We consider algebraic varieties canonically associated with any Lie superalgebra, and study them in detail for super-Poincaré algebras of physical interest. They are the locus of nilpotent elements in (the projectivized parity reversal of) the odd part of the algebra. Most of these varieties have appeared in various guises in previous literature, but we study them systematically here, from a new perspective:

We investigate the spectral properties of discrete one-dimensional Schrödinger operators whose potentials are generated by sampling along the elements of the Fibonacci subshift with a locally constant function. The fundamental trace map formalism for this model is presented and related to its spectral features via an extension of a multitude of works on the classical model, where the sampling function

In this work, we provide results on the long-time localization in space (dynamical localization) of certain two-dimensional magnetic quantum systems. The underlying Hamiltonian may have the form \(H=H_0+W\), where \(H_0\) is rotationally symmetric and has dense point spectrum and W is a perturbation that breaks the rotational symmetry. In the latter case, we also give estimates for the growth of the

Consider a finite triangulation of a surface M of genus g and assume that spin-less fermions populate the edges of the triangulation. The quantum dynamics of such particles takes place inside the algebra of canonical anti-commutation relations (CAR). Following Kitaev’s work on toric models, we identify a sub-algebra of CAR generated by elements associated to the triangles and vertices of the triangulation

We give a complete characterization of the reflectionless Schrödinger operators on the line with integrable potentials, solve the inverse scattering problem of reconstructing such potentials from the eigenvalues and norming constants, and derive the corresponding generalized soliton solutions of the Korteweg–de Vries equation.

The probability distribution of a function of a subsystem conditioned on the value of the function of the whole, in the limit when the ratio of their values goes to zero, has a limit law: It equals the unconditioned marginal probability distribution weighted by an exponential factor whose exponent is uniquely determined by the condition. We apply this theorem to explain the canonical equilibrium ensemble

One of the most striking features of gapped quantum phases that exhibit topological order is the presence of long-range entanglement that cannot be detected by any local order parameter. The formalism of projected entangled-pair states is a natural framework for the parameterization of gapped ground state wavefunctions which allows one to characterize topological order in terms of the virtual symmetries

We consider a family of globally stationary (horizonless), asymptotically flat solutions of five-dimensional supergravity. We prove that massless linear scalar waves in such soliton spacetimes cannot have a uniform decay rate faster than inverse logarithmically in time. This slow decay can be attributed to the stable trapping of null geodesics. Our proof uses the construction of quasimodes which are

We present the BFV and the BV extension of the Poisson sigma model (PSM) twisted by a closed 3-form H. There exist superfield versions of these functionals such as for the PSM and, more generally, for the AKSZ sigma models. However, in contrast to those theories, they depend on the Euler vector field of the source manifold and contain terms mixing data from the source and the target manifold. Using

We present a new conjectural symmetry of the colored Alexander polynomial, that is the specialization of the quantum \(\mathfrak {sl}_N\) invariant widely known as the colored HOMFLY-PT polynomial. We provide arguments in support of the existence of the symmetry by studying the loop expansion and the character expansion of the colored HOMFLY-PT polynomial. We study the constraints this symmetry imposes

We consider suspension (semi)flows over \({C}^{1+\alpha }\) full-branch Markov piecewise expanding interval maps and piecewise hyperbolic maps and prove exponential decay of correlations with respect to Gibbs measures associated with piecewise Hölder continuous potentials. As a consequence, typical codimension one attractors of \(C^{1+\alpha }\) Axiom A flows have exponential decay of correlations

In general, linear response theory expresses the relation between a driving and a physical system’s response only to first order in perturbation theory. In the context of charge transport, this is the linear relation between current and electromotive force expressed in Ohm’s law. We show here that in the case of the quantum Hall effect, all higher-order corrections vanish. We prove this in a fully

Quantum trajectories are Markov processes that describe the time evolution of a quantum system undergoing continuous indirect measurement. Mathematically, they are defined as solutions of the so-called Stochastic Schrödinger Equations, which are nonlinear stochastic differential equations driven by Poisson and Wiener processes. This paper is devoted to the study of the invariant measures of quantum

