Mathematics - Spectral Theory Publications (50)


Mathematics - Spectral Theory Publications

Let M be a compact Riemannian manifold with boundary. Let b>0 be the number of connected components of its boundary. For manifolds of dimension at least 3, we prove that it is possible to obtain an arbitrarily large (b+1)-th Steklov eigenvalue using a smooth conformal perturbation which is supported in a thin neighbourhood of the boundary, identically equal to 1 on the boundary. Read More

Consideration is given to the continuous-time supercritical branching random walk over a multidimensional lattice with a finite number of particle generation sources of the same intensity both with and without constraint on the variance of jumps of random walk underlying the process. Asymptotic behavior of the Green function and eigenvalue of the evolution operator of the mean number of particles under source intensity close to the critical one was established. Read More

Suppose we count the positive integer lattice points beneath a convex decreasing curve in the first quadrant having equal intercepts. Then stretch in the coordinate directions so as to preserve the area under the curve, and again count lattice points. Which choice of stretch factor will maximize the lattice point count? We show the optimal stretch factor approaches $1$ as the area approaches infinity. Read More

We consider an alloy-type random Schr\"odinger operator $H$ in multi-dimensional Euclidean space $\mathbb{R}^{d}$ and its perturbation $H^{\tau}$ by a bounded and compactly supported potential with coupling constant $\tau$. Our main estimate concerns separate exponential decay of the disorder-averaged Schatten-von Neumann $p$-norm of $\chi_{a}(f(H) - f(H^{\tau})) \chi_{b}$ in $a$ and $b$. Here, $\chi_{a}$ is the multiplication operator corresponding to the indicator function of a unit cube centred about $a\in\mathbb{R}^{d}$, and $f$ is in a suitable class of functions of bounded variation with distributional derivative supported in the region of complete localisation for $H$. Read More

In this paper we study the behaviour of the Neumann data of Dirichlet eigenfunctions on triangles. We prove that the $L^2$ norm of the (semi-classical) Neumann data on each side is equal to the length of the side divided by the area of the triangle. The novel feature of this result is that it is {\it not} an asymptotic, but an exact formula. Read More

Let $\Omega$ be an open set in a complete, non-compact, $m$-dimensional Riemannian manifold $M$ with non-negative Ricci curvature, and without boundary, and let $v_{\Omega}$ be the torsion function for $\Omega$. It is shown that $v_{\Omega}$ is bounded if and only if the bottom of the spectrum of the Dirichlet Laplacian acting in $\Leb^2(\Omega)$, and denoted by $\lambda(\Omega)$, is bounded away from $0$. An upper bound for the torsion function is obtained for planar, convex sets in the Euclidean space $M=\R^2$ which is sharp in the limit of elongation. Read More

For a Riemannian covering $M_1\to M_0$ of complete Riemannian manifolds with boundary (possibly empty) and respective fundamental groups $\Gamma_1\subseteq\Gamma_0$, we show that the bottoms of the spectra of $M_0$ and $M_1$ coincide if the right action of $\Gamma_0$ on $\Gamma_1\backslash\Gamma_0$ is amenable. Read More

We derive asymptotic expansion of the heat trace for the Laplace-Beltrami operator on an even-dimensional manifold with a metrically conic singularity. Then we investigate how the terms in the expansion reflect geometry of a manifold. In four-dimensional case the criterion for a neighbourhood of a singularity to be isometric to a disk is proven. Read More

We investigate the level spacing distribution for the quantum spectrum of the square billiard. Extending work of Connors--Keating, and Smilansky, we formulate an analog of the Hardy--Littlewood prime $k$-tuple conjecture for sums of two squares, and show that it implies that the spectral gaps, after removing degeneracies and rescaling, are Poisson distributed. Consequently, by work of Rudnick and Uebersch\"ar, the level spacings of arithmetic toral point scatterers, in the weak coupling limit, are also Poisson distributed. Read More

