Mathematical Results in Quantum Mechanics

The following sections are included:

• Preface
• Committees
• Contents

For an arbitrary nonempty, open set Ω ⊂ ℝⁿ, n ∈ ℕ of finite (Euclidean) volume, we consider the minimally defined higher-order Laplacian , m ∈ ℕ, and its Krein–von Neumann extension A_K,Ω,m in L²(Ω). with N(λ, A_K,Ω,m), λ > 0, denoting the eigenvalue counting function corresponding to the strictly positive eigenvalues of A_K,Ω,m, we derive the bound

The proof relies on variational considerations and exploits the fundamental link between the Krein–von Neumann extension and an underlying (abstract) buckling problem.

This note is based on the results contained in [1] and [2]. We consider the Landau Hamiltonian perturbed by an electric potential V that decays at infinity. The spectrum of the perturbed operator consists of eigenvalue clusters which accumulate to the Landau levels. We distinguish two regimes: a short-range and a long-range one, and estimate the rate at which the eigenvalue clusters shrink to the Landau levels, as the energy increases. Further, we obtain an explicit description of the asymptotic density of eigenvalues within the qth cluster as q → ∞. For short-range potentials this description is given in terms of the Radon transform of the potential V , whereas in the long-range regime we consider potentials V(x) = 𝕍(x) + O(|x|^-ρ-ε), with 𝕍 homogeneous of order -ρ, ρ ∈ (0, 1), and obtain a description in terms of the mean-value transform of 𝕍.

We review our recent results [2] proving van der Waals-London's law of decay of the van der Waals force for a collection of neutral atoms at large separation and present the proof is a special case.

We review the random loop representations of Tóth and Aizenman-Nachtergaele for quantum Heisenberg models. They can be combined and extended so as to include the quantum XY model and certain SU(2)-invariant spin 1 systems. We explain the calculations of correlation functions.

We discuss properties of L²-eigenfunctions of Schrödinger operators and elliptic partial differential operators. The focus is set on unique continuation principles and equidistribution properties. We review recent results and announce new ones.

Various approaches to quantization of vector fields, both massive (Proca) and massless (Maxwell), are described. The massless limit of the Proca fields is discussed.

We present a rigorous derivation of the BCS gap equation for superfluid fermionic gases with point interactions. Our starting point is the BCS energy functional, whose minimizer we investigate in the limit when the range of the interaction potential goes to zero.

In this note we consider the Dirichlet Laplacian in twisted three-dimensional tubes. In particular we study a related Hardy type inequality, which has been establish in previous works, and show how such an inequality can be improved. We also discuss the optimality of the corresponding integral weight.

We give a brief historical account on microscopic explanations of electrical conduction. One aim of this short review is to show that Thermodynamics is fundamental to the theoretical understanding of the phenomenon. We discuss how the 2nd law, implemented in the scope of Quantum Statistical Mechanics, can be naturally used to give mathematical sense to conductivity of very general quantum many–body models. This is reminiscent of original ideas of J.P. Joule. We start with Ohm and Joule's discoveries and proceed by describing the Drude model of conductivity. The impact of Quantum Mechanics and the Anderson model are also discussed. The exposition is closed with the presentation of our approach to electrical conductivity based on the 2nd law of Thermodynamics as passivity of systems at thermal equilibrium. It led to new rigorous results on linear conductivity of interacting fermions. One example is the existence of so–called AC–conductivity measures for such a physical system. These measures are, moreover, Fourier transforms of time correlations of current fluctuations in the system. I.e., the conductivity satisfies, for a large class of quantum mechanical microscopic models, Green–Kubo relations.

The question of whether given density operators for subsystems of a multipartite quantum system are compatible to one common total density operator is known as the quantum marginal problem. We briefly review the solution of a subclass of such problems found just recently. In particular, this provides the solution of the 1-body N-representability problem. Its solution, the so-called generalized Pauli constraints, restrict the set of mathematically possible fermionic occupation numbers significantly, and strengthens Pauli's exclusion principle. Moreover, we review the study of a concrete physical model of interacting fermions confined to a harmonic trap. There, we found occupation numbers close, but not exactly on the boundary of the allowed region. This new effect of quasipinning is physically relevant since it corresponds to a simplified structure of the corresponding N-fermion quantum state.

