Discrete States for Singular Potential Problems

The solutions of the quantum mechanical wave equations for singular potentials are re-examined. It is shown that a set of orthonormal wave functions with complex energy eigenvalues ($E_n=W_n\pm \frac{1}{2}i{\Gamma }_n$) is obtained if certain natural analyticity requirements are imposed on the form of the potentials. In general, the result is interpreted in the following way: $W_n$ is the most probable position of the energy level for various types of cutoff and ${\Gamma }_n$ is a measure of the probable error.