We classify chiral symmetric periodically driven quantum systems on a one-dimensional lattice. The driving process is local, can be continuous, or discrete in time, and we assume a gap condition for the corresponding Floquet operator. The analysis is in terms of the unitary operator at a half-period, the half-step operator. We give a complete classification of the connected classes of half-step operators

We introduce a new class of non-local games and corresponding densities, which we call bisynchronous. Bisynchronous games are a subclass of synchronous games and exhibit many interesting symmetries when the algebra of the game is considered. We develop a close connection between these non-local games and the theory of quantum groups which recently surfaced in studies of graph isomorphism games. When

We establish the future nonlinear stability of Friedmann–Lemaître–Robertson–Walker (FLRW) solutions to the Einstein–Euler equations of the universe filled with a large class of perfect fluids (the equations of state are allowed to be certain nonlinear or linear types both). Several previous results as specific examples can be covered in the results of this article. We emphasize that the future stability

We present a procedure for asymptotic gluing of hyperboloidal initial data sets for the Einstein field equations that preserves the shear-free condition. Our construction is modeled on the gluing construction in Isenberg et al. (Ann Henri Poincaré 11(5):881–927, 2010), but with significant modifications that incorporate the shear-free condition. We rely on the special Hölder spaces, and the corresponding

We construct a topological space to study contextuality in quantum mechanics. The resulting space is a classifying space in the sense of algebraic topology. Cohomological invariants of our space correspond to physical quantities relevant to the study of contextuality. Within this framework the Wigner function of a quantum state can be interpreted as a class in the twisted K-theory of the classifying

We consider N bosons in a box with volume one, interacting through a two-body potential with scattering length of the order \(N^{-1+\kappa }\), for \(\kappa >0\). Assuming that \(\kappa \in (0;1/43)\), we show that low-energy states exhibit Bose–Einstein condensation and we provide bounds on the expectation and on higher moments of the number of excitations.

The framework of dynamical C*-algebras for scalar fields in Minkowski space, based on local scattering operators, is extended to theories with locally perturbed kinetic terms. These terms encode information about the underlying spacetime metric, so the causality relations between the scattering operators have to be adjusted accordingly. It is shown that the extended algebra describes scalar quantum

We study the asymptotic behaviour of the eigenvalue counting function for self-adjoint elliptic linear operators defined through classical weighted symbols of order (1, 1), on an asymptotically Euclidean manifold. We first prove a two-term Weyl formula, improving previously known remainder estimates. Subsequently, we show that under a geometric assumption on the Hamiltonian flow at infinity, there

We study the correlation between the nodal length of random spherical harmonics and the length of a nonzero level set. We show that the correlation is asymptotically zero, while the partial correlation after removing the effect of the random \(L^2\)-norm of the eigenfunctions is asymptotically one.

We construct states describing Bose–Einstein condensates at finite temperature for a relativistic massive complex scalar field with \(|\varphi |^4\)-interaction. We start with the linearized theory over a classical condensate and construct interacting fields by perturbation theory. Using the concept of thermal masses, equilibrium states at finite temperature can be constructed by the methods developed

We consider the discrete Schrödinger operator \(H=-\Delta +V\) on a cube \(M\subset {{\mathbb {Z}}}^d\), with periodic or Dirichlet (simple) boundary conditions. We use a hidden landscape function u, defined as the solution of an inhomogeneous boundary problem with uniform right-hand side for H, to predict the location of the localized eigenfunctions of H. Explicit bounds on the exponential decay of

The emergence of the concept of a causal fermion system is revisited and further investigated for the vacuum Dirac equation in Minkowski space. After a brief recap of the Dirac equation and its solution space, in order to allow for the effects of a possibly nonstandard structure of spacetime at the Planck scale, a regularization by a smooth cutoff in momentum space is introduced, and its properties

We call a partially hyperbolic diffeomorphism partially volume expanding if the Jacobian restricted to any hyperplane that contains the unstable bundle \(E^u\) is larger than 1. This is a \(C^1\) open property. We show that any \(C^{1+}\) partially volume expanding diffeomorphisms admits finitely many physical measures, and the union of their basins has full volume.

