Vol. 4, No. 3-4, 2016

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A remark on eigenvalue perturbation theory at vanishing isolation distance

Fiorella Barone and Sandro Graffi

Vol. 4 (2016), No. 3-4, 297–309

Let T be a self-adjoint operator in a separable Hilbert space X, admitting compact resolvent and simple eigenvalues with possibly vanishing isolation distance, and let V be symmetric and bounded. Consider the self-adjoint operator family T(g) : g in X defined by T + gV on D(T). A simple criterion is formulated ensuring, for any eigenvalue of T(g), the existence to all orders of its perturbation expansion and its asymptotic nature near g = 0, with estimates independent of the eigenvalue index. An application to a class of Schrödinger operators is described.

isolation distance, eigenvalue perturbation theory
Mathematical Subject Classification 2010
Primary: 81Q05, 81Q10, 81Q15
Received: 26 October 2015
Revised: 28 October 2015
Accepted: 9 May 2016
Published: 17 December 2016

Communicated by Raffaele Esposito
Fiorella Barone
Dipartimento di Matematica
Università di Bari
70122 Bari
Sandro Graffi
Dipartimento di Matematica
Università di Bologna
40127 Bologna