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

We study equivariant linear maps between finite-dimensional matrix algebras, as introduced in Collins et al. (Linear Algebra Appl 555:398–411, 2018). These maps satisfy an algebraic property which makes it easy to study their positivity or k-positivity. They are therefore particularly suitable for applications to entanglement detection in quantum information theory. We characterize their Choi matrices

In this article we consider the ergodic optimization for hyperbolic flows and Lorenz attractors with respect to both continuous and Hölder continuous observables. In the context of hyperbolic flows we prove that a Baire generic subset of continuous observables have a unique maximizing measure, with full support and zero entropy, and that a Baire generic subset of Hölder continuous observables admit

The main aim of this paper is the study of the general solution of the exceptional Hermite differential equation with fixed partition \(\lambda = (1)\) and the construction of minimal surfaces associated with this solution. We derive a linear second-order ordinary differential equation associated with a specific family of exceptional polynomials of codimension two. We show that these polynomials can

We consider a one-dimensional continuum Anderson model where the potential decays in average like \(|x|^{-\alpha }\), \(\alpha >0\). We show dynamical localization for \(0<\alpha <\frac{1}{2}\) and provide control on the decay of the eigenfunctions.

We consider scattering matrix for Schrödinger-type operators on \(\mathbb {R}^d\) with perturbation \(V(x)=O(\langle x \rangle ^{-1})\) as \(|x|\rightarrow \infty \). We show that the scattering matrix (with time-independent modifiers) is a pseudodifferential operator and analyze its spectrum. We present examples of which the spectrum of the scattering matrices has dense point spectrum, and absolutely

It is the purpose of this note to study the inhomogeneous coupled Schrödinger system$$\begin{aligned} i\dot{u}_j +\Delta u_j\pm |x|^b\left( \displaystyle \sum _{k=1}^{m}a_{jk}|u_k|^p\right) |u_j|^{p-2}u_j=0. \end{aligned}$$Indeed, in the stationary case, the existence of ground states is proved and a sharp inhomogeneous Gagliardo–Nirenberg-type inequality is established. In the evolving case, the existence

We consider a construction of observables by using methods of supersymmetric field theories. In particular, we give an extension of AKSZ-type observables constructed in Mnev (Lett Math Phys 105:1735–1783, 2015) using the Batalin–Vilkovisky structure of AKSZ theories to a formal global version with methods of formal geometry. We will consider the case where the AKSZ theory is “split” which will give

We give explicit bounds for the Hausdorff dimension of the unique invariant measure of \(C^3\) multicritical circle maps without periodic points. These bounds depend only on the arithmetic properties of the rotation number, the number of critical points and their criticalities.

We prove that if \(\phi :\mathbb {R}^2 \rightarrow \mathbb {R}^{1+2}\) is a smooth, proper, timelike immersion with vanishing mean curvature, then necessarily \(\phi \) is an embedding, and every compact subset of \(\phi (\mathbb {R}^2)\) is a smooth graph. It follows that if one evolves any smooth, self-intersecting spacelike curve (or any planar spacelike curve whose unit tangent vector spans a closed

We introduce and investigate certain N player dynamic games on the line and in the plane that admit Coulomb gas dynamics as a Nash equilibrium. Most significantly, we find that the universal local limit of the equilibrium is sensitive to the chosen model of player information in one dimension but not in two dimensions. We also find that players can achieve game theoretic symmetry through selfish behavior

We study the asymptotic behaviour of Bianchi type VI\(_0\) spacetimes with orthogonal perfect fluid matter satisfying Einstein’s equations. In particular, we prove a conjecture due to Wainwright about the initial singularity on such backgrounds. Using the expansion-normalized variables of Wainwright–Hsu, we demonstrate that for a generic solution the initial singularity is vacuum dominated, anisotropic

It is physically expected that plane-wave configurations of the electron in QED induce disjoint representations of the algebra of the electromagnetic fields. This phenomenon of velocity superselection, which is one aspect of the infrared problem, is mathematically well established in non-relativistic (Pauli–Fierz type) models of QED. We show that velocity superselection can be resolved in such models

In the present paper, we make a rigorous study of the solitary wave solutions to a coupled Schrödinger system with quadratic and cubic nonlinearity. This kind of system of Schrödinger equations arises from optics theory. First, the existence and nonexistence of nontrivial solutions, respectively, in focusing and defocusing cases are considered. Second, we prove the existence of multiple nontrivial

