Spectral properties of differential operators using orthogonal polynomials
The classical theorem by Bochner classies the second order dierential operators having polynomial eigenfunctions.
Generalising Bochner's approach we look at dierential operators for which there exists a suitable
basis of functions tridiagonalising the dierential operator. This gives the opportunity to describe the spectrum
of the dierential operators involved. We illustrate the approach by several examples, and we discuss
generalisations to other types of operators. A particular well-known example is the Schrodinger with the
Morse potential studied by chemists. Other examples to be discussed are the Jacobi function transform
and the Whittaker transform as well as some q-analogues.
** This is a joint work with: M. Ismail.