Spectral Theory

This is a survey of accumulated spectral analysis observations spanning more than a century, referring to the double layer potential integral equation, also known as Neumann-Poincar\'e operator. The very notion of spectral analysis has…

We give a description of the growth rates of $L^2$-normalized Laplace eigenfunctions on the unit disk with Dirichlet and Neumann boundary conditions. In particular, we show that the growth rates of both Dirichlet and Neumann eigenfunctions…

We consider the cases where there is equality in Courant's nodal domain theorem for the Laplacian with a Robin boundary condition on the square. In our previous two papers, we treated the cases where the Robin parameter $h>0$ is large,…

We study perturbations of the discrete Laplacian associated to discrete analogs of cusps and funnels. We perturb the metric and the potential in a long-range way. We establish a propagation estimate and a Limiting Absorption Principle away…

We construct a one-parameter family of Laplacians on the Sierpinski Gasket that are symmetric and self-similar for the 9-map iterated function system obtained by iterating the standard 3-map iterated function system. Our main result is the…

We introduce the signless 1-Laplacians and the dual Cheeger constants on simplicial complexes. The connection of its spectrum to the combinatorial properties like independence number, chromatic number and dual Cheeger constant is…

We consider a family of Schr\"odinger operators supported by a periodic chain of loops connected either tightly or loosely through connecting links of the length $\ell>0$ with the vertex coupling which is non-invariant with respect to the…

The main focus of this paper is the following matrix Cauchy problem for the Dirac system on the interval $[0,1]:$ \[ D'(x)+\left[\begin{array}{cc} 0 & \sigma_1(x)\\ \sigma_2(x) & 0 \end{array} \right] D(x)=i\mu\left[\begin{array}{cc} 1 &…

The center of mass of a finite measure with respect to a radially increasing weight is shown to exist, be unique, and depend continuously on the measure.

We say that a list $\Lambda =\{ \lambda _{1},\ldots ,\lambda _{n}\}$ of complex numbers is realizable, if it is the spectrum of a nonnegative matrix $A$ (the realizing matrix). We say that $\Lambda $ is universally realizable if it is…

We present new methods to construct a Ruelle-Pollicott spectrum for the geodesic flow on manifolds with strictly negative curvature and a finite number of hyperbolic cusps.

Extremal problems concerning the number of complete subgraphs have a long story in extremal graph theory. Let $k_s(G)$ be the number of $s$-cliques in a graph $G$ and $m={{r_m}\choose s}+t_m$, where $0\le t_m\leq r_m$. Edr\H{o}s showed that…

We consider a negative Laplacian in multi-dimensional Euclidean space (or a multi-dimensional layer) with a weak disorder random perturbation. The perturbation consists of a sum of lattice translates of a delta interaction supported on a…

Given a Schr\"odinger operator with a real-valued potential on a bounded, convex domain or a bounded interval we prove inequalities between the eigenvalues corresponding to Neumann and Dirichlet boundary conditions, respectively. The…

This review paper summarizes the contents of the talk given by the author at the 8th International Congress of Chinese Mathematicians. Using examples of Schr\"odinger operators on metric graphs, it is shown that a nontrivial topology of the…

Here we prove the following result. Let $A = \{a_{ij}\}_{i,j\in \mathbb{N}}$ be a bounded operator. Then there exists a signing of $A$ such that $$||A\circ S||_2 < 2||A||_{l_\infty(l_2)},$$ where $A\circ S$ denotes the matrix generated by…

In this paper we provide sharp results for the Dixmier traceability of discrete pseudo-differential operators on $\ell^2(\mathbb{Z}^n)$. In this setting, we introduce a suitable notion of a class of classical symbols which provide a class…

We investigate the low-energy behavior of the resolvent of Schrodinger operators with finitely many point interactions in three dimensions. We also discuss the occurrence and the multiplicity of zero energy obstructions.

We extend the classical boundary values \begin{align*} & g(a) = - W(u_{a}(\lambda_0,.), g)(a) = \lim_{x \downarrow a} \frac{g(x)}{\hat u_{a}(\lambda_0,x)}, \\ &g^{[1]}(a) = (p g')(a) = W(\hat u_{a}(\lambda_0,.), g)(a) = \lim_{x \downarrow…

We derive closed formula for the heat kernel $K_\mathbb{H}$ associated to Maass-Laplacian operator $D_k$ for any real $k$ and prove that heat kernel $K_\mathbb{H}$ is strictly monotone decreasing function. We also derive some important…