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.

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

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

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

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.

In this paper, we analyse the boundedness of solutions \(\phi \) of the wave equation in the Oppenheimer–Snyder model of gravitational collapse in both the case of a reflective dust cloud and a permeating dust cloud. We then proceed to define the scattering map on this space-time and look at the implications of our boundedness results on this scattering map. Specifically, it is shown that the energy

We investigate entanglement breaking times of Markovian evolutions in discrete and continuous time. In continuous time, we characterize which Markovian evolutions are eventually entanglement breaking, that is, evolutions for which there is a finite time after which any entanglement initially present has been destroyed by the noisy evolution. In the discrete-time framework, we consider the entanglement

We study magnetic Schrödinger operators on graphs. We extend the notion of sparseness of graphs by including a magnetic quantity called the frustration index. This notion of magnetic-sparseness turns out to be equivalent to the fact that the form domain is an \(\ell ^{2}\) space. As a consequence, we get criteria of discreteness for the spectrum and eigenvalue asymptotics.

Hyun-Chul Jang observed that we dropped a term in our calculations in [1].

We recall the notions of Frölicher and diffeological spaces, and we build regular Frölicher Lie groups and Lie algebras of formal pseudo-differential operators in one independent variable. Combining these constructions with a smooth version of Mulase’s deep algebraic factorization of infinite-dimensional groups based on formal pseudo-differential operators, we present two proofs of the well-posedness

We consider the Schrödinger operator \(H_0\) with constant magnetic field B of scalar intensity \(b>0\), self-adjoint in \(L^2({{\mathbb {R}}}^3)\), and its perturbations \(H_+\) (resp., \(H_-\)) obtained by imposing Dirichlet (resp., Neumann) conditions on the boundary of the bounded domain \(\Omega _{\mathrm{in}} \subset {{\mathbb {R}}}^3\). We introduce the Krein spectral shift functions \(\xi (E;H_\pm

One can argue that on flat space \({\mathbb {R}}^d\), the Weyl quantization is the most natural choice and that it has the best properties (e.g., symplectic covariance, real symbols correspond to Hermitian operators). On a generic manifold, there is no distinguished quantization, and a quantization is typically defined chart-wise. Here we introduce a quantization that, we believe, has the best properties

We construct several examples of compactifications of Einstein metrics. We show that the Eguchi–Hanson instanton admits a projective compactification which is non-metric, and that a metric cone over any (pseudo)-Riemannian manifolds admits a metric projective compactification. We construct a para-c-projective compactification of neutral signature Einstein metrics canonically defined on certain rank-n

We show that the patterns in the Abelian sandpile are stable. The proof combines the structure theory for the patterns with the regularity machinery for non-divergence form elliptic equations. The stability results allow one to improve weak-\(*\) convergence of the Abelian sandpile to pattern convergence for certain classes of solutions.

We consider solutions of the massless scalar wave equation \(\Box _g\psi =0\), without symmetry, on fixed subextremal Kerr backgrounds \(({{\mathcal {M}}}, g)\). It follows from previous analyses in the Kerr exterior that for solutions \(\psi \) arising from sufficiently regular data on a two-ended Cauchy hypersurface, the solution and its derivatives decay suitably fast along the event horizon \({{\mathcal

We discuss realizations of \(L:=-\partial _x^2+V(x)\) as closed operators on \(L^2]a,b[\), where V is complex, locally integrable and may have an arbitrary behavior at (finite or infinite) endpoints a and b. The main tool of our analysis is Green’s operators, that is, various right inverses of L.

This paper is devoted to an averaging principle for multiscale stochastic fractional Schrödinger equation. This averaging principle can validate the effectiveness of the averaging method in fractional quantum mechanics; it ascertains the utility of the approximation obtained by averaging method. Unlike the stochastic heat equation, the smoothing effect of fractional Schrödinger semigroup is not enough

Properties of the energy–momentum relation for the Fröhlich polaron are of continuing interest, especially for large values of the coupling constant. By combining spectral theory with the available results on the central limit theorem for the polaron path measure, we prove that, except for an intermediate range of couplings, the inverse effective mass is strictly positive and coincides with the diffusion

We obtain an explicit formula for the full expansion of the spectral action on Robertson–Walker spacetimes, expressed in terms of Bell polynomials, using Brownian bridge integrals and the Feynman–Kac formula. We then apply this result to the case of multifractal Packed Swiss Cheese Cosmology models obtained from an arrangement of Robertson–Walker spacetimes along an Apollonian sphere packing. Using

We study homogenization for a class of generalized Langevin equations (GLEs) with state-dependent coefficients and exhibiting multiple time scales. In addition to the small mass limit, we focus on homogenization limits, which involve taking to zero the inertial time scale and, possibly, some of the memory time scales and noise correlation time scales. The latter are meaningful limits for a class of

When trying to cast the free fermion in the framework of functorial field theory, its chiral anomaly manifests in the fact that it assigns the determinant of the Dirac operator to a top-dimensional closed spin manifold, which is not a number as expected, but an element of a complex line. In functorial field theory language, this means that the theory is twisted, which gives rise to an anomaly theory

Quantum Markov semigroups characterize the time evolution of an important class of open quantum systems. Studying convergence properties of such a semigroup and determining concentration properties of its invariant state have been the focus of much research. Quantum versions of functional inequalities (like the modified logarithmic Sobolev and Poincaré inequalities) and the so-called transportation

The product of local operators in a topological quantum field theory in dimension greater than one is commutative, as is more generally the product of extended operators of codimension greater than one. In theories of cohomological type, these commutative products are accompanied by secondary operations, which capture linking or braiding of operators, and behave as (graded) Poisson brackets with respect

Classical field theory is insensitive to the split of the field into a background configuration and a dynamical perturbation. In gauge theories, the situation is complicated by the fact that a covariant (w.r.t. the background field) gauge fixing breaks this split independence of the action. Nevertheless, background independence is preserved on the observables, as defined via the BRST formalism, since

We consider a system of N interacting fermions in \( {\mathbb {R}}^3 \) confined by an external potential and in the presence of a homogeneous magnetic field. The intensity of the interaction has the mean-field scaling 1/N. With a semi-classical parameter \( \hbar \sim N^{-1/3} \), we prove convergence in the large N limit to the appropriate magnetic Thomas–Fermi-type model with various strength scalings

A hyperlink is a finite set of non-intersecting simple closed curves in \(\mathbb {R} \times \mathbb {R}^3\). Let R be a compact set inside \(\mathbb {R}^3\). The dynamical variables in General Relativity are the vierbein e and a \(\mathfrak {su}(2)\times \mathfrak {su}(2)\)-valued connection \(\omega \). Together with Minkowski metric, e will define a metric g on the manifold. Denote \(V_R(e)\) as