We consider the time-independent scattering theory for time evolution operators of one-dimensional two-state quantum walks. The scattering matrix associated with the position-dependent quantum walk naturally appears in the asymptotic behavior at the spatial infinity of generalized eigenfunctions. The asymptotic behavior of generalized eigenfunctions is a consequence of an explicit expression of the

In the presence of the homogeneous electric field E and the homogeneous perpendicular magnetic field B, the classical trajectory of a quantum particle on ℝ2 moves with drift velocity α which is perpendicular to the electric and magnetic fields. For such Hamiltonians, the absence of the embedded eigenvalues of perturbed Hamiltonian has been conjectured. In this paper, one proves this conjecture for

This article is devoted to the description of the eigenvalues and eigenfunctions of the magnetic Laplacian in the semiclassical limit via the complex WKB method. Under the assumption that the magnetic field has a unique and non-degenerate minimum, we construct the local complex WKB approximations for eigenfunctions on a general surface. Furthermore, in the case of the Euclidean plane, with a radially

We construct a family of q deformations of E(2) group for nonzero complex parameters |q|<1 as locally compact braided quantum groups over the circle group 𝕋 viewed as a quasitriangular quantum group with respect to the unitary R-matrix R(m,n):=(q/q̄)mn for all m,n∈ℤ. For real 0<|q|<1, the deformation coincides with Woronowicz’s Eq(2) groups. As an application, we study the braided analogue of the

We generalize gauge theory on a graph so that the gauge group becomes a finite-dimensional ribbon Hopf algebra, the graph becomes a ribbon graph, and gauge-theoretic concepts such as connections, gauge transformations and observables are replaced by linearized analogs. Starting from physical considerations, we derive an axiomatic definition of Hopf algebra gauge theory, including locality conditions

With the help of time-dependent scattering theory on the observable algebra of infinitely extended quasifree fermionic chains, we introduce a general class of so-called right mover/left mover states which are inspired by the nonequilibrium steady states for the prototypical nonequilibrium configuration of a finite sample coupled to two thermal reservoirs at different temperatures. Under the assumption

We propose several different types of construction principles for new classes of Toda field theories based on root systems defined on Lorentzian lattices. In analogy to conformal and affine Toda theories based on root systems of semi-simple Lie algebras, also their Lorentzian extensions come about in conformal and massive variants. We carry out the Painlevé integrability test for the proposed theories

We start with the task of discriminating finitely many multipartite quantum states using LOCC protocols, with the goal to optimize the probability of correctly identifying the state. We provide two different methods to show that finitely many measurement outcomes in every step are sufficient for approaching the optimal probability of discrimination. In the first method, each measurement of an optimal

We develop a new scheme of proofs for spectral theory of the N-body Schrödinger operators, reproducing and extending a series of sharp results under minimum conditions. Our main results include Rellich’s theorem, limiting absorption principle bounds, microlocal resolvent bounds, Hölder continuity of the resolvent and a microlocal Sommerfeld uniqueness result. We present a new proof of Rellich’s theorem

We study quantum channels that vary on time in a deterministic way, that is, they change in an independent but not identical way from one to another use. We derive coding theorems for the entanglement-assisted and unassisted classical capacities. We then specialize the theory to lossy bosonic quantum channels and show the existence of contrasting examples where capacities can or cannot be drawn from

Let L0=∑j=1nMj0Dj+M00,Dj=1i∂∂xj,x∈ℝn, be a constant coefficient first-order partial differential system, where the matrices Mj0 are Hermitian. It is assumed that the homogeneous part is strongly propagative. In the non-homogeneous case it is assumed that the operator is isotropic. The spectral theory of such systems and their potential perturbations is expounded, and a Limiting Absorption Principle

We perform generalizations of Witt and Virasoro algebras, and derive the corresponding Korteweg–de Vries equations from known ℛ(p,q)-deformed quantum algebras previously introduced in J. Math. Phys.51 (2010) 063518. Related relevant properties are investigated and discussed. Besides, we construct the ℛ(p,q)-deformed Witt n-algebra, and determine the Virasoro constraints for a toy model, which play

We present the mathematical construction of the physically relevant quantum Hamiltonians for a three-body system consisting of identical bosons mutually coupled by a two-body interaction of zero range. For a large part of the presentation, infinite scattering length will be considered (the unitarity regime). The subject has several precursors in the mathematical literature. We proceed through an operator-theoretic

This article proposes a new two-parameter generalized entropy, which can be reduced to the Tsallis and Shannon entropies for specific values of its parameters. We develop a number of information-theoretic properties of this generalized entropy and divergence, for instance, the sub-additive property, strong sub-additive property, joint convexity, and information monotonicity. This article presents an

