Stabilization of coupled Schrödinger and heat equations with boundary coupling

We study stability of a Schrödinger equation with a collocated boundary feedback compensator in the form of a heat equation with a collocated input/output pair. We show that the spectrum of the closed-loop system consists only of two branches along two parabolas which are asymptotically symmetric relative to the line Reλ = -Imλ (the 135° line in the second quadrant). The asymptotic expressions of both eigenvalues and eigenfunctions are obtained. The Riesz basis property and exponential stability of the system are then proved. Finally we show that the semigroup, generated by the system operator, is of Gevrey class δ >; 2. A numerical computation is presented for the distributions of the spectrum of the closed-loop system.