonians to the one on the whole space holds even at the points of discontinuity. For this we use an equicontinuity-from-the-right argument. The same statements hold for the restriction of the Hamiltonian to the infinite cluster. In this case we prove that the integrated density of states can be constructed using local data only. Finally we study some mixed Anderson-Quantum percolation models and establish results in the spirit of Wegner, and Delyon and Souillard. (See http://arxiv.org/math-ph/0405006)
Villegas-Blas, Carlos Contributed The Bargmann Transform, the Hopf fibration and regularization of the n=2,3,5 dimensional Kepler problem We introduce a Bargmann transform for the space $L^2(S^n)$ of square integrable functions on the $n=2,3,5$ dimensional unit sphere $S^n$ inmersed in ${\mathbb R}^{n+1}$. This is done on base of the Hopf fibration for the spheres $S^K\mapsto{S}^d$ with $(k,d)=(1,1),$ $(3,2),$ $(7,4)$ and a suitable canonical transformation relating two different ways to regularize the $n=2,3,5$ dimensional Kepler problem (for negative energy) involving the null quadrics in ${\mathbb C}^m$, $m=3,4,6$.The unitarity of the Bargmann transform onto a suitable space of analytical functions is showed. We also show reproducing kernels for these spaces of analytical functions. Since our Bargmann transform is actually a coherent states transform we provide sets of coherent states for both $L^2(S^n)$, $n=2,3,5$ and the negative energy Hilbert space of the $n=2,3,5$ dimensional hydrogen atom problem.
Vittot, michel Contributed Control of Hamiltonian Systems: global and local theory. This theory, based on Lie-perturbation method, has already some verifications, numerically as well as experimentally, mainly in classical mechanics. The modifications for the quantum applications are also exposed.
Vougalter, Vitali Contributed Spectra of positive and negative energies in the linearized NLS problem We study the spectrum of the linearized NLS equation in three dimensions, in association with the energy spectrum. We prove that unstable eigenvalues of the linearized NLS problem are related to negative eigenvalues of the energy spectrum, while neutrally stable eigenvalues may have both positive and negative energies. We show how the negative index of the problem can be reduced by going to the proper constrained subspace.
Vytras, Petr Contributed Aharonov-Bohm effect with a homogenous magnetic field The most general admissible boundary conditions are derived for an idealised Aharonov-Bohm flux intersecting the plane at the origin on the background of a homogeneous magnetic field. A standard technique based on self-adjoint extensions yields a four-parameter family of boundary conditions; other two parameters of the model are the Aharonov-Bohm flux and the homogeneous magnetic field. The generalised boundary conditions may be regarded as a combination of the Aharonov-Bohm effect with a point interaction. Spectral properties of the derived Hamiltonians are studied in detail.
Wang, Xue Ping Contributed Existence of the N-body Efimov effect It is commonly believed in physical litterature that there is no Efimov effect for quantum systems with four or more particles (Amado-Greenwood, 1973; Adhikari-Fonseca, 1980). For N-body Schroedinger operators, $N \ge 4$, under the assumptions that the bottom of the essential spectrum is attained by a unique three-cluster decomposition and that the three associated two-cluster Subhamiltonians have a resonance at this threshold, we prove that the discrete spectrum is infinite, even if the potentials are very weak. This result is similar to the well-known three-body Efimov effect.
Warzel, Simone Contributed Anderson Localization and Lifshits Tails for Random Surface Potentials We consider Schroedinger operators on $ L^2(R^d) $ with a random potential concentrated near the surface $ R^{d_1}\times\{0\}\subset R^d$. We prove that the integrated density of states of such operators exhibits Lifshits tails near the bottom of the spectrum. From this and the multiscale analysis by Boutet de Monvel and Stollmann [Arch. Math. 80 (2003) 87] we infer Anderson localization (pure point spectrum and dynamical localization) for low energies. Besides we analyze spectral properties of Schroedinger operators with partially periodic potentials. This is joint work with W. Kirsch (Bochum)
Weidl, Timo Contributed Lieb-Thirring inequalities in Quantum wires We discuss Lieb-Thirring type bounds on trapped modes in strip- und tubelike domains, perturbed by potentials, geometric deformations or local changes of boundary conditions. This is a joint work with P. Exner and in part also with H. Linde.
Yafaev, Dmitri Contributed A particle in an inhomogeneous magnetic field It is shown that a particle in a magnetic field of a straight wire propagates along the current.
Zagrebnov, Valentin Contributed Equilibrium Superradiance of Bosons We consider simple microscopic models describing interference of the two cooperative phenomena: Bose-Einstein Condensate (BEC) and superradiation. Our resuts confirm the presence of the experimentaly observed superradiant light interaction with BEC: (a) the equilibrium superradiance exists only below a certain transition temperature; (b) there is superradiance and matter-wave (BEC) enhancement due to the coherent coupling between the light and the matter.
Zhislin, Grigorii Contributed Point symmetry and spectral properties of pseudorelativistic electrons hamiltonians Abstract. We consider the pseudorelativistic energy operator $H=T+V(r)$ of n electrons in the field of m fixed nuclei; here T is relativistic operator of the electrons kinetic energy, V(r) is the sum of electrons-electrons and electrons-nuclei Coulomb interactions, $r=(r_1,r_2,...,r_n)$. Let F be such finite subgroup from O(3), that the transformations from F move identical NUCLEI one to the place of the other one. Then H is invariant under these transformations, when they act on the ELECTRONS as well. The group F is named by the group of the point symmetry of the system. Let S(n) be the group of the permutations of n electrons and $G=S(n)xF$. In this talk we investigate the spectrum structure of the restriction H(A) of the operator H to the subspace of functions f(r), which are transformed by the operators from G (when they act on the coordinates r) according to the group G irreducible representation of any fixed type A. Our results: 1)We dicovered the location of the essential spectrum of H(A), 2)We obtained two-sided estimates of the counting function of the discrete spectrum $s_d(H(A))of the operator H(A) in the terms of counting functions of the discrete spectrum of some effective one-electron operators, 3)For the systems with nonnegative total charge we proved(on the base 2)) the infiniteness of $s_d(H(A))for any type A. Early the results of the kind 1),3) were known only for nonrelativistic systems, the results of the kind 2) were unknown even for them. The investigation is supported by RFBR grant 03-01-00715
Zielinski, Lech Contributed Semiclassical Weyl asymptotics of eigenvalues for Hamiltotnians with critical points Let P(h)=p(x,hD;h) be a self-adjoint differential operator in L^2(R^d) with the principal symbol p_0 and let E be a real number such that the preimage of ]-\infty , E+\epsilon] by p_0 is compact. The aim of the talk is to present the semiclassical Weyl formula for the number of eigenvalues less that E with remainder estimates depending on the volume of the phase space obtained as the preimage of [E'-h, E'+h]. The estimates hold for symbols with critical points and it is possible to obtain similar results for elliptic problems with non-smooth coefficients.
Znojil, Miloslav Contributed A fragility of stability of a quantum system in three models The concept of stability is re-examined via simplified models. We start from an elementary scalar Klein-Gordon equation and illustrates how a collapse of the system may result from its attempted supersymmetrization. Our second example returns to non-relativistic kinematical regime and shows that the similar collapse may be caused by a controlled strengthening of a singularity in a ``regularized" potential V(x). The third example employs a fully regular and exactly solvable (viz., the simplest square well) potential V(x) and illustrates how an infinitesimally small ``regularization-like" deformation of its domain causes the sudden collapse of infinitely many of its bound states.