We study solitons of general topological charge over noncommutative tori from the perspective of time-frequency analysis. These solitons are associated with vector bundles of higher rank, expressed in terms of vector-valued Gabor frames. We apply the duality theory of Gabor analysis to show that Gaussians are such solitons for any value of a topological charge. Also they solve self/anti-self duality

In this pedagogical review, we summarize the mathematical basis and practical hints for the explicit analytical computation of spectral sums that involve the eigenvalues of the Laplace operator in simple domains such as d-dimensional balls (with d=1,2,3), an annulus, a spherical shell, right circular cylinders, rectangles and rectangular cuboids. Such sums appear as spectral expansions of heat kernels

We consider a molecule described by the Hartree–Fock model without the exchange term. We prove that the nuclei of total charge Z can bind at most Z+C electrons, where C is a constant that is independent of Z.

We consider a 2×2 system of 1D semiclassical differential operators with two Schrödinger operators in the diagonal part and small interactions of order hν in the off-diagonal part, where h is a semiclassical parameter and ν is a constant larger than 1/2. We study the absence of resonance near a non-trapping energy for both Schrödinger operators in the presence of crossings of their potentials. The

This paper is devoted to a collection of results on nonlinear interpolation inequalities associated with Schrödinger operators involving Aharonov–Bohm magnetic potentials, and to some consequences. As symmetry plays an important role for establishing optimality results, we shall consider various cases corresponding to a circle, a two-dimensional sphere or a two-dimensional torus, and also the Euclidean

In this work, a general Monte Carlo framework is proposed for applying numerical knot invariants in simulations of systems containing knotted one-dimensional ring-shaped objects like polymers and vortex lines in fluids, superfluids or other quantum liquids. A general prescription for smoothing the sharp corners appearing in discrete knots consisting of segments joined together is provided. Smoothing

We discuss the problem of recovering geometric objects from the spectrum of a quantum integrable system. In the case of one degree of freedom, precise results exist. In the general case, we report on the recent notion of good labelings of asymptotic lattices.

We consider the cubic-quintic nonlinear Schrödinger equation of up to three space dimensions. The cubic nonlinearity is thereby focusing while the quintic one is defocusing, ensuring global well-posedness of the Cauchy problem in the energy space. The main goal of this paper is to investigate the interplay between dispersion and orbital (in-)stability of solitary waves. In space dimension one, it is

We consider the free energy of a mean-field quantum spin glass described by a p-spin interaction and a transversal magnetic field. Recent rigorous results for the case p=∞, i.e. the quantum random energy model (QREM), are reviewed. We show that the free energy of the p-spin model converges in a joint thermodynamic and p→∞ limit to the free energy of the QREM.

We review old and new results on the Fröhlich polaron model. The discussion includes the validity of the (classical) Pekar approximation in the strong coupling limit, quantum corrections to this limit, as well as the divergence of the effective polaron mass.

We study the magnetic Laplacian and the Ginzburg–Landau functional in a thin planar, smooth, tubular domain and with a uniform applied magnetic field. We provide counterexamples to strong diamagnetism, and as a consequence, we prove that the transition from the superconducting to the normal state is non-monotone. In some nonlinear regime, we determine the structure of the order parameter and compute

We review the proofs of a theorem of Bloch on the absence of macroscopic stationary currents in quantum systems. The standard proof shows that the current in 1D vanishes in the large volume limit under rather general conditions. In higher dimensions, the total current across a cross-section does not need to vanish in gapless systems but it does vanish in gapped systems. We focus on the latter claim

The Askey–Wilson algebra is realized in terms of the elements of the quantum algebras Uq(𝔰𝔲(2)) or Uq(𝔰𝔲(1,1)). A new realization of the Racah algebra in terms of the Lie algebras 𝔰𝔲(2) or 𝔰𝔲(1,1) is also given. Details for different specializations are provided. The advantage of these new realizations is that one generator of the Askey–Wilson (or Racah) algebra becomes diagonal in the usual

This paper discusses some of the latest advances in the mathematical understanding of the nature of kinetic equations that describe the collective behavior of many-particle systems with collisional dynamics.

Deformations of geometric characteristics of statistical hypersurfaces governed by the law of growth of entropy are studied. Both general and special cases of deformations are considered. The basic structure of the statistical hypersurface is explored through a differential relation for the variables, and connections with the replicator dynamics for Gibbs’ weights are highlighted. Ideal and super-ideal

We review our recent relativistic generalization of the Gutzwiller–Duistermaat–Guillemin trace formula and Weyl law on globally hyperbolic stationary space-times with compact Cauchy hypersurfaces. We also discuss anticipated generalizations to non-compact Cauchy hypersurface cases.

