Laplacians with point interactions -- expected and unexpected spectral properties

Abstract

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 $L^2([0,\infty ))\oplus L^2([0,\infty ))$ 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.

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Cited by in corpus (4)

• Stability of (eventually) positive semigroups on spaces of continuous functions
• Criteria for eventual domination of operator semigroups and resolvents
• Hidden symmetries in non-self-adjoint graphs
• Eventually positive semigroups: spectral and asymptotic analysis