arXiv subjects
Publications and source records attributed to Leonid Friedlander.
I discuss a simple toy problem for the Dirichlet Laplacian in a sequence of domains where the contribution of the boundary to the spectral asymptotics is of the same order as the contribution from the interior
We establish the two-term spectral asymptotics for boundary value problems of linear elasticity on a smooth compact Riemannian manifold of arbitrary dimension. We also present some illustrative examples and give a historical overview of the subject. In particular, we correct erroneous results published in [J. Geom. Anal. 31 (2021), 10164--10193].
The paper by G. Liu [arxiv:2109.02561] contains an error. In this note, I give a brief review of the problem and indicate what the error is.
We give a simple proof of the Weyl asymptotic formula for eigenvalues of the Dirichlet Laplacian, the buckling problem, and the Dirichlet bi-Laplacian in Euclidean domains of finite volume, with no assumptions about the boundary.
The article is dedicated to thye memory of a distinguished mathematician Professor Misha Shubin
Let $A$ be an elliptic pseudodifferential operator of positive order on a compact closed manifold, and let $T$ be a pseudodifferential operator of negative order such that $T^m$ is of trace class. We compute $\log\det(A(I+T))-\log\det A-\log\det_m (I+T)$ where first two determinants are zeta function regularized, and the last one is a regularized Fredholm determinant.
For a compact, connected metric graphs with a boundary that consists of $k$ vertices, we prove that an arbitrary symmetric $k\times k$ matrix with real entries can be realized as the Dirichlet-to-Neumann operator for the Laplacian plus a constant.
This paper describes the construction of a canonical compactification of the space of trajectories and of the unstable/stable sets of a generic gradient like vector field on a closed manifold as well as a canonical structure of a smooth manifold with corners of these spaces. As an application we discuss the geometric complex associated with a gradient like vector field and show how differential forms can be integrated on its unstable/stable sets. Integration leads to a morphism between the de Rham complex and the geometric complex.
We consider the Dirichlet Laplacian in a family of narrow unbounded domains. As the width of these domains goes to 0, we study the asymptotic behavior of the eigenvalues that lie below the essential spectrum and the asymptotic behavior of the corresponding eigenfunctions.
We derive a two-terms asymptotics for eigenvalues of the Dirichlet Laplacian in a narrow strip of variable width. The asymptotics is taken with respect to a small paprameter that characterizes the width of the strip.
We show that the Szego regularized determinant of a zeroth order operator differs from the zeta regularized determinant of the same operator by a local term. In addition, we compute multiplicative anomalies for the zeta regularized determinant.
This paper achieves, among other things, the following: 1)It frees the main result of [BFKM] from the hypothesis of determinant class and extends this result from unitary to arbitrary representations. 2)It extends (and at the same times provides a new proof of) the main result of Bismut and Zhang [BZ] from finite dimensional representations of $Γ$ to representations on an ${\cal A}-$Hilbert module of finite type (${\cal A}$ a finite von Neumann algebra). The result of [BZ] corresponds to ${\cal A}=\bbc.$ 3)It provides interesting real valued functions on the space of representations of the fundamental group $Γ$ of a closed manifold M. These functions might be a useful source of topological and geometric invariants of M. These objectives are achieved with the help of the relative torsion $\cal R $, first introduced by Carey, Mathai and Mishchenko [CMM] in special cases. The main result of this paper calculates explicitly this relative torsion (cf Theorem 0.1).
The conjecture about relation between infinite-dimensional Grassmannian and string theory is based on the fact that moduli spaces of algebraic curves are embedded into Grassmannian via Krichever construction. We describe a multidimensional analog of Krichever construction, that can be used in the attempt to relate Grassmannian to membranes.