Eigenvalue of $(\partial_\tau-\frac{1}{2}\triangledown^2_x+1-\xi(\vec{x},\tau))\Psi(\vec{x},\tau)=\lambda\Psi(\vec{x},\tau)$

Question

Originally, I want to evaluate the path integral $\int D\Psi^{\dagger}D\Psi e^{-S}$, where S is the imaginary time action $S=\Psi^{\dagger}(\partial_\tau-\frac{1}{2}\triangledown^2_x+1-\xi(\vec{x},\tau))\Psi(\vec{x},\tau)$, $\Psi$ is the electron field, and $\xi$ is the columb field, which is known. Obviously, this path integral gives the fermionic determinant $det(\partial_\tau-\frac{1}{2}\triangledown^2_x+1-\xi(\vec{x},\tau))$. Now my problems reduces to finding the eigenvalues of $(\partial_\tau-\frac{1}{2}\triangledown^2_x+1-\xi(\vec{x},\tau))\Psi(\vec{x},\tau)=\lambda\Psi(\vec{x},\tau)$, express it in terms of $\xi$, assuming $\Psi(\vec{x},\tau)$ is antiperiodic in $\tau$ ($\Psi(\vec{x},0)=-\Psi(\vec{x},T)$ ), and periodic in $\vec{x}$ $(\Psi(\vec{x},\tau)=\Psi(\vec{x}+\vec{a},\tau), \vec{a}=(a_x,0,0) or(0,a_y,0) or(0,0,a_z))$. The domain of $\vec{x},\tau$ is $\vec{x}\in [0,a_x]\times [0,a_y]\times [0,a_z], \tau \in [0,T]$