On inverse spectral theory for selfadjoint extensions: Mixed types of spectra
Let \(H\) be a symmetric operator in a separable Hilbert space \({\mathcal H}\). Suppose that \(H\) has some gap \(J=(a,b), -\infty\leq a < b < \infty\), i.e \[ (Hf,f) \geq b\|f\|^2, f\in{\mathcal D}(H), \text{ if} \;a=-\infty, \] \[ \|(H-{{a+b}\over 2})f\|\geq {{b-a}\over 2}\|f\|, f\in{\mathcal D}(H), \text{ if} \;\infty < a. \] The authors investigate spectral properties of selfadjoint extensions of \(H\) inside \(J\). Under some weak assumptions about the operator \(H\) which are satisfied. e.g.. provided the deficiency indices of \(H\) are infinite and the operator \((H-\lambda)^{-1} \) is compact for one regular point \(\lambda\) of \(H\), it is proved that for every (auxiliary) selfadjoint operator \(M'\) in the Hilbert space \(\mathcal H\) and every open subset \(J_0\) of the gap \(J\) there exists a selfadjoint extension \(\hat H\) of \(H\) such that inside \(J\) the operator \(\hat H\) has the same absolutely continuous and the same point spectrum as the given operator \(M'\) and the singular continuous spectrum of \(\hat H\) on \(J\) equals the closure of \(J_0\) in \(J\). Moreover a method of construction of such a selfadjoint extension \(\hat H\) is proposed. This method is applied to the construction of new kinds of selfadjoint realizations of the Laplacian on a bounded domain \(\Omega\) in \({\mathbf R}^d, d > 1\) with spectral properties very different from the spectral properties of the selfadjoint realizations known before.
selfadjoint realizations of the Laplacian on a bounded domain
scientific article; zbMATH DE number 1213982
scientific article