Laplacians with point interactions -- expected and unexpected spectral properties

We study the one-dimensional Laplace operator with point interactions on the real line identified with two copies of the half-line $[0,\infty)$. All possible boundary conditions that define generators of $C_0$-semigroups on $L^2\big([0,\infty)\big)\oplus L^2\big([0,\infty)\big)$ are characterized. Here, the Cayley transform of the boundary conditions plays an important role and using an explicit representation of the Green's functions, it allows us to study invariance properties of semigroups

Inverse operator of the generator of a C-0-semigroup

Let $A$ be the generator of a uniformly bounded $C_0$-semigroup in a Banach space $X$ such that $A$ has a trivial kernel and a dense range. The question whether $A^{-1}$ is a generator of a $C_0$-semigroup is considered. It is shown that the answer is negative in general for $X = \ell_p$, $p \in (1, 2) \cap (2,\infty)$. In the case when $X$ is a Hilbert space it is proved that there exist $C_0$-semigroups ($e^{tA})$, $t > 0$, of arbitrarily slow growth at infinity such that the densely defined operator $A^{-1}$ is not the generator of a $C_0$-semigroup