A Krein-like formula for singular perturbations of self-adjoint operators and applications (Q5940315)

Given a selfadjoint operator \(A: D(A)\subset {\mathcal H}\to {\mathcal H}\) and a continuous linear operator \(\tau:D(A)\to {\mathcal X}\) with \(\text{Ran } \tau '\cap{H}'=\{0\}\), \(\mathcal X\) a Banach space, author explicitly constructs a family \(A^\tau_\theta\) of selfadjoint operators such that any \(A^{\tau}_{\theta}\) coincides with the original \(A\) on the kernel of \(\tau\). Such a family is obtained by giving a Krein-like resolvent formula where the role of the deficiency spaces is played by the dual pair \(({\mathcal X},{\mathcal X}')\); the parameter \(\theta\) belongs to the space of symmetric operators from \({\mathcal X}'\) to \({\mathcal X}\). NEWLINENEWLINENEWLINEConsidering the situation in which \({\mathcal H}=L^2({\mathbb R}^n)\) and \(\tau\) is the trace (restriction) operator along some null subset, author gives unified approach to the various applications including singular perturbations of the Laplacian supported by regular curves, singular perturbations given by \(d\)-sets and \(d\)-measures, singular perturbations of the d'Alemberian supported by time-like straight lines, singular perturbations given by traces on Malgrange space.