Schrodinger equations

This is not a note for the original Schrodinger equation of Quantum Mechanics, but in the context of ODEs. They are equations in the form

\frac{\partial^{2}}{\partial x^{2}} Ψ = g (x) Ψ .

I found them first time appearing from a Riccati equation.

It is motivated by the time-independent Schrodinger equation in Quantum Mechanics, i.e., the search of the eigenstates of the Hamiltonian operator

- \frac{ℏ^{2}}{2 m} \frac{d^{2} Ψ}{d x^{2}} + u Ψ = E Ψ