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2017 | Book

Shell Interactions for Dirac Operators Read chapter Read first chapter

Advances in Quantum Mechanics

Contemporary Trends and Open Problems

Editors: Prof. Alessandro Michelangeli, Prof. Dr. Gianfausto Dell'Antonio

Publisher: Springer International Publishing

Book Series : Springer INdAM Series

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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About this book

This volume collects recent contributions on the contemporary trends in the mathematics of quantum mechanics, and more specifically in mathematical problems arising in quantum many-body dynamics, quantum graph theory, cold atoms, unitary gases, with particular emphasis on the developments of the specific mathematical tools needed, including: linear and non-linear Schrödinger equations, topological invariants, non-commutative geometry, resonances and operator extension theory, among others. Most of contributors are international leading experts or respected young researchers in mathematical physics, PDE, and operator theory. All their material is the fruit of recent studies that have already become a reference in the community. Offering a unified perspective of the mathematics of quantum mechanics, it is a valuable resource for researchers in the field.

Table of Contents

Frontmatter
Shell Interactions for Dirac Operators
Abstract
In this notes we gather the latest results on spectral theory for the coupling H + V, where H = −iα ⋅ ∇ + mβ is the free Dirac operator in \(\mathbb{R}^{3}\), m > 0 and V is a measure-valued potential. The potentials under consideration are given in terms of surface measures on the boundary of bounded regular domains in \(\mathbb{R}^{3}\). We give three main results. We study the self-adjointness. We give a criterion for the existence of point spectrum, with applications to electrostatic shell potentials, V λ , which depend on a parameter \(\lambda \in \mathbb{R}\). Finally, we prove an isoperimetric-type inequality for the admissible range of λ’s for which the coupling H + V λ generates pure point spectrum in (−m, m). The ball is the unique optimizer of this inequality.
Naiara Arrizabalaga
Correlation Inequalities for Classical and Quantum XY Models
Abstract
We review correlation inequalities of truncated functions for the classical and quantum XY models. A consequence is that the critical temperature of the XY model is necessarily smaller than that of the Ising model, in both the classical and quantum cases. We also discuss an explicit lower bound on the critical temperature of the quantum XY model.
Costanza Benassi, Benjamin Lees, Daniel Ueltschi
Dissipatively Generated Entanglement
Abstract
Given two non-interacting 2-level systems weakly coupled to an environment and thus evolving according to a statistically mixing dissipative reduced dynamics, we provide necessary and sufficient conditions for the generator of the time-evolution to entangle the two systems.
Fabio Benatti
Abelian Gauge Potentials on Cubic Lattices
Abstract
The study of the properties of quantum particles in a periodic potential subjected to a magnetic field is an active area of research both in physics and mathematics, and it has been and is yet deeply investigated. In this chapter we discuss how to implement and describe tunable Abelian magnetic fields in a system of ultracold atoms in optical lattices. After reviewing two of the main experimental schemes for the physical realization of synthetic gauge potentials in ultracold set-ups, we study cubic lattice tight-binding models with commensurate flux. We finally discuss applications of gauge potentials in one-dimensional rings.
M. Burrello, L. Lepori, S. Paganelli, A. Trombettoni
Relative-Zeta and Casimir Energy for a Semitransparent Hyperplane Selecting Transverse Modes
Abstract
We study the relative zeta function for the couple of operators A 0 and A α , where A 0 is the free unconstrained Laplacian in L 2(R d ) (d ≥ 2) and A α is the singular perturbation of A 0 associated to the presence of a delta interaction supported by a hyperplane. In our setting the operatorial parameter α, which is related to the strength of the perturbation, is of the kind α = α(−Δ ∥), where −Δ ∥ is the free Laplacian in L 2(R d−1). Thus α may depend on the components of the wave vector parallel to the hyperplane; in this sense A α describes a semitransparent hyperplane selecting transverse modes.
