WIAS Preprint No. 1595, (2011)

Sturm--Liouville boundary value problems with operator potentials and unitary equivalence



Authors

  • Malamud, Mark
  • Neidhardt, Hagen

2010 Mathematics Subject Classification

  • 34G10 47E05 47F05 47A20 47B25

Keywords

  • Sturm-Liouville operators, operator potentials, elliptic partial differential operators, boundary value problems, self-adjoint extensions, unitary equivalence, direct sums of symmetric operators

Abstract

Consider the minimal Sturm-Liouville operator $A = A_rm min$ generated by the differential expression $cA := -fracd^2dt^2 + T$ in the Hilbert space $L^2(R_+,cH)$ where $T = T^*ge 0$ in $cH$. We investigate the absolutely continuous parts of different self-adjoint realizations of $cA$. In particular, we show that Dirichlet and Neumann realizations, $A^D$ and $A^N$, are absolutely continuous and unitary equivalent to each other and to the absolutely continuous part of the Krein realization. Moreover, if $infsigma_ess(T) = infgs(T) ge 0$, then the part $wt A^acE_wt A(gs(A^D))$ of any self-adjoint realization $wt A$ of $cA$ is unitarily equivalent to $A^D$. In addition, we prove that the absolutely continuous part $wt A^ac$ of any realization $wt A$ is unitarily equivalent to $A^D$ provided that the resolvent difference $(wt A - i)^-1- (A^D - i)^-1$ is compact. The abstract results are applied to elliptic differential expression in the half-space.

Appeared in

  • J. Differential Equations, 252 (2012) pp. 5875--5922.

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