Perturbation determinants for singular perturbations
Abstract
Let A be a densely defined symmetric operator and let { Ã', Ã} be an ordered pair of proper extensions of A such that their resolvent difference is of trace class. We study the perturbation determinant ΔÃ'/Ã(·) of the singular pair { Ã', Ã} by using the boundary triplet approach. We show that, under additional mild assumptions on { Ã', Ã, the perturbation determinant ΔÃ'/Ã(·) is the ratio of two ordinary determinants involving the Weyl function and boundary operators. In particular, if the deficiency indices of A are finite, then we obtain ΔÃ'/Ã( z) = det ( B' - M( z))/det ( B - M ( z)), z ∈ ρ( Ã), where M(·) stands for the Weyl function and B' and B for the boundary operators corresponding to Ã' and à with respect to a chosen boundary triplet Π. The results are applied to ordinary differential operators and to second-order elliptic operators.
- Publication:
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Russian Journal of Mathematical Physics
- Pub Date:
- March 2014
- DOI:
- 10.1134/S1061920814010051
- Bibcode:
- 2014RJMP...21...55M