Schrodinger operators with singular boundary conditions

In order to define a differential operator on a domain with boundary one often needs
to specify boundary conditions. In the presence of singular terms this can be quite tricky. It may lead to interesting and surprising phase diagrams.
I will illustrate this with the so-called perturbed Bessel operators,
that is one-dimensional Schrodinger operators on the half-line of the form
$$-\partial_x^2+(m^2-1/4) 1/x^2+ Q(x).$$
The talk will be based on my joint work with Jeremy Faupin.