We show that the spectrum of a discrete two-dimensional periodic Schr\"odinger operator on a square lattice with a sufficiently small potential is an interval, provided the period is odd in at least one dimension. In general, we show that the spectrum may consist of at most two intervals and that a gap may only open at energy zero. This sharpens several results of Kr\"uger and may be thought of as a discrete version of the Bethe--Sommerfeld conjecture. Read More

The Sturm-Liouville operator on a star-shaped graph is considered. We assume that the potential is known a priori on all the edges except one, and study the partial inverse problem, which consists in recovering the potential on the remaining edge from the part of the spectrum. A constructive method is developed for the solution of this problem, based on the Riesz-basicity of some sequence of vector functions. Read More

The first terms of the small volume asymptotic expansion for the splitting of Neumann boundary condition Laplacian eigenvalues due to a grounded inclusion of size {\epsilon} are derived. An explicit formula to compute the first term from the eigenvalues and eigenfunctions of the unperturbed domain, the inclusion size and position is given. As a consequence, when an eigenvalue of double multiplicity splits in two distinct eigenvalues, one decays like O(1/log({\epsilon})), the other like O({\epsilon}^2). Read More

For all convex co-compact hyperbolic surfaces, we prove the existence of an essential spectral gap, that is a strip beyond the unitarity axis in which the Selberg zeta function has only finitely many zeroes. We make no assumption on the dimension $\delta$ of the limit set, in particular we do not require the pressure condition $\delta\leq {1\over 2}$. This is the first result of this kind for quantum Hamiltonians. Read More

M. F. Atiyah proved that the index of a transversally elliptic operator relative to a free action can be computed by using indices of elliptic operators on the orbit manifold. Read More

This is a chapter of a forthcoming Lecture Notes in Mathematics "Modern Approaches to Discrete Curvature" edited by L. Najman and P. Romon. Read More

Let $f: X\to {\Bbb C}P^1$ be a meromorphic function of degree $N$ with simple poles and simple critical points on a compact Riemann surface $X$ of genus $g$ and let $\mathsf m$ be the standard round metric of curvature $1$ on the Riemann sphere ${\Bbb C}P^1$. Then the pullback $f^*\mathsf m$ of $\mathsf m$ under $f$ is a metric of curvature $1$ with conical singularities of conical angles $4\pi$ at the critical points of $f$. We study the $\zeta$-regularized determinant of the Laplace operator on $X$ corresponding to the metric $f^*\mathsf m$ as a functional on the moduli space of the pairs $(X, f)$ (i. Read More

Let $\Omega \subset \mathbb{R}^n$ be a convex domain with inradius $\rho$ and diameter $D$ and let $u:\Omega \rightarrow \mathbb{R}$ be the first eigenfunction of the Laplacian $-\Delta$ on $\Omega$ with Dirichlet boundary conditions on $\partial \Omega$. A conjecture of van den Berg states that $$ \| u \|_{L^{\infty}} \lesssim \frac{1}{\rho^{n/2}} \left( \frac{\rho}{D} \right)^{1/6} \|u\|_{L^2}.$$ We prove the conjecture for $n=2$. Read More

We investigate anomalous localized resonance on the circular coated structure and cloaking related to it in the context of elasto-static systems. The structure consists of the circular core with constant Lam\'e parameters and the circular shell of negative Lam\'e parameters proportional to those of the core. We show that the eigenvalues of the Neumann-Poincar\'e operator corresponding to the structure converges to certain non-zero numbers determined by Lam\'e parameters and derive precise asymptotics of the convergence. Read More

Under the weak interaction regime, we prove the one and the two volumes Wegner type bounds for one dimensional multi-particle models on the lattice and for very singular probability distribution functions such as the Bernoulli measures. The results imply the Anderson loclalization in both the spectral exponential and the strong dynamical localization for the one dimensional multi-particle Bernoulli-Anderson model with weak interaction. Read More

We consider the class of compact n-dimensional Riemannian manifolds with cylindrical boundary, Ricci curvature bounded below by a given constant and injectivity radius bounded below by a positive constant, away from the boundary. For a manifold M of this class, we introduce a notion of discretization, leading to a graph with boundary which is roughly isometric to M, with constants depending only on the dimension and bounds on curvature and injectivity radius. In this context, we prove a uniform spectral comparison inequality between the Steklov eigenvalues of the manifold M and those of its discretization. Read More

