I have simplified my problem here, in my actual problem the matrix is much bigger which makes it impossible to find eigenvalues analytically. So, I chose standard BCS problem (2x2 matrix) to demonstrate my problem with Mathematica. But, with even this simple case, it takes forever to integrate...
Bound state implies the "classically expected" state, so bound state energy should be smaller than V(r = infinity). For definition, Shankar says "a particle's wave function should go to zero as x -> infinity in bound state"