Spectrum of singular Sturm-Liouville operators with singular interior point

[dukonian]

Hi,

I have a singular Sturm-Liouville problem with LCNO end-points, but also one limit circle point in the interior of the interval. Suppose I take boundary conditions that get me a self-adjoint extension of the differential operator, does anyone know if that gives me a discrete spectrum?

The SL operator is:
L(d,dx) f(x) = (P(x) f'(x))' + 2x^2 f(x), x in (-2,1)

P(x) = (x^2-1)(x^2-4)

So the boundaries +1 and -2 are limit circle end-points, but there is one more singularity at -1.

Thanks

 

[jvicens]

for any help!



Thank you for sharing your question with the community. I can offer some insights into your inquiry. The short answer to your question is that it is possible for a self-adjoint extension of a Sturm-Liouville operator with LCNO end-points and a limit circle point in the interior to have a discrete spectrum. However, this is not always the case and it depends on the specific boundary conditions chosen for the self-adjoint extension.

First, let's define some terms for those who may not be familiar with Sturm-Liouville operators. A Sturm-Liouville operator is a second-order differential operator with variable coefficients, named after mathematicians J.C.F. Sturm and J. Liouville. LCNO stands for "locally contained in the norm of the operator," which means that the end-points of the interval are not singular points of the differential equation. A limit circle point is a point in the interior of the interval where the coefficients of the differential equation become singular.

Now, to address your question, it is important to note that the discrete spectrum of a self-adjoint extension is determined by the boundary conditions imposed on the operator. In your case, the boundary conditions may or may not lead to a discrete spectrum, depending on their specific form. For example, if the boundary conditions are such that they satisfy the Weyl-Titchmarsh-Kodaira (WTK) formula, then the self-adjoint extension will have a discrete spectrum. However, if the boundary conditions do not satisfy the WTK formula, then the spectrum may be continuous.

To determine whether the WTK formula is satisfied, one would need to know the form of the boundary conditions chosen for the self-adjoint extension. Without this information, it is difficult to say for certain whether your operator will have a discrete spectrum.

In conclusion, it is possible for a self-adjoint extension of a Sturm-Liouville operator with LCNO end-points and a limit circle point in the interior to have a discrete spectrum, but it depends on the specific boundary conditions chosen for the extension. I hope this helps answer your question. If you would like further clarification or assistance, please do not hesitate to reach out. Good luck with your research!