Related papers: Unexpected Spectral Asymptotics for Wave Equations…

Under certain assumptions we derive a complete semiclassical asymptotics of the spectral function $e_{h,\varepsilon}(x,x,\lambda)$ for a scalar operator \begin{equation*} A_\varepsilon (x,hD)= A^0(hD) + \varepsilon B(x,hD), \end{equation*}…

We study the spectra of $N\times N$ Toeplitz band matrices perturbed by small complex Gaussian random matrices, in the regime $N\gg 1$. We prove a probabilistic Weyl law, which provides an precise asymptotic formula for the number of…

Let $M$ be a compact boundaryless Riemannian manifold, carrying an effective and isometric action of a compact Lie group $G$, and $P_0$ an invariant elliptic classical pseudodifferential operator on $M$. Using Fourier integral operator…

We study the leading coefficient in the asymptotical formula $ N (R) = \frac{W}{\pi} R + O (1) $, $ R \to \infty $, for the resonance counting function $ N (R)$ of Schr\"odinger Hamiltonians with point interactions. For such Hamiltonians,…

For a Riemannian manifold $(M,g)$ which is isometric to the Euclidean space outside of a compact set, and whose trapped set has Liouville measure zero, we prove Weyl type asymptotics for the scattering phase with remainder depending on the…

We discuss an enhancement of the Brown-Henneaux boundary conditions in three-dimensional AdS General Relativity to encompass Weyl transformations of the boundary metric. The resulting asymptotic symmetry algebra, after a field-dependent…

In this work we study the asymptotic behavior of solutions for a general linear second-order evolution differential equation in time with fractional Laplace operators in $\mathbb{R}^n$. We obtain improved decay estimates with less demand on…

We study the wave equation in the exterior of a bounded domain $K$ with dissipative boundary condition $\partial_{\nu} u - \gamma(x) \partial_t u = 0$ on the boundary $\Gamma$ and $\gamma(x) > 0.$ The solutions are described by a…

We calculate the mean and almost-sure leading order behaviour of the high frequency asymptotics of the eigenvalue counting function associated with the natural Dirichlet form on $\alpha$-stable trees, which lead in turn to short-time heat…

We develop an asymptotic expansion of the spectral measures on a degenerating family of hyperbolic Riemann surfaces of finite volume. As an application of our results, we study the asymptotic behavior of weighted counting functions, which,…

We study stochastic wave equations in the sense of Walsh defined by fractal Laplacians on Cantor-like sets. For this purpose, we give an improved estimate on the uniform norm of eigenfunctions and approximate the wave propagator using the…

We study a variety of problems in the spectral theory of automorphic forms using entirely analytic techniques such as Selberg trace formula, asymptotics of Whittaker functions and behavior of heat kernels. Error terms for Weyl's law and an…

Let $\Omega$ be a bounded domain with $C^2$-smooth boundary in an $n$-dimensional oriented Riemannian manifold. It is well-known that for the bi-harmonic equation $\Delta^2 u=0$ in $\Omega$ with the $0$-Dirichlet boundary condition, there…

Spectral asymptotics of the Sturm-Liouville problem with an arithmetically self-similar singular weight is considered. Previous results by A. A. Vladimirov and I. A. Sheipak, and also by the author, rely on the spectral periodicity…

We investigate the Stark operator restricted to a bounded domain $\Omega\subset\mathbb{R}^2$ with Dirichlet boundary conditions. In the semiclassical limit, a three-term asymptotic expansion for its individual eigenvalues has been…

We consider the three-dimensional sloshing problem on a triangular prism whose angles with the sloshing surface are of the form $\frac{\pi}{2q}$, where $q$ is an integer. We are interested in finding a two-term asymptotic expansion of the…

We prove a weak version of Hardy's uncertainty principle using properties of the prolate spheroidal wave functions (PSWFs). We describe the eigenvalues of the sum of a time limiting operator and a band limiting operator acting on L2(R). A…

We consider the resonances of a quantum graph $\mathcal G$ that consists of a compact part with one or more infinite leads attached to it. We discuss the leading term of the asymptotics of the number of resonances of $\mathcal G$ in a disc…

In this paper, we investigate the variance of the nodal length for band-limited spherical random waves. When the frequency window includes a number of eigenfunctions that grows linearly, the variance of the nodal length is linear with…

We study asymptotical expansion as $\nu\to0$ for integrals over ${ \mathbb{R} }^{2d}=\{(x,y)\}$ of quotients of the form $F(x,y) \cos(\lambda x\cdot y) \big/ \big( (x\cdot y)^2+\nu^2\big)$, where $\lambda\ge 0$ and $F$ decays at infinity…