Eigenvalues of bounded differential operators

I'm trying to understand the following passage from Sidney Coleman's "Aspects of Symmetry" (pg 340 for those who have it):

We wish to study the equation

[; (\partial_{t}^2 - W)\psi = \lambda \psi;]

where W is some bounded function of [; t ;]. Let us define [; \psi_\lambda(t) ;] as the solution of this equation obeying boundary conditions

[; \psi_\lambda(-T/2) = 0 ;]

and

[; \partial_{t} \psi_{\lambda}(-T/2) = 1 ;].

The operator [; -\partial_{t}^2 + W ;] (acting on the space of functions vanishing at [; \pm T/2 ;]) has an eigenvalue [; \lambda_n ;] if and only if

[; \psi_\lambda_n (T/2) = 0 ;]

My question is essentially: what role do the boundary conditions have on this problem? My understanding is that if [; \psi_\lambda ;] is a solution to the differential equation above, then it automatically has the associated eigenvalue irrespective of boundary conditions. What am I missing?