In this note, we will consider semiclassical scattering for compactly supported non-trapping perturbations of the Laplacian on $\mathbb{R}^d$. We will define a family of Gaussian states on $\mathbb{S}^{d-1}$, parametrized by points in $T^*\mathbb{S}^{d-1}$, and show that the action of the scattering matrix on a Gaussian state of parameter $\rho\in T^*\mathbb{S}^{d-1}$ is still a Gaussian state, with parameter $\kappa(\rho)$, where $\kappa$ is the (classical) scattering map. This is one way of saying that \emph{the scattering matrix quantizes the scattering map}, complementary to a previous result of Alexandrova in terms of Fourier Integral Operators. Read More

Motivated by Gilkey's local formulae for asymptotic expansion of heat kernels in spectral geometry, we propose a definition of Ricci curvature in noncommutative settings. The Ricci operator of an oriented closed Riemannian manifold can be realized as a spectral functional, namely the functional defined by the zeta function of the full Laplacian of the de Rham complex, localized by smooth endomorphisms of the cotangent bundle and their trace. We use this formulation to introduce the Ricci functional in a noncommutative setting and in particular for curved noncommutative tori. Read More

We give a reformulation of Salem's conjecture about the absence of Salem numbers near one in terms of a uniform spectral gap for certain arithmetic hyperbolic surfaces. Read More

We improve the classical discrete Hardy inequality \begin{equation*}\label{1} \sum _{{n=1}}^{\infty }a_{n}^{2}\geq \left({\frac {1}{2}}\right)^{2} \sum _{{n=1}}^{\infty }\left({\frac {a_{1}+a_{2}+\cdots +a_{n}}{n}}\right)^{2}, \end{equation*} where $\{a_n\}_{n=1}^\infty$ is any sequence of non-negative real numbers. Read More

We study the ground state of a large bosonic system trapped in a symmetric double-well potential, letting the distance between the two wells increase to infinity with the number of particles. In this context, one should expect an interaction-driven transition between a delocalized state (particles are independent and all live in both wells) and a localized state (particles are correlated, half of them live in each well). We start from the full many-body Schr{\"o}dinger Hamiltonian in a large-filling situation where the on-site interaction and kinetic energies are comparable. Read More

We study direct and inverse scattering problem for systems of interacting particles, having web-like structure. Such systems consist of a finite number of semi-infinite chains attached to the central part formed by a finite number of particles. We assume that the semi-infinite channels are homogeneous at infinity, but the limit values of the coefficients may vary from one chain to another. Read More

A deterministic sequence of real numbers in the unit interval is called \emph{equidistributed} if its empirical distribution converges to the uniform distribution. Furthermore, the limit distribution of the pair correlation statistics of a sequence is called Poissonian if the number of pairs $x_k,x_l \in (x_n)_{1 \leq n \leq N}$ which are within distance $s/N$ of each other is asymptotically $\sim 2sN$. A randomly generated sequence has both of these properties, almost surely. Read More

We study the spectrum of an operator matrix which is a generalization of the energy operator of a lattice spin-boson model with at most two bosons. An analytic description of the essential spectrum is established. Further, a criterion for the finiteness of the number of eigenvalues below the bottom of the essential spectrum is derived. Read More

We consider discrete Schr\"odinger operators with periodic potentials on periodic graphs perturbed by guided non-positive potentials, which are periodic in some directions and finitely supported in other ones. The spectrum of the unperturbed operator is a union of a finite number of non-degenerate bands and eigenvalues of infinite multiplicity. We show that the spectrum of the perturbed operator consists of the unperturbed one plus the additional guided spectrum, which is a union of a finite number of bands. Read More

We study the spectrum of a system of second order differential operator perturbed by a non-selfadjoint matrix valued potential. We prove that eigenvalues of the perturbed operator are located near the edges of the spectrum of the unperturbed operator. Read More