Hartree–Fock theory has been justified as a mean-field approximation for fermionic systems. However, it suffers from some defects in predicting physical properties, making necessary a theory of quantum correlations. Recently, bosonization of many-body correlations has been rigorously justified as an upper bound on the correlation energy at high density with weak interactions. We review the bosonic

Non-relativistic quantum particles bounded to a curve in ℝ2 by attractive contact δ-interaction are considered. The interval between the energy of the transversal bound state and zero is shown to belong to the absolutely continuous spectrum, with possible embedded eigenvalues. The existence of the wave operators is proved for the mentioned energy interval using the Hamiltonians with the interaction

We prove a Koszul formula for the Levi-Civita connection for any pseudo-Riemannian bilinear metric on a class of centered bimodule of noncommutative one-forms. As an application to the Koszul formula, we show that our Levi-Civita connection is a bimodule connection. We construct a spectral triple on a fuzzy sphere and compute the scalar curvature for the Levi-Civita connection associated to a canonical

We develop a quantization method, that we name decomposable Weyl quantization, which ensures that the constants of motion of a prescribed finite set of Hamiltonians are preserved by the quantization. Our method is based on a structural analogy between the notions of reduction of the classical phase space and diagonalization of selfadjoint operators. We obtain the spectral decomposition of the emerging

In this note, we review some results on localization and related properties for random Schrödinger operators arising in aperiodic media. These include the Anderson model associated to disordered quasicrystals and also the so-called Delone operators, operators associated to deterministic aperiodic structures.

Increasing tensor powers of the k×k matrices Mk(ℂ) are known to give rise to a continuous bundle of C∗-algebras over I={0}∪1/ℕ⊂[0,1] with fibers A1/N=Mk(ℂ)⊗N and A0=C(Xk), where Xk=S(Mk(ℂ)), the state space of Mk(ℂ), which is canonically a compact Poisson manifold (with stratified boundary). Our first result is the existence of a strict deformation quantization of Xk à la Rieffel, defined by perfectly

We consider the entanglement evolution of two qubits embedded into disordered multiconnected environment. We model the environment and its interaction with qubits by large random matrices allowing for a possibility to describe environments of meso- and even nanosize. We obtain general formulas for the time dependent reduced density matrix of the qubits corresponding to several cases of the qubit-environment

This paper aims at explaining some incarnations of the idea of topological recursion: in two-dimensional quantum field theories (2d TQFTs), in cohomological field theories (CohFT), and in the computation of volumes of the moduli space of curves. It gives an introduction to the formalism of quantum Airy structures on which the topological recursion is based, which is seen at work in the above topics

We complement our work on the causality of upper semi-continuous distributions of cones with some results on Cauchy hypersurfaces. We prove that every locally stably acausal Cauchy hypersurface is stable. Then we prove that the signed distance dS from a spacelike hypersurface S is, in a neighborhood of it, as regular as the hypersurface, and by using this fact we give a proof that every Cauchy hypersurface

We consider a family of Schrödinger operators supported by a periodic chain of loops connected either tightly or loosely through connecting links of the length ℓ>0 with the vertex coupling which is non-invariant with respect to the time reversal. The spectral behavior of the model illustrates that the high-energy behavior of such vertices is determined by the vertex parity. The positive spectrum of

These notes give an introduction to the mathematical framework of the Batalin–Vilkovisky and Batalin–Fradkin–Vilkovisky formalisms.

We consider a gas of interacting bosons trapped in a box of side length one in the Gross–Pitaevskii limit. We review the proof of the validity of Bogoliubov’s prediction for the ground state energy and the low-energy excitation spectrum. This note is based on joint work with C. Brennecke, S. Cenatiempo and B. Schlein.

Renault, Wassermann, Handelman and Rossmann (early 1980s) and Evans and Gould (1994) explicitly described the K-theory of certain unital AF-algebras A as (quotients of) polynomial rings. In this paper, we show that in each case the multiplication in the polynomial ring (quotient) is induced by a ∗-homomorphism A⊗A→A arising from a unitary braiding on a C*-tensor category and essentially defined by

Joint spectral measures associated to the rank two Lie group G2, including the representation graphs for the irreducible representations of G2 and its maximal torus, nimrep graphs associated to the G2 modular invariants have been studied. In this paper, we study the joint spectral measures for the McKay graphs (or representation graphs) of finite subgroups of G2. Using character theoretic methods we

For parity-conserving fermionic chains, we review how to associate ℤ2-indices to ground states in finite systems with quadratic and higher-order interactions as well as to quasifree ground states on the infinite CAR algebra. It is shown that the ℤ2-valued spectral flow provides a topological obstruction for two systems to have the same ℤ2-index. A rudimentary definition of a ℤ2-phase label for a class

