Related papers: Spectral asymptotics for the semiclassical Dirichl…

For a family of second-order elliptic operators with rapidly oscillating periodic coefficients, we study the asymptotic behavior of the Green and Neumann functions, using Dirichlet and Neumann correctors. As a result we obtain asymptotic…

In these notes we study the Dirichlet problem for critical points of a convex functional of the form \[ F(u)=\int_{\Omega}\phi\left( \left\vert \nabla u\right\vert \right) , \] where $\Omega$ is a bounded domain of a complete Riemannian…

We propose simple conditions equivalent to the discreteness of the spectrum of the Laplace-Beltrami operator on a class of Riemannian manifolds close to warped products. For this class of manifolds we establish a relationship between…

On a compact manifold $M^{n}$ ($n\geq 3$) with boundary, we study the asymptotic behavior as $\epsilon$ tends to zero of solutions $u_{\epsilon}: M \to \mathbb{C}$ to the equation $\Delta u_{\epsilon} + \epsilon^{-2}(1 -…

Let $\Omega \subset {\bf R}^d$ be an open bounded set with Lipschitz boundary $\Gamma$. Let $D_V$ be the Dirichlet-to-Neumann operator with respect to a purely second-order symmetric divergence form operator with real Lipschitz continuous…

We obtain the strong asymptotics of polynomials $p_n(\lambda)$, $\lambda\in\mathbb{C}$, orthogonal with respect to measures in the complex plane of the form $$ e^{-N(|\lambda|^{2s}-t\lambda^s-\overline{t\lambda}^s)}dA(\lambda), $$ where $s$…

We consider operators of the form $\mathbf{T}=\mathbf{A^*}(V\mu)\mathbf{A}$ in $\mathbb{R}^\mathbf{N}$, where $\mathbf{A}$ is a pseudodifferential operator of order $-l$, $\mu$ is a compactly supported singular measure, order $s>0$…

We consider a general form of L-function L(s) defined by an Euler product and satisfies several analytic assumptions. We show several asymptotic formulas for L(1) and log L(1). In particular those asymptotic formulas are valid for Dirichlet…

For $2a$-order strongly elliptic operators $P$ generalizing $(-\Delta )^a$, $0<a<1$, the treatment of the homogeneous Dirichlet problem on a bounded open set $\Omega \subset R^n$ by pseudodifferential methods, has been extended in a recent…

Sturm-Liouville problem with generalized derivative of self-similar Cantor type function as a weight is considered. Under Neumann and mixed boundary conditions the oscillating properties of the eigenfunctions are studied. The spectral…

We address the problems in applying cycle expansions to bound chaotic systems, caused by e.g. intermittency and incompleteness of the symbolic dynamics. We discuss zeta functions associated with weighted evolution operators and in…

We continue our study of a magnetic Schr\"odinger operator on a two-dimensional compact Riemannian manifold in the case when the minimal value of the module of the magnetic field is strictly positive. We analyze the case when the magnetic…

Let $\Omega$ be an unbounded two dimensional strip on a ruled surface in $\mathbb{R}^d$, $d\geq2$. Consider the Laplacian operator in $\Omega$ with Dirichlet and Neumann boundary conditions on opposite sides of $\Omega$. We prove some…

In this paper we study the asymptotic behavior of solutions of fractional differential equations of the form $ D^{\alpha}_Cu(t)=Au(t)+f(t), u(0)=x, 0<\alpha\le1, ( *) $ where $D^{\alpha}_Cu(t)$ is the derivative of the function $u$ in the…

We consider a nonlocal differential--difference Schr\"odinger operator on a segment with the Neumann conditions and two translations in the free term. The values of the translations are denoted by $\alpha$ and $\beta$ and are treated as…

This is the first of a series of papers dealing with the asymptotic behavior of certain integrals occuring in the description of the spectrum of an invariant elliptic operator on a compact Riemannian manifold carrying the action of a…

We prove the complete asymptotic expansion of the spectral function (the integral kernel of the spectral projection) of a Schrodinger operator $H=-\Delta+b$ acting in $R^d$ when the potential $b$ is real and either smooth periodic, or…

A representation for the Riemann zeta function valid for arbitrary complex $s=\sigma+it$ is $\zeta(s)=\sum_{n=0}^\infty A(n,s)$, where \[A(n,s)=\frac{2^{-n-1}}{1-2^{1-s}} \sum_{k=0}^n \left(\!\begin{array}{c}n\\k\end{array}\!\right)…

The main results of this article provide asymptotics at infinity of the Green's functions near and at the spectral gap edges for "generic" periodic second-order elliptic operators on noncompact Riemannian co-compact coverings with abelian…

In this article we study the semiclassical spectral measures associated with Schr\"odinger operators on $R^n$. In particular we compute the first few coefficients of the asymptotic expansions of these measures and, as an application, give…