The purpose of this paper is to study the essential spectrum of non-self-adjoint singular matrix differential operators in the Hilbert space $L^2(\mathbb{R})\oplus L^2(\mathbb{R})$ induced by matrix differential expressions of the form \begin{align}\label{abstract:mdo} \left(\begin{array}{cc} \tau_{11}(\,\cdot\,,D) & \tau_{12}(\,\cdot\,,D)\\[3.5ex] \tau_{21}(\,\cdot\,,D) & \tau_{22}(\,\cdot\,,D) \end{array}\right), \end{align} where $\tau_{11}$, $\tau_{12}$, $\tau_{21}$, $\tau_{22}$ are respectively $m$-th, $n$-th, $k$-th and 0 order ordinary differential expressions with $m=n+k$ being even. Under suitable assumptions on their coefficients, we establish an analytic description of the essential spectrum. Read More

The goal of this paper is to combine ideas from the theory of mixed spectral problems for differential operators with new results in the area of the Uncertainty Principle in Harmonic Analysis (UP). Using recent solutions of Gap and Type Problems of UP we prove a version of Borg's two-spectra theorem for Schr\"odinger operators, allowing uncertainty in the placement of the eigenvalues. We give a formula for the exact 'size of uncertainty', calculated from the lengths of the intervals where the eigenvalues may occur. Read More

The known upper bounds for the multiplicities of the Laplace-Beltrami operator eigenvalues on the real projective plane are improved for the eigenvalues with even indexes. Upper bounds for Dirichlet, Neumann and Steklov eigenvalues on the real projective plane with holes are also provided. Read More

A binary tensor consists of $2^n$ entries arranged into hypercube format $2 \times 2 \times \cdots \times 2$. There are $n$ ways to flatten such a tensor into a matrix of size $2 \times 2^{n-1}$. For each flattening, $M$, we take the determinant of its Gram matrix, ${\rm det }(M M^T)$. Read More

This note is devoted to prove that the de Gennes function has a holomorphic extension on a strip containing the real axis. Read More

For a given subcritical discrete Schr\"odinger operator $H$ on a weighted infinite graph $X$, we construct a Hardy-weight $w$ which is optimal in the following sense. The operator $H - \lambda w$ is subcritical in $X$ for all $\lambda < 1$, null-critical in $X$ for $\lambda = 1$, and supercritical near any neighborhood of infinity in $X$ for any $\lambda > 1$. As a side product, we present a criticality theory for Schr\"odinger operators on general weighted graphs. Read More

Let $\sigma_n$ denote the largest mode-$n$ multilinear singular value of an $I_1\times\dots \times I_N$ tensor $\mathcal T$. We prove that $$ \sigma_1^2+\dots+\sigma_{n-1}^2+\sigma_{n+1}^2+\dots+\sigma_{N}^2\leq (N-2)\|\mathcal T\|^2 + \sigma_n^2,\quad n=1,\dots,N. $$ We also show that at least for third-order cubic tensors the inverse problem always has a solution. Read More

In the context of an infinite locally finite weighted graph, we give a necessary and sufficientcondition for semi-Fredholmness of the Gauss-Bonnet operator. This result is a discrete version of thetheorem of Gilles Carron in the continuous case [5]. In addition, using a criterion of Anghel [2], we givea sufficient condition to have an operator of Gauss-Bonnet with closed range. Read More

Given a Riemannian spin^c manifold whose boundary is endowed with a Riemannian flow, we show that any solution of the basic Dirac equation satisfies an integral inequality depending on geometric quantities, such as the mean curvature and the O'Neill tensor. We then characterize the equality case of the inequality when the ambient manifold is a domain of a K\"ahler-Einstein manifold or a Riemannian product of a K\"ahler-Einstein manifold with R (or with the circle S^1). Read More

We extend the classical Ambarzumyan's theorem to the quasi-periodic boundary value problems by using only a part knowledge of one spectrum. We also weaken slightly the Yurko's conditions on the first eigenvalue. Read More

