Removing cut-offs from singular perturbations: An abstract result
Let \(H_ 0\geq 0\) and V be selfadjoint operators in a Hilbert space such that for some dense subspace D with \(D\subset D(H_ 0)\) and for \(0<a<1\) and \(b>0\), \[ \| Vf\| \leq a\| H_ 0f\| +b\| f\|,\quad f\in D, \] and \((H_ 0+V)\upharpoonright_ D\) has deficiency indices (m,m), \(m<\infty\). Then the author proves that if \(\{V_ n\}\) is a sequence of selfadjoint operators converging to V in some sense such that \(H_ 0+V\) are selfadjoint and bounded from below, then \(\{H_ 0+V_ n\}\) converges to the Friedrichs extension of \((H_ 0+V)\upharpoonright_{D(H_ 0)\cap D(V)}\) in the norm resolvent sense, even if the spectra of \(H_ 0+V_ n\) are not uniformly bounded from below. The proof relies on his result in [J. Oper. Theory 10, 209-218 (1983)].
selfadjoint operators in a Hilbert space
scientific article; zbMATH DE number 3897633