2007 | OriginalPaper | Buchkapitel
Spectral Analysis of Differential Operators with Indefinite Weights and a Local Point Interaction
verfasst von : Ilia Karabash, Aleksey Kostenko
Erschienen in: Operator Theory in Inner Product Spaces
Verlag: Birkhäuser Basel
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We consider quasi-self-adjoint extensions of the symmetric operator
$$ A = - (\operatorname{sgn} x)\frac{{d^2 }} {{dx^2 }},dom(A) = \{ f \in W_2^2 (\mathbb{R}):f(0) = f'(0) = 0\} $$
, in the Hilbert space
L
2
(ℝ). The main result is a criterion of similarity to a normal operator for operators of this class. The spectra and resolvents of these extensions are described. As an application we describe the main spectral properties of the operators
$$ (\operatorname{sgn} x)\left( { - \tfrac{{d^2 }} {{dx^2 }} + c\delta } \right)and (\operatorname{sgn} x)\left( { - \tfrac{{d^2 }} {{dx^2 }} + c\delta '} \right) $$
.