Eigenvalue and eigenfunction of Hamiltonian on interval $[0,1]$

I am seeking the minimum eigenvalue and the eigenfunction of the following Hamiltonian on interval $[0,1]$;

$$hat{P}^2+ c hat{Q}^{-1} + c (I-hat{Q})^{-1}$$

where $c $ is an arbitrary positive constant.
$hat{P} $ is the momentum operator and $hat{Q}$ is the position operator.