Spectral analysis and stabilization of the dissipative Schrödinger operator on the tadpole graph

Abstract

We consider the damped Schrödinger semigroup $e^{-it\frac{d^2}{d{x}^2}}$ on the tadpole graph $\mathcal{R}$. We first give a careful spectral analysis and an appropriate decomposition of the kernel of the resolvent. As a consequence and by showing that the generalized eigenfunctions form a Riesz basis of some subspace of $L^2(\mathcal{R})$, we prove that the corresponding energy decay exponentially.

arXiv admin note: text overlap with arXiv:1512.05269