This paper intends to provide new, simple, and self-contained proofs of the equivalence of various different descriptions of the uniformly hyperbolic SL$(2,\mathbb R)$. While in the scenario of the Schrödinger cocyles, they may in turn be applied to give new and simple proofs of theorems regarding their relation with the spectral analysis of one-dimensional discrete Schrödinger operators. Concretely

We consider general symmetric systems of first order linear partial differential operators on domains $\Omega \subset \mathbb R^d$ , and we seek sufficient conditions on the coefficients which ensure essential self-adjointness. The coefficients of the first order terms are only required to belong to $C^1 (\Omega)$ and there is no ellipticity condition. Our criterion writes as the completeness of an

Using the factorizations of suitable operators, we establish several identities that give simple and direct understandings as well as provide the remainders and “virtual” optimizers of several the Hardy and Hardy–Rellich type inequalities.

Let $H = H_0 + P$ denote the harmonic oscillator on $\mathbb{R}^d$ perturbed by an isotropic pseudodifferential operator $P$ of order $1$ and let $U(t) = \operatorname{exp}(- it H)$. We prove a Gutzwiller–Duistermaat–Guillemin type trace formula for $\operatorname{Tr} U(t).$ The singularities occur at times $t \in 2 \pi \mathbb{Z}$ and the coefficients involve the dynamics of the Hamilton flow of the

Oscillation theory locates the spectrum of a differential equation by counting the zeros of its solutions. We present a version of this theory for canonical systems $Ju'=-zHu$ and then use it to discuss semibounded operators from this point of view. Our main new result is a characterization of systems with purely discrete spectrum in terms of the asymptotics of their coefficient functions; we also

In this paper, we obtain Hardy, Hardy–Rellich and refined Hardy inequalities on general stratified groups and weighted Hardy inequalities on general homogeneous groups using the factorization method of differential operators, inspired by the recent work of Gesztesy and Littlejohn [9]. We note that some of the obtained inequalities are new also in the usual Euclidean setting. We also obtain analogues

In this paper we study an interacting two-particle system on the half-line. We focus on spectral properties of the Hamiltonian for a large class of two-particle potentials. We characterize the essential spectrum and prove, as a main result, the existence of eigenvalues below the bottom of it. We also prove that the discrete spectrum contains only finitely many eigenvalues.

We consider Sturm–Liouville problems with a discontinuity in an interior point, which are motivated by the inverse problems for the torsional modes of the Earth. We assume that the potential on the right half-interval and the coefficient in the right boundary condition are given. Half-inverse problems are studied, that consist in recovering the potential on the left half-interval and the left boundary

Matrix-valued measures provide a natural language for the theory of finite rank perturbations. In this paper we use this language to prove some new perturbation theoretic results. Our main result is a generalization of the Aronszajn–Donoghue theorem about the mutual singularity of the singular parts of the spectrum for rank one perturbations to the case of finite rank perturbations. Simple direct sum

This paper is devoted to the spectral analysis of the Neumann realization of the 2D magnetic Laplacian with semiclassical parameter $h > 0$ in the case when the magnetic field vanishes along a smooth curvewhich crosses itself inside a bounded domain. We investigate the behavior of its eigenpairs in the limit $h \to 0$. We show that each crossing point acts as a potential well, generating a new decay

We prove and apply two theorems: first, a quantitative, scale-free unique continuation estimate for functions in a spectral subspace of a Schrödinger operator on a bounded or unbounded domain; second, a perturbation and lifting estimate for edges of the essential spectrum of a self-adjoint operator under a semi-definite perturbation. These two results are combined to obtain lower and upper Lipschitz

We prove exponential dynamical localization in expectation for the one dimensional Anderson model via positivity and uniform LDT for the Lyapunov exponent. The ideas are based on the methods developed in [29].

