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

Question

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

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

where c is an arbitrary positive constant.
P is the momentum operator and Q is the position operator.

Comments

Try a WKB approximation for your $H=P^2+ \frac{c} {Q(1-Q)}$