Let X be a closed 6-dimensional manifold with a half-closed SU(3)-structure. On the product manifold \(X\times S^{1}\), with respect to the product \(G_{2}\)-structure and on a pullback vector bundle from X, we show that any \(G_{2}\)-instanton is equivalent to a Hermitian Yang–Mills connection on X via a “broken gauge”. This result reveals the topological type of the moduli of \(G_{2}\)-instantons

We discuss how much information on a Friedrichs model operator (a finite rank perturbation of the operator of multiplication by the independent variable) can be detected from ‘measurements on the boundary’. The framework of boundary triples is used to introduce the generalised Titchmarsh–Weyl M-function and the detectable subspaces which are associated with the part of the operator which is ‘accessible

We prove boundedness and decay statements for solutions to the spin \(\pm \,2\) generalized Teukolsky system on a Reissner–Nordström background with very small charge. The first equation of the system is the generalization of the Teukolsky equation in Schwarzschild for the extreme component of the curvature \(\alpha \). The second equation, coupled with the first one, is a new equation for a new gauge-invariant

In the present paper, we consider the Cauchy problem of the system of quadratic derivative nonlinear Schrödinger equations. This system was introduced by Colin and Colin (Differ Integral Equ 17:297–330, 2004). The first and second authors obtained some well-posedness results in the Sobolev space \(H^{s}({\mathbb R}^d)\). We improve these results for conditional radial initial data by rewriting the

We consider the Laguerre partition function and derive explicit generating functions for connected correlators with arbitrary integer powers of traces in terms of products of Hahn polynomials. It was recently proven in Cunden et al. (Ann. Inst. Henri Poincaré D, to appear) that correlators have a topological expansion in terms of weakly or strictly monotone Hurwitz numbers that can be explicitly computed

In this paper, we consider the minimization of interaction functional$$\begin{aligned} E[\rho ]= & {} \int _{{\mathbb {R}}^N}\frac{1}{a-1}\rho ^a(x)\mathrm{d}x+\frac{1}{2}\int _{{\mathbb {R}}^N} \int _{{\mathbb {R}}^N}K(x-y)\rho (x)\rho (y)\mathrm{d}x\mathrm{d}y\nonumber \\&+\int _{{\mathbb {R}}^N}F(x)\rho (x)\mathrm{d}x, \end{aligned}$$(0.1)where \(a>1\), the term \(\frac{1}{a-1}\rho ^a\) is homogeneous

In this paper, the spectral and scattering properties of a family of self-adjoint Dirac operators in \(L^2(\Omega ; \mathbb {C}^4)\), where \(\Omega \subset \mathbb {R}^3\) is either a bounded or an unbounded domain with a compact \(C^2\)-smooth boundary, are studied in a systematic way. These operators can be viewed as the natural relativistic counterpart of Laplacians with boundary conditions as

In this paper, the translation invariant massless Nelson model with ultraviolet cutoff is investigated. It is proven that the fiber operators does not have a ground state if the interaction is not infrared regular. Similar results have been obtained by other authors under severe assumptions on the regularity of the mass shell. The method introduced in this paper works under very general conditions

We show decay of the local energy of solutions of the charged Klein–Gordon equation in the exterior De Sitter–Reissner–Nordström spacetime by means of a resonance expansion of the local propagator.

The ground-states of the spin-S antiferromagnetic chain \(H_{{\text {AF}}}\) with a projection-based interaction and the spin-1/2 XXZ-chain \( H_{{\text {XXZ}}}\) at anisotropy parameter \(\Delta =\cosh (\lambda ) \) share a common loop representation in terms of a two-dimensional functional integral which is similar to the classical planar Q-state Potts model at \( \sqrt{Q}= 2S+1 =2\cosh (\lambda

We present a new approach to the eigensystem multiscale analysis (EMSA) for random Schrödinger operators that relies on the Wegner estimate. The EMSA treats all energies of the finite volume operator in an energy interval at the same time, simultaneously establishing localization of all eigenfunctions with eigenvalues in the energy interval with high probability. It implies all the usual manifestations

We prove that for various impurity models, in both classical and quantum settings, the self-energy matrix is a sparse matrix with a sparsity pattern determined by the impurity sites. In the quantum setting, such a sparsity pattern has been known since Feynman. Indeed, it underlies several numerical methods for solving impurity problems, as well as many approaches to more general quantum many-body problems