We show that the set of $n \times n$ complex symmetric matrix pencils of rank at most $r$ is the union of the closures of $\lfloor r/2\rfloor +1$ sets of matrix pencils with some, explicitly described, complete eigenstructures. As a consequence, these are the generic complete eigenstructures of $n \times n$ complex symmetric matrix pencils of rank at most $r$. We also show that the irreducible components

We study of the directional distribution function of nodal lines for eigenfunctions of the Laplacian on a planar domain. This quantity counts the number of points where the normal to the nodal line points in a given direction. We give upper bounds for the flat torus, and compute the expected number for arithmetic random waves.

L. A. Bunimovich and B. Z. Webb developed a theory for transforming a finite weighted graph while preserving its spectrum, referred as isospectral reduction theory. In this workwe extend this theory to a class of operators on Banach spaces that include Markov type operators. We apply this theory to infinite countable weighted graphs admitting a finite structural set to calculate the stationary measures

We find classes of nonlocal operators of Schrödinger type on a locally compact noncompact Abelian group $\mathfrak G$, for which there exists a ground state. In particular, such a result is obtained for the case where the principal part of our operator generates a recurrent random walk. Explicit conditions for the existence of a ground state are obtained for the case $\mathfrak G =\mathbb Q_p^n$ where

In the proof of the BFK-gluing formula for zeta-determinants of Laplacians there appears a real polynomial whose constant term is an important ingredient in the gluing formula. This polynomial is determined by geometric data on an arbitrarily small collar neighborhood of a cutting hypersurface. In this paper we express the coefficients of this polynomial in terms of the scalar and principal curvatures

We investigate how bounds of resonance counting functions for Schottky surfaces behave under transitions to covering surfaces of finite degree. We consider the classical resonance counting function asking for the number of resonances in large (and growing) disks centered at the origin of $\mathcal C$, as well as the (fractal) resonance counting function asking for the number of resonances in boxes

We study three-body Schrödinger operators in one and two dimensions modelling an exciton interacting with a charged impurity. We consider certain classes of multiplicative interaction potentials proposed in the physics literature. We show that if the impurity charge is larger than some critical value, then three-body bound states cannot exist. Our spectral results are confirmed by variational numerical

We study the one-dimensional discrete Schrödinger operator with the skew-shift potential $2\lambda\cos\big(2\pi \big(\binom{j}{2} \omega + jy + x\big)\big)$. This potential is long conjectured to behave like a random one, i.e., it is expected to produce Anderson localization for arbitrarily small coupling constants $\lambda > 0$. In this paper, we introduce a novel perturbative approach for studying

We discuss the essential self-adjointness of wave operators, as well as the limiting absorption principle, in generalizations of asymptotically Minkowski settings. This is obtained via using a Fredholm framework for inverting the spectral family first, and then refining its conclusions to show dense range of $\Box-\lambda$, $\lambda\notin\mathbb R$, in $L^2_\mathrm {sc}$ when acting on an appropriate

We consider the problem of identifying a unitary Yang–Mills connection $\nabla$ on a Hermitian vector bundle from the Dirichlet-to-Neumann (DN) map of the connection Laplacian $\nabla^*\nabla$ over compact Riemannian manifolds with boundary. We establish uniqueness of the connection up to a gauge equivalence in the case of trivial line bundles in the smooth category and for the higher rank case in

It is shown that in a large class of disordered systems with singular alloy-type disorder and non-local media-particle interactions, the marginal measures of the induced random potential and the finite-volume integrated density of states (IDS) are infinitely differentiable in higher dimensions. The proposed approach complements the classical Wegner estimate which says that the IDS in the short-range

We consider resonances associated with excited eigenvalues of the cavity of a general Helmholtz resonator with straight neck. Under the assumption that the neck stays away from the nodal set of the corresponding eigenstate, we generalise the optimal exponential lower bound on the width of the resonance, that we have obtained in a previous paper for the ground resonance only.

In this paper we consider Anderson models with a large number of sites with zero interaction. For such models we study the spectral statistics in the region of complete localization. We show that Poisson statistics holds for such energies, by proving the Minami estimate.

This paper establishes by employing analytic and probabilistic techniques estimates concerning the heat content for the fractional Schrödinger operator $(-\Delta)^\frac {\alpha}{2} + c1_{\Omega}, c > 0$ with $0 < \alpha \leq 2$ in $\mathbb R^d$, $d \geq 2$ and $\Omega$ a Lebesgue measure set satisfying some regularity conditions.

