We study a family of differential operators L[alpha] in two variables, depending on the coupling parameter [alpha][greater-or-equal, slanted]0 that appears only in the boundary conditions. Our main concern is the spectral properties of L[alpha], which turn out to be quite different for [alpha]<1 and for [alpha]>1. In particular, L[alpha] has a unique self-adjoint realization for [alpha]<1 and many such realizations for [alpha]>1. In the more difficult case [alpha]>1 an analysis of non-elliptic pseudodifferential operators in dimension one is involved.
Journal of Computational and Applied Mathematics
- Pub Date:
- November 2007
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