As an application we give an expression for the associated thermal Casimir energy. Whenever α = χ I (−Δ ∥), where χ I is the characteristic function of an interval I, the thermal Casimir energy can be explicitly computed.
Claudio Cacciapuoti, Davide Fermi, Andrea Posilicano
Analysis of Fluctuations Around Non-linear Effective Dynamics
Abstract
We consider the derivation of effective equations approximating the many-body quantum dynamics of a large system of N bosons in three dimensions, interacting through a two-body potential N 3β−1 V (N β x). For any 0 ≤ β ≤ 1 well known results establish the trace norm convergence of the k-particle reduced density matrices associated with the solution of the many-body Schrödinger equation towards products of solutions of a one-particle non linear Schrödinger equation, as N → ∞. In collaboration with C. Boccato and B. Schlein we studied fluctuations around the approximate non linear Schrödinger dynamics, obtaining for all 0 < β < 1 a norm approximation of the evolution of an appropriate class of data on the Fock space.
Serena Cenatiempo
Logarithmic Sobolev Inequalities for an Ideal Bose Gas
Abstract
The aim of this work is to derive logarithmic Sobolev inequalities, with respect to the Fock vacuum state and for the second quantized Hamiltonian \(d\varGamma (H^{\varLambda } -\mu \mathbb{I})\) of an ideal Bose gas with Dirichlet boundary conditions in a finite volume Λ, from the free energy variation with respect to a Gibbs temperature state and from the monotonicity of the relative entropy. Hypercontractivity of the semigroup \(e^{-\beta d\varGamma (H^{\varLambda }) }\) is also deduced.
Fabio Cipriani
Spherical Schrödinger Hamiltonians: Spectral Analysis and Time Decay
Abstract
In this survey, we review recent results concerning the canonical dispersive flow e itH led by a Schrödinger Hamiltonian H. We study, in particular, how the time decay of space L p -norms depends on the frequency localization of the initial datum with respect to the some suitable spherical expansion. A quite complete description of the phenomenon is given in terms of the eigenvalues and eigenfunctions of the restriction of H to the unit sphere, and a comparison with some uncertainty inequality is presented.
Luca Fanelli
On the Ground State for the NLS Equation on a General Graph
Abstract
We review some recent results on the existence of the ground state for a nonlinear Schrödinger equation (NLS) posed on a graph or network composed of a generic compact part to which a finite number of half-lines are attached. In particular we concentrate on the main theorem in Cacciapuoti et al. (Ground state and orbital stability for the NLS equation on a general starlike graph with potentials, preprint arXiv:1608.01506) which covers the most general setting and we compare it with similar results.
Domenico Finco
Self-Adjoint Extensions of Dirac Operator with Coulomb Potential
Abstract
In this note we give a concise review of the present state-of-art for the problem of self-adjoint realisations for the Dirac operator with a Coulomb-like singular scalar potential V (x) = ϕ(x)I 4. We try to follow the historical and conceptual path that leads to the present understanding of the problem and to highlight the techniques employed and the main ideas. In the final part we outline a few major open questions that concern the topical problem of the multiplicity of self-adjoint realisations of the model, and which are worth addressing in the future.
Matteo Gallone
Dispersive Estimates for Schrödinger Operators with Point Interactions in ℝ3
Abstract
The study of dispersive properties of Schrödinger operators with point interactions is a fundamental tool for understanding the behavior of many body quantum systems interacting with very short range potential, whose dynamics can be approximated by non linear Schrödinger equations with singular interactions. In this work we proved that, in the case of one point interaction in \(\mathbb{R}^{3}\), the perturbed Laplacian satisfies the same L p − L q estimates of the free Laplacian in the smaller regime q ∈ [2, 3). These estimates are implied by a recent result concerning the L p boundedness of the wave operators for the perturbed Laplacian. Our approach, however, is more direct and relatively simple, and could potentially be useful to prove optimal weighted estimates also in the regime q ≥ 3.
