Leonid Friedlander

Analytic and Reidemeister torsion for representations in finite type Hilbert modules

For a closed Riemannian manifold (M, g) we extend the definition of analytic and Reidemeister torsion associated to a unitary representation of π1 (M) on a finite dimensional vector space to a representation on aA-Hilbert moduleW of finite type whereA is a finite von Neumann algebra. If (M,W) is of determinant class we prove, generalizing the Cheeger-Müller theorem, that the analytic and Reidemeister torsion are equal. In particular, this proves the conjecture that for closed Riemannian manifolds with positive Novikov-Shubin invariants, theL2-analytic andL2-Reidemeister torsions are equal.

Meyer-vietoris type formula for determinants of elliptic differential operators☆

Abstract For a closed codimension one submanifold Γ of a compact manifold M, let MΓ be the manifold with boundary obtained by cutting M along Γ. Let A be an elliptic differential operator on M and B and C be two complementary boundary conditions on Γ. If (A, B) is an elliptic boundary valued problem on MΓ, then one defines an elliptic pseudodifferential operator R of Neumann type on Γ and prove the following factorization formula for the ζ-regularized determinants: Det A Det (A, B) = K Det R , with K a local quantity depending only on the jets of the symbols of A, B and C along Γ. The particular case when M has dimension 2, A is the Laplace-Beltrami operator, and B resp. C is the Dirichlet resp. Neumann boundary condition is considered.

On the spectrum of narrow periodic waveguides

This is a continuation of [1] and [2]. We consider the spectrum of the Dirichlet Laplacian on the domain {(x, y) : 0 < y < εh(x)}, where h(x) is a positive periodic function. The main assumption is that h(x) has one point of global maximum on the period interval. We study the location of bands and prove that the band lengths decay exponentially as ε → 0.

On the determination of elliptic differential and finite difference operators in vector bundles overS1

AbstractFor an elliptic differential operatorA overS1,

Genericity of simple eigenvalues for a metric graph

A = \sum\limits_{k = 0}^n {A_k (x)D^k }

On the spectrum of the periodic problem for the Schrödinger operator

, withAk(x) in END(ℂr) and θ as a principal angle, the ζ-regularized determinant DetθA is computed in terms of the monodromy mapPA, associated toA and some invariant expressed in terms ofAn andAn−1. A similar formula holds for finite difference operators. A number of applications and implications are given. In particular we present a formula for the signature ofA whenA is self adjoint and show that the determinant ofA is the limit of a sequence of computable expressions involving determinants of difference approximation ofA.

On the second eigenvalue of the dirichlet laplacian

ABSTRACT Let Ω0 be a domain in the cube , and let be a function that equals 1 inside Ω0, equals τ in , and that is extended periodically to R n . It is known that, in the limit , the spectrum of the operator exhibits the band-gap structure. We establish the asymptotic behavior of the density of states function in the bands.

A summation method for the Rayleigh–Schrödinger series for the anharmonic oscillator

We prove that, for a metric graph different from a polygon, the spectrum of the Laplacian is generically simple.