Stabilization and gevrey regularity of a schrdinger equation in boundary feedback with a heat equation

We study stability of a Schrdinger equation with a collocated boundary feedback compensator in the form of a heat equation with a collocated input/output pair. Remarkably, exponential stability is achieved for both positive and negative gains, namely, as long as the gain is non-zero. 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 ${\rm Re}\lambda=-{\rm Im}\lambda$ (the 135$ ^\circ 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 $\delta2$. A numerical computation is presented for the distributions of the spectrum of the closed-loop system.