Spectral averaging for trace compatible operators

Trace Formulae and Inverse Spectral Theory for Schrödinger Operators

We extend the well-known trace formula for Hill's equation to general one-dimensional Schrodinger operators. The new function <J , which we introduce, is used to study absolutely continuous spectrum and inverse problems. In this note we will consider one-dimensional Schrodinger operators d2 (IS) H = -j-1 + V(x) onL2(R;dx)

Trace Formulas and Inverse Spectral Theory for Jacobi Operators

Based on high energy expansions and Herglotz properties of Green and Weyl m-functions we develop a self-contained theory of trace formulas for Jacobi operators. In addition, we consider connections with inverse spectral theory, in particular uniqueness results. As an application we work out a new approach to the inverse spectral problem of a class of reflectionless operators producing explicit ...

Inverse spectral problems for Sturm-Liouville operators with transmission conditions

Abstract: This paper deals with the boundary value problem involving the differential equation                      -y''+q(x)y=lambda y                                 subject to the standard boundary conditions along with the following discontinuity conditions at a point              y(a+0)=a1y(a-0),    y'(a+0)=a2y'(a-0)+a3y(a-0).  We develop the Hochestadt-Lieberman’s result for Sturm-Lio...

On Extreme Averaging Operators

Extension and averaging operators for finite fields

In this paper we study L − L estimates of both extension operators and averaging operators associated with the algebraic variety S = {x ∈ Fdq : Q(x) = 0} where Q(x) is a nondegenerate quadratic form over the finite field Fq. In the case when d ≥ 3 is odd and the surface S contains a (d − 1)/2-dimensional subspace, we obtain the exponent r where the L − L extension estimate is sharp. In particul...