The spectra of n-Laplacian operators \((-\Delta )^n\) on finite metric graphs are studied. An effective secular equation is derived and the spectral asymptotics are analysed, exploiting the fact that the secular function is close to a trigonometric polynomial. The notion of the quasispectrum is introduced, and its uniqueness is proved using the theory of almost periodic functions. To achieve this,

We investigate the trigonometric real form of the spin Ruijsenaars–Schneider system introduced, at the level of equations of motion, by Krichever and Zabrodin in 1995. This pioneering work and all earlier studies of the Hamiltonian interpretation of the system were performed in complex holomorphic settings; understanding the real forms is a non-trivial problem. We explain that the trigonometric real

This paper is concerned with the asymptotic properties of the small data solutions to the massless Vlasov–Maxwell system in 3d. We use vector field methods to derive almost optimal decay estimates in null directions for the electromagnetic field, the particle density and their derivatives. No compact support assumption in x or v is required on the initial data, and the decay in v is in particular initially

We investigate some foundational issues in the quantum theory of spin transport, in the general case when the unperturbed Hamiltonian operator \(H_0\) does not commute with the spin operator in view of Rashba interactions, as in the typical models for the quantum spin Hall effect. A gapped periodic one-particle Hamiltonian \(H_0\) is perturbed by adding a constant electric field of intensity \(\varepsilon

We deal with the dynamical system properties of a Gorini–Kossakowski–Sudarshan–Lindblad equation with mean-field Hamiltonian that models a simple laser by applying a mean-field approximation to a quantum system describing a single-mode optical cavity and a set of two-level atoms, each coupled to a reservoir. We prove that the mean-field quantum master equation has a unique regular stationary solution

Relying on the hyperboloidal foliation method, we establish the nonlinear stability of the ground state of the U(1) standard model of electroweak interactions. This amounts to establishing a global-in-time theory for the initial value problem for a nonlinear wave–Klein–Gordon system that couples (Dirac, scalar, gauge) massive equations together. In particular, we investigate here the Dirac equation

Let \(\mathcal {G}\) be a metric non-compact connected graph with finitely many edges. The main object of the paper is the Hamiltonian \(\mathbf{H}_{\alpha }\) associated in \(L^2(\mathcal {G};\mathbb {C}^m)\) with a matrix Sturm–Liouville expression and boundary delta-type conditions at each vertex. Assuming that the potential matrix is summable and applying the technique of boundary triplets and

We study the spectral properties of ergodic Schrödinger operators that are associated with a certain family of non-primitive substitutions on a binary alphabet. The corresponding subshifts provide examples of dynamical systems that go beyond minimality, unique ergodicity and linear complexity. In some parameter region, we are naturally in the setting of an infinite ergodic measure. The almost sure

We consider the complex eigenvalues of a Wishart type random matrix model \(X=X_1 X_2^*\), where two rectangular complex Ginibre matrices \(X_{1,2}\) of size \(N\times (N+\nu )\) are correlated through a non-Hermiticity parameter \(\tau \in [0,1]\). For general \(\nu =O(N)\) and \(\tau \), we obtain the global limiting density and its support, given by a shifted ellipse. It provides a non-Hermitian

We study the distribution of the eigenvalue condition numbers \(\kappa _i=\sqrt{ ({\mathbf{l}}_i^* {\mathbf{l}}_i)({\mathbf{r}}_i^* {\mathbf{r}}_i)}\) associated with real eigenvalues \(\lambda _i\) of partially asymmetric \(N\times N\) random matrices from the real Elliptic Gaussian ensemble. The large values of \(\kappa _i\) signal the non-orthogonality of the (bi-orthogonal) set of left \({\mathbf{l}}_i\)

