Smoothing properties of the heat semigroups associated to Hamiltonians describing point interactions in one and two dimensions (Q5951918)

The authors define certain Hamiltonians describing point interaction on \(L^p\)-spaces in one and two dimensions and describe their spectra. Let \(e^{-tH}\), \(t>0\) be the heat semigroup generated by one such Hamiltonian \(H\). It is shown that for every \(t> 0\) and every \(1\leq p\leq q\leq\infty\), the operator \(e^{-tH}: L^p\to L^q\) is bounded. The proof is based on the explicit form of the heat kernel for the point interaction [see \textit{S. Albeverio}, \textit{Z. Brzeźniak} and \textit{L. Dabrowski}, J. Funct. Anal., 130, No. 1, 220-254 (1995; Zbl 0822.35002)] and also uses the Riesz-Thorin theorem. A related result, based on pseudo-resolvents was given in \textit{S. Albevrio}, \textit{Z. Brzeźniak} and \textit{L. Dabrowski} [Integral Equations Operator Theory 21, No. 2, 127-138 (1995; Zbl 0835.47036)].