Felice Iandoli, Raffaele Scandone
Chern and Fu–Kane–Mele Invariants as Topological Obstructions
Abstract
The use of topological invariants to describe geometric phases of quantum matter has become an essential tool in modern solid state physics. The first instance of this paradigmatic trend can be traced to the study of the quantum Hall effect, in which the Chern number underlies the quantization of the transverse Hall conductivity. More recently, in the framework of time-reversal symmetric topological insulators and quantum spin Hall systems, a new topological classification has been proposed by Fu, Kane and Mele, where the label takes value in \(\mathcal{Z}_{2}\).
We illustrate how both the Chern number \(c \in \mathbb{Z}\) and the Fu–Kane–Mele invariant \(\delta \in \mathbb{Z}_{2}\) of 2-dimensional topological insulators can be characterized as topological obstructions. Indeed, c quantifies the obstruction to the existence of a frame of Bloch states for the crystal which is both continuous and periodic with respect to the crystal momentum. Instead, δ measures the possibility to impose a further time-reversal symmetry constraint on the Bloch frame.
Domenico Monaco
Norm Approximation for Many-Body Quantum Dynamics and Bogoliubov Theory
Abstract
We review some recent results on the norm approximation to the Schrödinger dynamics. We consider N bosons in \(\mathbb{R}^{3}\) with an interaction potential of the form N 3β−1 w(N β (x − y)) with 0 ≤ β < 1∕2, and show that in the large N limit, the fluctuations around the condensate can be effectively described using Bogoliubov approximation.
Phan Thành Nam, Marcin Napiórkowski
Effective Non-linear Dynamics of Binary Condensates and Open Problems
Abstract
We report on a recent result concerning the effective dynamics for a mixture of Bose-Einstein condensates, a class of systems much studied in physics and receiving a large amount of attention in the recent literature in mathematical physics; for such models, the effective dynamics is described by a coupled system of non-linear Schödinger equations. After reviewing and commenting our proof in the mean-field regime from a previous paper, we collect the main details needed to obtain the rigorous derivation of the effective dynamics in the Gross-Pitaevskii scaling limit.
Alessandro Olgiati
Remarks on the Derivation of Gross-Pitaevskii Equation with Magnetic Laplacian
Abstract
The effective dynamics for a Bose-Einstein condensate in the regime of high dilution and subject to an external magnetic field is governed by a magnetic Gross-Pitaevskii equation. We elucidate the steps needed to adapt to the magnetic case the proof of the derivation of the Gross-Pitaevskii equation within the “projection counting” scheme.
Alessandro Olgiati
On the Inverse Spectral Problems for Quantum Graphs
Abstract
We review some aspects of inverse spectral problems for quantum graphs. Under hypothesis of rational independence of lengths of edges it is possible, thanks to trace formulas, to reconstruct information on compact and non compact graphs from the knowledge, respectively, of the spectrum of Laplacian and of the scattering phase. In the case of Sturm-Liouville operators defined on compact graphs and in general for differential operators on compact star-graphs, unknown potentials can be recovered from the knowledge of the spectrum of operators obtained imposing different boundary conditions.
M. Olivieri, D. Finco
Double-Barrier Resonances and Time Decay of the Survival Probability: A Toy Model
Abstract
In this talk we consider the time evolution of a one-dimensional quantum system with a double barrier given by a couple of repulsive Dirac’s deltas. In such a pedagogical model we give, by means of the theory of quantum resonances, the asymptotic behavior of 〈ψ, e −itH ϕ〉 for large times, where H is the double-barrier Hamiltonian operator and where ψ and ϕ are two test functions. In particular, when ψ is close to a resonant state then explicit expression of the dominant terms of the survival probability defined as | 〈ψ, e −itH ψ〉 |2 is given.
Andrea Sacchetti
Metadata
Title
Advances in Quantum Mechanics
Editors
Prof. Alessandro Michelangeli
Prof. Dr. Gianfausto Dell'Antonio
Copyright Year
2017
Electronic ISBN
978-3-319-58904-6
Print ISBN
978-3-319-58903-9
DOI
https://doi.org/10.1007/978-3-319-58904-6

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