The oscillator Racah algebra \(\mathcal {R}_n(\mathfrak {h})\) is realized by the intermediate Casimir operators arising in the multifold tensor product of the oscillator algebra \(\mathfrak {h}\). An embedding of the Lie algebra \(\mathfrak {sl}_{n-1}\) into \(\mathcal {R}_n(\mathfrak {h})\) is presented. It relates the representation theory of the two algebras. We establish the connection between

We consider discrete Schrödinger operators on \(\mathbb {Z}^d\) for which the perturbation consists of the sum of a long-range-type potential and a Wigner–von Neumann-type potential. Still working in a framework of weighted Mourre theory, we improve the limiting absorption principle (LAP) that was obtained in Mandich (J Funct Anal 272(6):2235–2272, 2017). To our knowledge, this is a new result even

It is well known that the spacetime \(\text {AdS}_2\times S^2\) arises as the ‘near-horizon’ geometry of the extremal Reissner–Nordstrom solution, and for that reason, it has been studied in connection with the AdS/CFT correspondence. Motivated by a conjectural viewpoint of Juan Maldacena, Galloway and Graf (Adv Theor Math Phys 23(2):403–435, 2019) studied the rigidity of asymptotically \(\text {AdS}_2\times

Polynomial ensembles are a sub-class of probability measures within determinantal point processes. Examples include products of independent random matrices, with applications to Lyapunov exponents, and random matrices with an external field, that may serve as schematic models of quantum field theories with temperature. We first analyse expectation values of ratios of an equal number of characteristic

We consider the quantum mechanical many-body problem of a single impurity particle immersed in a weakly interacting Bose gas. The impurity interacts with the bosons via a two-body potential. We study the Hamiltonian of this system in the mean-field limit and rigorously show that, at low energies, the problem is well described by the Fröhlich polaron model.

We prove that the Aizenman–Lieb theorem on ferromagnetism in the Hubbard model holds true even if the electron–phonon interactions and the electron–photon interactions are taken into account. Our proof is based on path integral representations of the partition functions.

We study the spectra of general \(N\times N\) Toeplitz matrices given by symbols in the Wiener Algebra perturbed by small complex Gaussian random matrices, in the regime \(N\gg 1\). We prove an asymptotic formula for the number of eigenvalues of the perturbed matrix in smooth domains. We show that these eigenvalues follow a Weyl law with probability sub-exponentially close to 1, as \(N\gg 1\), in particular

We consider a multi-dimensional continuum Schrödinger operator which is given by a perturbation of the negative Laplacian by a compactly supported potential. We establish both an upper bound and a lower bound on the bipartite entanglement entropy of the ground state of the corresponding quasi-free Fermi gas. The bounds prove that the scaling behaviour of the entanglement entropy remains a logarithmically

Integrability condition of Hamiltonian perturbations of integrable Hamiltonian PDEs of hydrodynamic type up to the second-order approximation is considered. Under a nondegeneracy assumption, we show that the Hamiltonian perturbation at the first-order approximation is integrable if and only if it is trivial, and that under a further assumption, the Hamiltonian perturbation at the second-order approximation

We propose a field-theoretic interpretation of Ruelle zeta function and show how it can be seen as the partition function for BF theory when an unusual gauge-fixing condition on contact manifolds is imposed. This suggests an alternative rephrasing of a conjecture due to Fried on the equivalence between Ruelle zeta function and analytic torsion, in terms of homotopies of Lagrangian submanifolds.

In this paper, we study a hierarchical supersymmetric model for a class of gapless, three-dimensional, weakly disordered quantum systems, displaying pointlike Fermi surface and conical intersections of the energy bands in the absence of disorder. We use rigorous renormalization group methods and supersymmetry to compute the correlation functions of the system. We prove algebraic decay of the two-point

The purpose of this paper is twofold. In one direction, we extend the spectral method for random piecewise expanding and hyperbolic (Anosov) dynamics developed by the first author et al. to establish quenched versions of the large deviation principle, central limit theorem and the local central limit theorem for vector-valued observables. We stress that the previous works considered exclusively the

We address the spectral problem of the formally normal quantum mechanical operator associated with the quantised mirror curve of the toric (almost) del Pezzo Calabi–Yau threefold called local \({\mathbb {P}}^2\) in the case of complex values of Planck’s constant. We show that the problem can be approached in terms of the Bethe ansatz-type highly transcendental equations.

