I'm dealing with some discrete operators and I'm having some difficulties to find an expression or a method to find a set of eigenfunctions and eigenvalues for the bidimensional operator $S$ defined below. I'm rather novice to the subject.
Consider $\gamma_a,\gamma_b,\gamma_c,V \in R$, and $c,a \in Z^+ $ where $Z^+ = (0,1,2, \ldots)$.
I denote by
\begin{equation} \Delta^c_h=f(c+h,a)-f(c,a) \end{equation}
and
\begin{equation} \Delta^a_h=f(c,a+h)-f(c,a) \end{equation}
the one $h$ step finite difference operator applied to the subspace specified by the apex.
The operator has the following structure:
\begin{equation} S=\gamma_b^{-1}\gamma_a \gamma_c\left[\frac{V}{\gamma_a \gamma_c} \Delta^c_1 + \frac{c}{\gamma_a}\Delta^c_{-1} + \frac{c}{\gamma_a\gamma_c}\Delta^a_1 + \frac{a}{\gamma_c} \Delta^a_{-1}\right] \end{equation}
An eigenfunction and eigenvalues related to the $c$ subspace of the operator can be found as follows:
\begin{equation} S\left[c-\frac{V}{\gamma_c}\right] = -\gamma_b^{-1}\gamma_c\left(c-\frac{V}{\gamma_c}\right) \end{equation}
however I would not even know how to proceed for the eigenfunction and eigenvalue of the whole operator or the eigenfunction and values of the part of the operator realted to $a$.
Does any one happen to have any experience/hints?