We review recent results [3] concerning the evolution of fermionic systems. We are interested in the mean field regime, where particles experience many weak collisions. For fermions, the mean field regime is naturally linked with a semiclassical limit. Assuming some regularity of the interaction potential we show that the many body evolution of initial states close to Slater determinants exhibiting the appropriate semiclassical structure can be approximated by the Hartree-Fock equation. Our method provides precise bounds on the rate of the convergence.

We discuss two models for the multipolaron: the Fröhlich Hamiltonian and the Pekar-Tomasevich functional. The ground state energies of the multipolaron in the two models coincide to leading order in α in the so-called strong coupling limit α → ∞. We outline the history of this result and we explain some of the main ideas of our proof given in [2].

We discuss a model which modifies the system studied first by Smilansky. In our case a singular potential ‘channel’ is replaced by a regular, below unbounded potential which shrinks as it becomes deeper. We demonstrate that, similarly to the original model, such a system exhibits a spectral transition with respect to the coupling constant, and determine the critical value above which a new spectral branch opens. The result is generalized to situations with multiple potential ‘channels’.

In experiments, Bose-Einstein condensates are prepared by cooling a dilute Bose gas in a trap. After the phase transition has been reached, the trap is switched off and the evolution of the condensate observed. The evolution is macroscopically described by the Gross-Pitaevskii equation. On the microscopic level, the dynamics of Bose gases are described by the N-body Schrödinger equation. We review our article [1] in which we construct a class of initial data in Fock space which are energetically close to the ground state and prove that their evolution approximately follows the Gross-Pitaevskii equation. The key idea is to model two-particle correlations with a Bogoliubov transformation.

Boundary triplets approach is applied to the construction of self-adjoint extensions of the Dirac operator. Previous results on the extension of tensor product of operators are discussed.

We present results on Bose-Einstein condensation (BEC) on general compact quantum graphs, i.e., one-dimensional systems with a (potentially) complex topology. We first investigate non-interacting many-particle systems and provide a complete classification of systems that exhibit condensation. We then consider models with interactions that consist of a singular part as well as a hardcore part. In this way we obtain generalisations of the Tonks-Girardeau gas to graphs. For this we find an absence of phase transitions which then indicates an absence of BEC.

A method to describe classical and quantum evolution systems is presented. The method is based on the so-called Feynman formulae, i.e. on representations of solutions of evolution equations as limits of n-fold iterated integrals of some (usually elementary) functions when n tends to infinity. Such formulae are suitable for direct calculations and numeric modeling. On the other hand, these formulae are related to some functional (or path) integrals with respect to probability measures and Feynman type pseudomeasures. We outline some results related to classical and quantum systems on different geometrical structures.

We review recent results on the existence of asymptotic observables in algebraic QFT. The problem of asymptotic completeness is discussed from this perspective.

We review recent probabilistic results on covariant Schrödinger operators on vector bundles over (possibly locally infinite) weighted graphs, and explain applications like semiclassical limits. We also clarify the relationship between these results and their formal analogues on smooth (possibly noncompact) Riemannian manifolds.

We describe a classification of threshold properties for one-dimensional discrete Schrödinger operators and associated asymptotic expansions of the resolvent for a general class of interactions.

We consider a generalized Schrödinger operator in L²(ℝ²) describing an attractive δ′ interaction in a strong coupling limit. δ′ interaction is characterized by a coupling parameter β and it is supported by a C⁴-smooth infinite asymptotically straight curve Γ without self-intersections. It is shown that in the strong coupling limit, β → 0₊, the eigenvalues for a non-straight curve behave as , where μ_j is the j-th eigenvalue of the Schrödinger operator on L²(ℝ) with the potential where γ is the signed curvature of Γ.

Let Ω be a periodic unbounded domain in ℝⁿ. It is well-known that the spectrum of the Neumann Laplacian in L₂(Ω) is a locally finite union of compact intervals called bands. The neighbouring bands may overlap, otherwise we have a gap in the spectrum. Our goal is to construct such periodic domain Ω that the spectrum of the corresponding Neumann Laplacian has at least m gaps (m ∈ ℕ is a given number) and these gaps are close to predefined intervals.

Quasi-one-dimensional model for the Stokes flow concentrated near a network is suggested. It is analogous to the quantum graph model in quantum mechanics. A version of the Harnack's inequality for the flow is obtained.