We define a mass-type invariant for asymptotically hyperbolic manifolds with a non-compact boundary which are modelled at infinity on the hyperbolic half-space and prove a sharp positive mass inequality in the spin case under suitable dominant energy conditions. As an application, we show that any such manifold which is Einstein and either has a totally geodesic boundary or is conformally compact and

We prove a version of Talagrand’s concentration inequality for subordinated sub-Laplacians on a compact Riemannian manifold using tools from noncommutative geometry. As an application, motivated by quantum information theory, we show that on a finite-dimensional matrix algebra the set of self-adjoint generators satisfying a tensor stable modified logarithmic Sobolev inequality is dense.

We introduce a class of UV-regularized two-body interactions for fermions in \({\mathbb {R}}^d\) and prove a Lieb–Robinson estimate for the dynamics of this class of many-body systems. As a step toward this result, we also prove a propagation bound of Lieb–Robinson type for Schrödinger operators. We apply the propagation bound to prove the existence of infinite-volume dynamics as a strongly continuous

In this work, we study the number of small eigenvalues and the convergence to the equilibrium of the Bouncy Particle Sampler process and the zigzag process generators in the small temperature regime. Such processes, which fall in the class of Piecewise Deterministic Markov Processes, are non-diffusive and non-reversible. They have recently been used a lot for simulation issues, falling in the domain

For hyperbolic flows \(\varphi _t\), we examine the Gibbs measure of points w for which $$\begin{aligned} \int _0^T G(\varphi _t w) \hbox {d}t - a T \in (- e^{-\epsilon n}, e^{- \epsilon n}) \end{aligned}$$ as \(n \rightarrow \infty \) and \(T \ge n\), provided \(\epsilon > 0\) is sufficiently small. This is similar to local central limit theorems. The fact that the interval \((- e^{-\epsilon n}, e^{-

We study uncertainty principles for function classes on the torus. The classes are defined in terms of spectral subspaces of the energy or the momentum, respectively. In our main theorems, the support of the Fourier transform of the considered functions is allowed to be contained in (a finite number of) d-dimensional cubes. The estimates we obtain do not depend on the size of the torus and the position

For a given finite index inclusion of strongly additive conformal nets \(\mathcal {B}\subset \mathcal {A}\) and a compact group \(G < {{\,\mathrm{Aut}\,}}(\mathcal {A}, \mathcal {B})\), we consider the induction and the restriction procedures for twisted representations. Let \(G' < {{\,\mathrm{Aut}\,}}(\mathcal {B})\) be the group obtained by restricting each element of G to \(\mathcal {B}\). We introduce

Unfortunately, the corresponding author name was incorrectly published in the original version.

In this article, we prove a local energy estimate for the linear wave equation on metrics with slow decay to a Kerr metric with small angular momentum. As an application, we study the quasilinear wave equation \(\Box _{g(u, t, x)} u = 0\) where the metric g(u, t, x) is close (and asymptotically equal) to a Kerr metric with small angular momentum g(0, t, x). Under suitable assumptions on the metric

We consider a time-dependent two-level quantum system interacting with a free Boson reservoir. The coupling is energy conserving and depends slowly on time, as does the system Hamiltonian, with a common adiabatic parameter \(\varepsilon \). Assuming that the system and reservoir are initially decoupled, with the reservoir in equilibrium at temperature \(T\ge 0\), we compute the transition probability

We study conformal properties of local terms such as contact terms and semi-local terms in correlation functions of a conformal field theory. Not all of them are universal observables, but they do appear in physically important correlation functions such as (anomalous) Ward–Takahashi identities or Schwinger–Dyson equations. We develop some tools such as embedding space delta functions and effective