We develop a technique for proving number rigidity (in the sense of Ghosh and Peres in Duke Math J 166(10):1789–1858, 2017) of the spectrum of general random Schrödinger operators (RSOs). Our method makes use of Feynman–Kac formulas to estimate the variance of exponential linear statistics of the spectrum in terms of self-intersection local times. Inspired by recent results concerning Feynman–Kac formulas

We study Maxwell’s equation as a theory for smooth k-forms on globally hyperbolic spacetimes with timelike boundary as defined by Aké et al. (Structure of globally hyperbolic spacetimes with timelike boundary. arXiv:1808.04412 [gr-qc]). In particular, we start by investigating on these backgrounds the D’Alembert–de Rham wave operator \(\Box _k\) and we highlight the boundary conditions which yield

We establish the charged Penrose inequality$$\begin{aligned} m\ge \frac{1}{2}\left( \rho + \frac{q^2}{\rho } \right) \end{aligned}$$for time-symmetric initial data sets having an outermost minimal surface boundary and finitely many asymptotically cylindrical ends, with an appropriate rigidity statement. This is accomplished by a doubling argument based on the work of Weinstein and Yamada (Commun Math

We provide quantitative estimates on the location of eigenvalues of one-dimensional discrete Dirac operators with complex \(\ell ^p\)-potentials for \(1\le p \le \infty \). As a corollary, subsets of the essential spectrum free of embedded eigenvalues are determined for small \(\ell ^1\)-potential. Further possible improvements and sharpness of the obtained spectral bounds are also discussed.

We introduce a dynamically defined class of unbounded, connected, equilateral metric graphs on which the Kirchhoff Laplacian has zero Lebesgue measure spectrum and a non-trivial singular continuous part. A local Borg–Marchenko uniqueness result is obtained in order to utilize Kotani theory for aperiodic subshifts satisfying Boshernitzan’s condition.

We consider the XXZ spin chain, characterized by an anisotropy parameter \(\Delta >1\), and normalized such that the ground state energy is 0 and the ground state given by the all-spins-up state. The energies \(E_K = K(1-1/\Delta )\), \(K=1,2,\ldots \), can be interpreted as K-cluster breakup thresholds for down-spin configurations. We show that, for every K, the bipartite entanglement of all states

We study the small-mass (overdamped) limit of Langevin equations for a particle in a potential and/or magnetic field with matrix-valued and state-dependent drift and diffusion. We utilize a bootstrapping argument to derive a hierarchy of approximate equations for the position degrees of freedom that are able to achieve accuracy of order \(m^{\ell /2}\) over compact time intervals for any \(\ell \in

We prove the positive mass theorem for manifolds with distributional curvature which have been studied in Lee and LeFloch (Commun Math Phys 339(1):99–120, 2015) without spin condition. In our case, the manifold M has asymptotically flat metric \(g\in C^0\bigcap W^{1,p}_{-q}\), \(p>n\), \(q>\frac{n-2}{2}\). We show that the generalized ADM mass \(m_{ADM}(M,g)\) is nonnegative as long as \(q=n-2\), and

We consider for discrete Schrödinger operators with dimension three or larger on the square lattice. We prove that the asymptotic behavior at infinity of resonances at elliptic thresholds is similar to those of continuous Schrödinger operators and that resonances are absent at hyperbolic thresholds. Moreover, we show the limiting absorption principle near the hyperbolic thresholds with the optimal

The classical Lorenz flow, and any flow which is close to it in the \(C^2\)-topology, satisfies a Central Limit Theorem (CLT). We prove that the variance in the CLT varies continuously.

We prove that localization near band edges of multi-dimensional ergodic random Schrödinger operators with periodic background potential in \(L^2({\mathbb {R}}^d)\) is universal. By this, we mean that localization in its strongest dynamical form holds without extra assumptions on the random variables and independently of regularity or degeneracy of the Floquet eigenvalues of the background operator

In this paper, we consider the NLS equation with focusing nonlinearities in the presence of a potential. We investigate the compact soliton motions that correspond to a free soliton escaping the well created by the potential. We exhibit the dynamical system driving the exiting trajectory and construct associated nonlinear dynamics for untrapped motions. We show that the nature of the potential/soliton

In this paper we first provide the proof of \(\hbox {SU}(2)\)Y-Map convergence. Then, by using \(\hbox {SU}(1,1)\) LQG simplicity constraints, we define \(\hbox {SU}(1,1)\)Y-Map from infinitely differentiable with a compact support functions on \(\hbox {SU}(1,1)\) to the functions (not necessarily square integrable) on \(\hbox {SL}(2,C)\), and prove its convergence as well.