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
$$ \frac{\partial^2}{\partial x^2} \Psi=g(x)\Psi. $$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
$$ -\frac{\hbar^2}{2 m} \frac{d^2 \Psi}{d x^2}+u \Psi=E \Psi $$________________________________________
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Author of the notes: Antonio J. Pan-Collantes
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