Eigenfunction expansions for selfadjoint integro-differential operators

Robert Charles Carlson

Vol. 81 (1979), No. 2, 327–347

DOI: 10.2140/pjm.1979.81.327

Abstract

Let ℒ be a self-adjoint ordinary differential operator on a Hilbert space L²(ℐ), ℐ an open interval, while 𝒦 will denote a bounded self-adjoint operator on the same space. An eigenfunction expansion associated with H = ℒ + 𝒦 is developed when 𝒦 is an integral operator whose kernel K(x,y) has compact support in ℐ×ℐ. It is assumed that if ℒf = ∑ _k=0ⁿa_k(x)D^kf, where D = (d∕dx), then a_k ∈ C^k(ℐ), k = 0,⋯,n, and a_n(x)≠0 for x ∈ℐ.

Mathematical Subject Classification 2000

Primary: 47G05, 47G05

Secondary: 45J05

Milestones

Received: 21 January 1978

Published: 1 April 1979