We prove an averaging formula for the derivative of the absolutely continuous part of the density of states measure for an ergodic family of CMV matrices. As a consequence, we show that the spectral type of such a family is almost surely purely absolutely continuous if and only if the density of states is absolutely continuous and the Lyapunov exponent vanishes almost everywhere with respect to the same. Both of these results are CMV operator analogues of theorems obtained by Kotani for Schr\"odinger operators. Read More

We prove that the Dirichlet eigenvalues and Neumann boundary data of the corresponding eigenfunctions of the operator $-\Delta + q$, determine the potential $q$, when $q \in L^{n/2}(\Omega,\mathbb{R})$ and $n \geq 3$. We also consider the case of incomplete spectral data, in the sense that the above spectral data is unknown for some finite number of eigenvalues. In this case we prove that the potential $q$ is uniquely determined for $q \in L^p(\Omega,\mathbb{R})$ with $p=n/2$, for $n\geq4$ and $p>n/2$, for $n=3$. Read More

In this note, we exhibit a three dimensional structure that permits to guide waves. This structure is obtained by a geometrical perturbation of a 3D periodic domain that consists of a three dimensional grating of equi-spaced thin pipes oriented along three orthogonal directions. Homogeneous Neumann boundary conditions are imposed on the boundary of the domain. Read More

Conformal blocks for correlation functions of tensor operators play an increasingly important role for the conformal bootstrap programme. We develop a universal approach to such spinning blocks through the harmonic analysis of certain bundles over a coset of the conformal group. The resulting Casimir equations are given by a matrix version of the Calogero-Sutherland Hamiltonian that describes the scattering of interacting spinning particles in a 1-dimensional external potential. Read More

A first order trace formula is obtained for a regular differential operator perturbed by a finite signed measure multiplication operator. Read More

In this paper we prove that the Dirac operator with an electrostatic $\delta$-shell interaction of critical strength $\eta = \pm 2$ supported on a $C^2$-smooth compact surface $\Sigma$ is self-adjoint in $L^2(\mathbb{R}^3;\mathbb{C}^4)$ and we describe its domain explicitly in terms of traces and jump conditions in $H^{-1/2}(\Sigma; \mathbb{C}^4)$. While the non-critical interaction strengths $\eta \not= \pm 2$ have received a lot of attention in the recent past, the critical case $\eta = \pm 2$ remained open. Our approach is based on abstract techniques in extension theory of symmetric operators, in particular, boundary triples and their Weyl functions. Read More

In this paper, we will consider generalised eigenfunctions of the Laplacian on some surfaces of infinite area. We will be interested in lower bounds on the number of nodal domains of such eigenfunctions which are included in a given bounded set. We will first of all consider finite sums of plane waves, and give a criterion on the amplitudes and directions of propagation of these plane waves which guarantees an optimal lower bound, of the same order as Courant's upper bound. Read More

We study the spectrum of two kinds of operators involving a conical geometry: the Dirichlet Laplacian in conical layers and Schr\"odinger operators with attractive $\delta$-interactions supported by infinite cones. Under the assumption that the cones have smooth cross-sections, we prove that such operators have infinitely many eigenvalues accumulating below the threshold of the essential spectrum and we express the accumulation rate in terms of the eigenvalues of an auxiliary one-dimensional operator with a curvature-induced potential. Read More

Recently, several authors have proved inequalities on the spectral radius $\rho$, operator norm $\|\cdot\|$ and numerical radius of Hadamard products and ordinary products of non-negative matrices that define operators on sequence spaces, or of Hadamard geometric mean and ordinary products of positive kernel operators on Banach function spaces. In the present article we generalize and refine several of these results. In particular, we show that for a Hadamard geometric mean $A ^{\left( \frac{1}{2} \right)} \circ B ^{\left( \frac{1}{2} \right)}$ of positive kernel operators $A$ and $B$ on a Banach function space $L$, we have $$ \rho \left(A^{(\frac{1}{2})} \circ B^{(\frac{1}{2})} \right) \le \rho \left((AB)^{(\frac{1}{2})} \circ (BA)^{(\frac{1}{2})}\right)^{\frac{1}{2}} \le \rho (AB)^{\frac{1}{2}}. Read More