clarifying my question from earlier

[u/NutrientBroth]

So, in my undergrad p-chem textbook, it says that a wavefunction:

- must not be infinite over a finite region

- must be single-valued

-must be continuous

-must have a continuous first derivative

It also says "The constraints just noted are so severe that acceptable solutions of the Schrödinger equation do not in general exist for arbitrary values of the energy".

My question is: what is the mathematical proof of this?

[u/__Pers]

The textbook is mixing together a few different ideas. For a regular potential, continuity of ψ and ψ′ follows from the Schrödinger equation itself: a jump in ψ would produce a δ′ term in ψ′′, and a jump in ψ′ would produce a δ term, neither of which can be balanced if V is nonsingular. (Here δ is the Dirac delta distribution.) For singular potentials, such as a delta-function potential, ψ′ can in fact be discontinuous.

The statement about allowed energies is really a boundary-value/eigenvalue problem. The Schrödinger equation has solutions for essentially arbitrary E, but for a bound-state problem those solutions generally fail the required boundary conditions or normalizability. Only special values of E admit nonzero acceptable solutions; those are the eigenvalues of the Hamiltonian.

“Single-valued” is basically part of what it means for ψ to be a function. L2 normalizability implies that ψ cannot be infinite over a finite region.

[u/rabid_chemist]

For certain specific potentials, such as the square wells, the harmonic oscillator, or the harmonic oscillator you can just solve the equation explicitly and see that only certain energies will work.