We consider a 2-dimensional Bloch–Landau–Pauli Hamiltonian for a spinful electron in a constant magnetic field subject to a periodic background potential. Assuming that the z-component of the spin operator is conserved, we compute the linear response of the associated spin density of states to a small change in the magnetic field, and identify it with the spin Hall conductivity. This response is in

A natural way to obtain a system of partial differential equations on a manifold is to vary a suitably defined sesquilinear form. The sesquilinear forms we study are Hermitian forms acting on sections of the trivial ℂn-bundle over a smooth m-dimensional manifold without boundary. More specifically, we are concerned with first order sesquilinear forms, namely, those generating first order systems. Our

We first review the problem of a rigorous justification of Kubo’s formula for transport coefficients in gapped extended Hamiltonian quantum systems at zero temperature. In particular, the theoretical understanding of the quantum Hall effect rests on the validity of Kubo’s formula for such systems, a connection that we review briefly as well. We then highlight an approach to linear response theory based

We report our recent results on the existence and uniqueness of unitary propagators for N-particle Schrödinger equations which may be applied to most interesting problems in physics.

We consider the Laplacian on a periodic metric graph and obtain its decomposition into a direct fiber integral in terms of the corresponding discrete Laplacian. Eigenfunctions and eigenvalues of the fiber metric Laplacian are expressed explicitly in terms of eigenfunctions and eigenvalues of the corresponding fiber discrete Laplacian and eigenfunctions of the Dirichlet problem on the unit interval

In this paper, we consider Anderson type operators on a separable Hilbert space where the random perturbations are finite rank and the random variables have full support on ℝ. We show that spectral multiplicity has a uniform lower bound whenever the lower bound is given on a set of positive Lebesgue measure on the point spectrum away from the continuous one. We also show a deep connection between the

We review recent results obtained in the scattering theory of dissipative quantum systems representing the long-time evolution of a system S interacting with another system S′ and susceptible of being absorbed by S′. The effective dynamics of S is generated by an operator of the form H=H0+V−iC∗C on the Hilbert space of the pure states of S, where H0 is the self-adjoint generator of the free dynamics

On a connected, oriented, smooth Riemannian manifold without boundary we consider a real scalar field whose dynamics is ruled by E, a second-order elliptic partial differential operator of Laplace type. Using the functional formalism and working within the framework of algebraic quantum field theory and of the principle of general local covariance, first we construct the algebra of locally covariant

We consider ensembles of real symmetric band matrices with entries drawn from an infinite sequence of exchangeable random variables, as far as the symmetry of the matrices permits. In general, the entries of the upper triangular parts of these matrices are correlated and no smallness or sparseness of these correlations is assumed. It is shown that the eigenvalue distribution measures still converge

We extend Howland time-independent formalism to the case of completely positive and trace preserving dynamics of finite-dimensional open quantum systems governed by periodic, time-dependent Lindbladian in Weak Coupling Limit, expanding our result from previous papers. We propose the Bochner space of periodic, square integrable matrix-valued functions, as well as its tensor product representation, as

Wigner’s Theorem states that bijections of the set 𝒫1(H) of one-dimensional projections on a Hilbert space H that preserve transition probabilities are induced by either a unitary or an anti-unitary operator on H (which is uniquely determined up to a phase). Since elements of 𝒫1(H) define pure states on the C*-algebra B(H) of all bounded operators on H (though typically not producing all of them)

In the limit ℏ→0, we analyze a class of Schrödinger operators Hℏ=ℏ2L+ℏW+V⋅idℰ acting on sections of a vector bundle ℰ over a Riemannian manifold M where L is a Laplace type operator, W is an endomorphism field and the potential energy V has a non-degenerate minimum at some point p∈M. We construct quasimodes of WKB-type near p for eigenfunctions associated with the low-lying eigenvalues of Hℏ. These

We examine the solutions of the semilinear wave equation, and, in particular, of the φq model of quantum field theory in the curved spacetime. More exactly, for 2

We present homotopy theoretic and geometric interpretations of the Kane–Mele invariant for gapped fermionic quantum systems in three dimensions with time-reversal symmetry. We show that the invariant is related to a certain 4-equivalence which lends it an interpretation as an obstruction to a block decomposition of the sewing matrix up to non-equivariant homotopy. We prove a Mayer–Vietoris Theorem

We study the existence and the asymptotic behavior of the solution of an abstract viscoelastic system submitted to non-local initial data. utt+Au−∫0tg(t−s)Bu(s)ds=0u(0)=ξ(u)in V,ut(0)=η(u)in H, where A and B are differential operators satisfying B≈Aα for 0≤α≤1. We prove that the model is well-posed. Concerning the asymptotic behavior, we show that the exponential decay holds if and only if α=1 and