Schrödinger operator

Summary

A Schrödinger operator is a differential operator of the form $-\Delta + V$, where $\Delta$ is the Laplacian and $V$ is a real function called the potential. In one dimension it is the operator $-\frac{d^2}{dx^2} + V(x)$. Its eigenvalue equation is the time-independent Schrödinger equation of quantum mechanics, and its eigenvalues are the allowed energies of a particle in the potential $V$. In the theory of ordinary differential equations it is the main example of a Sturm–Liouville operator.