Do non-vanishing boundaries conditions necessarily force a discrete/quantized set of solutions for the Schrödinger Equation?

Question

I was reviewing past exams and I found a question where I could give no satisfactory answer. The core of the question is the following: If there are boundary conditions, does this necessarily imply, that the solutions to the time-independent Schrödinger equation are quantized, i.e. there are only countably many solutions? By the expression "has a boundary conditions" I want to specifically exclude the free particle i.e. $V=0$.
Is there a rigid mathematical proof for this?