Stabilisation of Schrödinger equation in dynamic boundary feedback with a memory-typed heat equation

In this work, we study the dynamic behaviour for a heat equation with exponential polynomial kernel memory to be a controller for a Schrödinger system. By introducing some new variables, the time-variant system is transformed into a time-invariant one. Remarkably, the resolvent of the closed-loop system operator is not compact anymore. The residual spectrum is shown to be empty and the continuous spectrum consisting of finite isolated points are obtained. It is shown that the sequence of generalised eigenfunctions forms a Riesz basis for the Hilbert state space. This deduces the spectrum-determined growth condition for the C ₀-semigroup, and the exponential stability is then established.