Exponential spectral formula conjecture for the Bloch–Torrey fiber semigroup

Fix $q\in\mathbb R$ and let $\mathcal B^D_q$ be the corresponding fiber operator. Set

$$K_{g,q}:=\exp(-t_g\mathcal B^D_q),\qquad t_g=\frac{2\pi}{g},$$

where $g$ is the parameter in the Bloch–Torrey operator. Let $\lambda_k(q)$ denote the eigenvalue sequences introduced for $\mathcal B^D_q$. Exponential spectral formula conjecture.

$$\sigma(K_{g,q})\setminus\{0\}=\bigcup_k\exp(-t_g\lambda_k(q)).$$

The formula is motivated by the recovered commutation relation of the semigroup with the translation operator. Establishing it requires understanding the spectrum of $K_{g,q}$; compactness would be immediate in the self-adjoint or sectorial case, but the fiber operator is not sectorial in the relevant non-self-adjoint setting.