We suggest an abstract approach for point contact problems in the framework of boundary triples. Using this approach we obtain the perturbation series for a simple eigenvalue in the discrete spectrum of the model self-adjoint extension with weak point coupling. An example of a two-level quantum model is provided.

The paper gives an overview on trace formulas for pairs of self-adjoint, maximal dissipative and other types of resolvent comparable operators. In particular, recent results on trace formulas for singular pairs of resolvent comparable operators are presented. The results improve and generalize certain classical results of M.G. Krein for pairs of selfadjoint and dissipative operators, the results of A. Rybkin for such pairs, as well as the results of V. Adamyan, B. Pavlov, and M.G. Krein for pairs {A, A*} with a maximal dissipative operator A.

In this paper we review our recent attempt to understand the physical meaning of quantum graph models through analysis of their vertex coupling approximations. We find a family of vertex couplings, generalizing the δ′-interaction on the line, and show that with a suitable choice of the parameters they can be approximated in the norm-resolvent sense by families of operators with singularly scaled rank-one interactions. These couplings are of interest mainly because their properties contrast in a sense to those of more regular couplings, for instance, a δ′-junction is opaque at high energies. Finally, our new generalization of the δ′-interaction on the line is different from the two known ones; in contrast to those it leans on the permutation asymmetry of the approximating operators.

We summarize recent results on stochastic differential equations associated with the standard model of non-relativistic quantum electrodynamics for an electron in an electrostatic potential interacting with the quantized electromagnetic field. Moreover, we present a Feynman-Kac formula for the corresponding semi-group – which is new in case the electron-spin is taken into account – and a new Feynman-Kac formula for an operator-valued integral kernel of the semi-group. Finally, we announce a proof of the norm-continuity of the integral kernel.

This article is devoted to the spectral problem for a branched chain of weakly coupled ball resonators. The solution for such a problem in the framework of theory of extensions is obtained. The spectral equations are derived for two types of the resonators coupling (δ-coupling and δ′-coupling).

Quantum mechanics can be formulated as a Hamiltonian system in terms of the Liouville equation for the density matrix. Dissipative effects are modeled via coupling to a macroscopic system, where the coupling operators act via commutators. Following Öttinger (2010) we use GENERIC to construct thermodynamically consistent evolution equations as a sum of a Hamiltonian and a gradient-flow contribution, which satisfy a particular non-interaction condition. We give three applications: First, we consider a finite-dimensional quantum system that is coupled to a finite number of simple and finite heat baths, each of which is described by a scalar temperature variable. Second, we model quantum system given by a one-dimensional Schrödinger operator connected to a one-dimensional heat equation on the left and on the right. Finally, we consider thermo-opto-electronics, where the Maxwell-Bloch system of optics is coupled to the energy-drift-diffusion system for semiconductor electronics.

We review recent developments in the spectral theory of continuum one-dimensional quasicrystals, yielding purely singular continuous spectrum for these Schrödinger operators. Allowing measures as potentials we can generalize some results to very singular potentials, including Kronig-Penney type models.

We present adiabatic theorems with and without spectral gap condition for operators A(t) : D(A(t)) ⊂ X → X in a Banach space X. In the case with spectral gap, the considered spectral values λ(t) ∈ σ(A(t)) are not required to be semisimple and, in particular, need not be eigenvalues. In the case without spectral gap, the considered eigenvalues λ(t) ∈ ∂σ(A(t)) are not required to be weakly semisimple. We also discuss adiabatic theorems for operators A(t) with time-dependent domains and an application to operators defined by symmetric sesquilinear forms.

We present an estimate on the number of eigenvalues of a matrix in a small interval, phrased in terms of the properties of a submatrix. We then apply this result to prove a theorem on the eigenvalue statistics of certain random Schrödinger operators. Specifically, we establish Wegner, Minami, and higher-order estimates for a class of random block operators modeling quasiparticles in the Bogoliubov-de Gennes equation from the theory of dirty superconductors.

We discuss the stability problem for a system of N identical fermions with unit mass interacting with a different particle of mass m via zero-range interactions in dimension three. We find a stability parameter m*(N) > 0, increasing with N, such that the Hamiltonian of the system is self-adjoint and bounded from below for m > m*(N).

The following sections are included:

• List of registered participants
• Author index