Schrodinger equation with parameters

I need to know the ground state energy $E_0$ defined by the following stationary Schrodinger equation:
$$
-frac{a}{2}phi”(xi) + left(frac1{2a}sinh^2(2xi) + (2b-1)cosh(2xi)right)phi(xi) = E_0 phi(xi).
$$
Here $xi$ is a dimensionless coordinate, $a>0$ and $b$ are dimensionless parameters. I know the answer $E_0=0$ for the $b=0$ case. I have doubts exact answers are possible for other cases. So I want to consider limiting cases. In my opinion, there are three simple cases:

$ato +0$,

$b to 0$,

and $bto +infty$.

The case $a to +infty$ looks nontrivial to me. I have qualitative arguments in favor of $E_0 sim C a/log^2(a)$ asymptotic in this case, where $C$ is an unknown constant. But that’s not enough. So the question is: what is the exact asymptotic of $E_0$ in the $ato+infty$ limit, with $C$ multiplier expressed explicitly or as a solution to some equation?