# Laplacians with point interactions -- expected and unexpected spectral properties

## TL;DR

This paper characterizes all boundary conditions for the one-dimensional Laplace operator with point interactions on the real line, exploring their spectral properties and the invariance of associated semigroups using Green's functions.

## Contribution

It provides a complete classification of boundary conditions that generate $C_0$-semigroups for Laplacians with point interactions, highlighting the role of the Cayley transform and Green's functions.

## Key findings

- Complete characterization of boundary conditions for semigroup generation
- Explicit representation of Green's functions for spectral analysis
- Analysis of invariance properties of semigroups

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