Singular perturbation of a time-dependent Cauchy problem in a Hilbert space

Larry Eugene Bobisud and James Calvert

Vol. 54 (1974), No. 1, 45–53

DOI: 10.2140/pjm.1974.54.45

Abstract

Let A be a self-adjoint operator, not necessarily bounded, in the Hilbert space H, with resolution of the identity E_λ. Define h(t,A) = ∫ _−∞^∞h(t,λ)dE_λ. It is shown that as 𝜖 → 0+ the solution of the abstract problem 𝜖U_𝜖′′ + bU_𝜖′ + h(t,A)U_𝜖 = 0,U_𝜖(0) = x₀,U_𝜖′(0) = x₁ tends in the norm of H to the solution of bU₀′ + h(t,A)U₀ = 0,U₀(0) = x₀ for data x₀,x₁ in a dense subset of H.

Mathematical Subject Classification

Primary: 34G05

Milestones

Received: 2 April 1973

Published: 1 September 1974