Illinois Journal of Mathematics

Spectrally unstable domains

Gerardo A. Mendoza

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Let $H$ be a separable Hilbert space, $A_{c}:\mathcal{D}_{c}\subset H\to H$ a densely defined unbounded operator, bounded from below, let $\mathcal{D}_{\min}$ be the domain of the closure of $A_{c}$ and $\mathcal{D}_{\max}$ that of the adjoint. Assume that $\mathcal{D}_{\max}$ with the graph norm is compactly contained in $H$ and that $\mathcal{D}_{\min}$ has finite positive codimension in $\mathcal{D}_{\max}$. Then the set of domains of selfadjoint extensions of $A_{c}$ has the structure of a finite-dimensional manifold $\mathfrak{SA}$ and the spectrum of each of its selfadjoint extensions is bounded from below. If $\zeta$ is strictly below the spectrum of $A$ with a given domain $\mathcal{D}_{0}\in\mathfrak{SA}$, then $\zeta$ is not in the spectrum of $A$ with domain $\mathcal{D}\in\mathfrak{SA}$ near $\mathcal{D}_{0}$. But $\mathfrak{SA}$ contains elements $\mathcal{D}_{0}$ with the property that for every neighborhood $U$ of $\mathcal{D}_{0}$ and every $\zeta\in\mathbb{R}$ there is $\mathcal{D}\in U$ such that $\operatorname{spec}(A_{\mathcal{D} })\cap(-\infty,\zeta)\neq\emptyset$. We characterize these “spectrally unstable” domains as being those satisfying a nontrivial relation with the domain of the Friedrichs extension of $A_{c}$.

Article information

Illinois J. Math., Volume 59, Number 4 (2015), 979-997.

Received: 8 April 2016
First available in Project Euclid: 27 February 2017

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Zentralblatt MATH identifier

Primary: 47B25: Symmetric and selfadjoint operators (unbounded) 47A10: Spectrum, resolvent
Secondary: 47F05: Partial differential operators [See also 35Pxx, 58Jxx] (should also be assigned at least one other classification number in section 47) 58J05: Elliptic equations on manifolds, general theory [See also 35-XX] 35P05: General topics in linear spectral theory


Mendoza, Gerardo A. Spectrally unstable domains. Illinois J. Math. 59 (2015), no. 4, 979--997. doi:10.1215/ijm/1488186017.

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