We give examples of semiclassical Schrödinger operators with exponentially large cutoff resolvent norms, even when the supports of the cutoff and potential are very far apart. The examples are radial, which allows us to analyze the resolvent kernel in detail using ordinary differential equation techniques. In particular,we identify a threshold spatial radius where the resolvent behavior changes. We

We give an elementary proof of a weighted resolvent estimate for semiclassical Schrödinger operators in dimension $n \ge 1$. We require the potential belong to $L^\infty(\mathbb{R}^n)$ and have compact support, but do not require that it have distributional derivatives in $L^\infty(\mathbb{R}^n)$. The weighted resolvent norm is bounded by $e^{Ch^{-4/3}\log(h^{-1})}$, where $h$ is the semiclassical

We work with very general Banach spaces of analytic functions in the disk or other domains which satisfy a minimum number of natural axioms. Among the preliminary results, we discuss some implications of the basic axioms and identify all functional Banach spaces in which every bounded analytic function is a pointwise multiplier. Next, we characterize (in various ways) the weighted composition operators

In this paper, we investigate the anisotropic Calderón problem on cylindrical Riemannian manifolds with boundary having two ends and equipped with singular metrics ofwarped product type, that iswhose coefficients only depend on the horizontal direction of the cylinder. By singular, we mean that these coefficients are positive almost everywhere and belong to some $L^p, 1 \leq p \leq \infty$ spaces only

Let $H$ and $H'$ be the complex Hilbert spaces. For $p\in]1,\infty[\backslash\{2\}$ we consider the Banach space $C_p(H)$ of all $p$-Schatten von Neumann operators, whose unit sphere is denoted by $S(C_p(H))$. In this paper we prove that every surjective isometry $\Delta\colon S(C_p(H))\to S(C_p(H'))$ can be extended to a complex linear or to a conjugate linear surjective isometry $T\colon C_p(H)\to

The main focus of this paper is the following matrix Cauchy problem for the Dirac system on the interval [0,1]: $$D'(x)+\begin{2bmatrix} 0 & \sigma_1(x)\\ \sigma_2(x) & 0 \end{2bmatrix} D(x)=i\mu\begin{2bmatrix} 1 & 0\\ 0 &-1 \end{2bmatrix}D(x),\quad D(0)=\begin{2bmatrix} 1 & 0\\ 0 & 1 \end{2bmatrix},$$ where $\mu\in\mathbb{C}$ is a spectral parameter, and $\sigma_j\in L_2[0,1]$, $j=1,2$. We propose

This article is a first attempt to obtain weak limit formulas for weighted means of orthogonal polynomials. For this, we introduce a new mean Nevai class that guarantees the existence of a limiting measure for the weighted means. We show that for a family of measures in this mean Nevai class also the means of the Christoffel–Darboux kernels and the asymptotic distribution of the roots converge weakly

Let $H$ be a Schrödinger operator defined on a noncompact Riemannian manifold $\Omega$, and let $W\in L^\infty(\Omega;\mathbb{R})$. Suppose that the operator $H+W$ is critical in $\Omega$, and let $\varphi$ be the corresponding Agmon ground state. We prove that if $u$ is a generalized eigenfunction of $H$ satisfying $|u|\leq \varphi$ in $\Omega$, then the corresponding eigenvalue is in the spectrum

In this paper we discuss the relationship between the numerical range of an extensive class of unbounded operator functions and the joint numerical range of the operator coefficients. Furthermore, we derive methods on how to find estimates of the joint numerical range. Those estimates are used to obtain explicitly computable enclosures of the numerical range of the operator function and resolvent estimates

We provide examples of operators $T(D)+V$ in $L^2(\mathbb R^d)$ with decaying potentials that have embedded eigenvalues. The decay of the potential depends on the curvature of the (Fermi) surfaces of constant kinetic energy $T$. We make the connection to counterexamples in Fourier restriction theory.