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

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

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

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

We study the spectrum of the spin-boson Hamiltonian with two bosons for arbitrary coupling α>0 in the case when the dispersion relation is a bounded function. We derive an explicit description of the essential spectrum which consists of the so-called two- and three-particle branches that can be separated by a gap if the coupling is sufficiently large. It turns out, that depending on the location of

In the paper, the Belavkin weighted square root measurement in infinite dimension is investigated. The question of uniqueness of such measurement is analyzed and some estimates for the probability of detection are obtained. Moreover, the asymptotics of the probability of detection and the probability of failure are derived in the situation when the pure states approach an orthonormal basis. The results

We review the BV formalism in the context of 0-dimensional gauge theories. For a gauge theory (X0,S0) with an affine configuration space X0, we describe an algorithm to construct a corresponding extended theory (X̃,S̃), obtained by introducing ghost and anti-ghost fields, with S̃ a solution of the classical master equation in 𝒪X̃. This construction is the first step to define the (gauge-fixed) BRST

We establish the Level-1 and Level-3 Large Deviation Principles (LDPs) for invariant measures on shift spaces over finite alphabets under very general decoupling conditions for which the thermodynamic formalism does not apply. Such decoupling conditions arise naturally in multifractal analysis, in Gibbs states with hard-core interactions, and in the statistics of repeated quantum measurement processes

We construct the moduli space of Spin(7)-instantons on a hermitian complex vector bundle over a closed 8-dimensional manifold endowed with a (possibly non-integrable) Spin(7)-structure. We find suitable perturbations that achieve regularity of the moduli space, so that it is smooth and of the expected dimension over the irreducible locus.

Schramm–Loewner evolution (SLE) is a random process that gives a useful description of fractal curves. After its introduction, many works concerning the connection between SLE and conformal field theory (CFT) have been carried out. In this paper, we develop a new method of coupling SLE with a Wess–Zumino–Witten (WZW) model for SU(2), an example of CFT, relying on a coset construction of Virasoro minimal

There are two groups which act in a natural way on the bundle TP tangent to the total space P of a principal G-bundle P(M,G): the group Aut0TP of automorphisms of TP covering the identity map of P and the group TG tangent to the structural group G. Let AutTGTP⊂Aut0TP be the subgroup of those automorphisms which commute with the action of TG. In the paper, we investigate G-invariant symplectic structures

We study a generalization of Kitaev’s abelian toric code model defined on CW complexes. In this model, qudits are attached to n-dimensional cells and the interaction is given by generalized star and plaquette operators. These are defined in terms of coboundary and boundary maps in the locally finite cellular cochain complex and the cellular chain complex. We find that the set of energy-minimizing ground

We consider a 3D quantum system of N identical bosons in a trapping potential |x|p, with p≥0, interacting via a Newton potential with an attractive interaction strength aN. For a fixed large N and the coupling constant aN smaller than a critical value a∗ (Chandrasekhar limit mass), in an appropriate sense, the many-body system admits a ground state. We investigate the blow-up behavior of the ground

This article is concerned with compositions in the context of three standard quantizations in the framework of Fock spaces, namely, anti-Wick, Wick and Weyl quantizations. The first one is a composition of states also known as a Wick product and is closely related to the standard scattering identification operator encountered in Quantum Electrodynamics for issues on time dynamics (see [29, 13]). Anti-Wick

Let H be a separable Hilbert space and T be a self-adjoint bounded linear operator on H⊗2 with norm ≤1, satisfying the Yang–Baxter equation. Bożejko and Speicher ([10]) proved that the operator T determines a T-deformed Fock space ℱ(H)=⊕n=0∞ℱn(H). We start with reviewing and extending the known results about the structure of the n-particle spaces ℱn(H) and the commutation relations satisfied by the

Higher gauge theory is a higher order version of gauge theory that makes possible the definition of 2-dimensional holonomy along surfaces embedded in a manifold where a gauge 2-connection is present. In this paper, we study Hamiltonian models for discrete higher gauge theory on a lattice decomposition of a manifold. We show that a construction for higher lattice gauge theory is well-defined, including

The spectral properties of the singularly perturbed self-adjoint Landau Hamiltonian Aα=(i∇+A)2+αδΣ in L2(ℝ2) with a δ-potential supported on a finite C1,1-smooth curve Σ are studied. Here A=12B(−x2,x1)T is the vector potential, B>0 is the strength of the homogeneous magnetic field, and α∈L∞(Σ) is a position-dependent real coefficient modeling the strength of the singular interaction on the curve Σ

Quantum Gaussian states on Boson Fock spaces are quantum versions of Gaussian distributions. In this paper, we explore infinite mode quantum Gaussian states. We extend many of the results of Parthasarathy in [14, 16] to the infinite mode case, which includes various characterizations, convexity and symmetry properties.

We investigate spreading rates of one-dimensional quantum states under the Schrödinger time-evolution. The focus of this paper is on the states that either have finite support or decay exponentially at ±∞. In particular, we extend results of Damanik and Tcheremchantsev on estimating transport exponents that were originally proved to hold for the initial states supported on a single site. These general

In his book on noncommutative geometry, Connes constructed a differential graded algebra out of a spectral triple. Lack of monoidality of this construction is investigated. We identify a suitable monoidal subcategory of the category of spectral triples and show that when restricted to this subcategory the construction of Connes is monoidal. Richness of this subcategory is exhibited by establishing

The p-wave resonances induce an infinite number of negative eigenvalues accumulating at the origin for the system of three identical particles in two dimensions, provided that the energy operator is restricted on the subspace of wave functions which are antisymmetric with respect to the permutations. This quantum phenomenon is called the super Efimov effect and corresponds to the Efimov effect in three

The relativistic Lippmann–Schwinger equation was earlier formulated in terms of the limit values of the corresponding resolvent. In the present paper, we write down the limit values of the resolvent in an explicit form, and so the relativistic Lippmann–Schwinger equation is presented as an integral equation. Using this integral equation, we investigate the stationary scattering problems (relativistic

Product vacua with boundary states (PVBS) are cousins of the Heisenberg XXZ spin model and feature n particle species on ℤd. The PVBS models were originally introduced as toy models for the classification of ground state phases. A crucial ingredient for this classification is the existence of a spectral gap above the ground state sector. In this work, we derive a spectral gap for PVBS models at arbitrary

We present examples in three symmetry classes of topological insulators in one or two dimensions where the proof of the bulk-edge correspondence is particularly simple. This serves to illustrate the mechanism behind the bulk-edge principle without the overhead of the more general proofs which are available. We also give a new formula for the ℤ2-index of our time-reversal invariant systems inspired