Are solutions to the particle in a (finite) box problem orthonormal?

Question

Solving the particle in a box problem is fairly straight-forward for both, the finite and infinite potential well. While it is well known that the solutions to the infinite potential well must be orthogonal, it is not so obvious whether that should be the case for a potential well with finite height.

I tried to find references in which orthogonality of those states is shown, but I was not very successful so far. Does anyone here know how one could show orthogonality of these states and/or tell me about papers in which this property has been established?

G. Smith · Accepted Answer

The time-independent Schrodinger equation is in the class of real second-order linear ordinary differential equations covered by 19th-century Sturm-Liouville theory. The eigenfunctions of such equations are orthogonal with respect to an appropriate inner product on the function space.
So one does not have to prove orthogonality for each potential one is interested in, such as a finite square well. It is a general result for all potentials.

Are solutions to the particle in a (finite) box problem orthonormal?

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