Singular Sturm-Liouville problems: criterion for discrete spectrum for zero potential ($q=0$) and Hermite Polynomials

Question

There are some known criteria for the Sturm-Liouville Problem

\begin{equation} \tag{1} \frac {\mathrm {d} }{\mathrm {d} x}\left[p(x){\frac {\mathrm {d} y}{\mathrm {d} x}}\right]+q(x)y=-\lambda w(x)y \end{equation}

to have a spectrum discrete and bounded below (BD) in the singular case: If singular endpoints are Limit-Circle and Non-Oscillating (LCNO), results from (Niessen, H.-D. and Zettl, A. (1992). Singular Sturm-Liouville Problems: The Friedrichs extension and comparison of eigenvalues.) apply. There also is the famous criterion by Molchanov and generalizations for $w, p \neq 1$ in (Kwong and Zettl - Discreteness conditions for the spectrum of ordinary differential operators). Various other generalizations exist.

As I understand it, the Molchanov criterion and its generalizations are not applicable to the case $q = 0$ or $q$ constant. Inherently, $q$ must run to infinity at the endpoint in a certain way (details depending on $p$ and $w$). (See related questions For which type of potentials a Schrödinger operator will have discrete spectrum?, Harmonic oscillator discrete spectrum)

My questions:

Is there a theory and criteria for (1) to be BD for the case $q = 0$ if endpoints are in the Limit-Point (LP) case? I specifically appreciate pointers to literature. Or do I overlook some tweak, allowing to apply Molchanov-style criteria to the case $q = 0$?

What I read and reasoned so far:

As far as I can assess, transformations to normal form are not suitable to work around this particular issue. Also, according to Zettl (Zettl, A. (2010). Sturm-Liouville Theory), it is questionable whether such a transformation preserves the desired properties of the spectrum (discussion in Chapter 10.13).

I looked into various books and papers and googled around, but I almost exclusively found Molchanov-style criteria, most usually setted for the half-line $[a > -\infty, \infty)$. It appears that Schrödinger operators are the main interest in this field and the Molchanov criterion is a good fit for them. I am more interested in self-adjoint operators similar to the one yielding Hermite polynomials:

\begin{equation} \tag{2} q = 0, \quad w = p = e^{-{\frac {x^{2}}{2}}} \end{equation}

on the real line with two singular LP endpoints. I used to perceive Hermite Polynomials as an important standard example and it puzzles me that it seems to be so hard to find criteria for their spectrum. I mean criteria that are not totally specific to equation (1) with values (2), but also hold if some other (normal-like?) probability distribution is used in (2), e.g. with adjusted variance, skewness, kurtosis etc. (not expecting the solutions to be (orthogonal) polynomials anymore, but still to have BD spectrum). (Edit: It occurred to me that the case of (2) with adjusted variance is known as the generalized Hermite polynomials.) E.g. I skimmed through some books by L. L. Littlejohn and A.M. Krall about orthogonal polynomials in context of singular SL theory, e.g. (L. L. Littlejohn and A.M. Krall, Orthogonal polynomials and singular Sturm-Liouville Systems, I) but they do not seem to discuss the discreteness of the spectrum (sorry if I overlooked it, pointers welcome).

Secondary question: What about the full line vs. the Half line?

It appears that in his original work, Molchanov considered the full line (A.M. Molchanov, Conditions for the discreteness of the spectrum of self-adjoint second-order differential equations.) however, I cannot read Russian and only got this impression from the math terms. Also (Inge Brinck, Self-Adjointness and spectra of Sturm-Liouville operators) assumes the full line, but (Kwong and Zettl - Discreteness conditions for the spectrum of ordinary differential operators) and (D. Hinton, Molchanov’s discrete spectra criterion for a weighted operator) assume the half line. This gives me the impression that the distinction between full line and half line is not essential, but I did not find it discussed anywhere. A pointer to literature that specifically discusses in what sense results are transferable between the half line and full line case would be welcome. (Niessen, H.-D. and Zettl, A. (1992). Singular Sturm-Liouville Problems: The Friedrichs extension and comparison of eigenvalues.) discuss it a bit, but not in context of criteria for BD in the LP case (or I don't understand how to conclude it).

Thanks in advance, also for hints, comments and improvements!

Answer 1

The case $q=0$ actually has a complete answer, and the reason this works is that we can solve the equation for $\lambda =0$ explicitly then, by $u=1$ and $v=\int_0^x \frac{dt}{p(t)}$. The spectrum is purely discrete if and only if ($w\in L^1$ and) $$ \lim_{x\to\infty}\int_0^x wv^2\, dt \int_x^{\infty} w\, dt =0 . \quad\quad\quad\quad (1) $$ (This holds in your example, if I managed the asymptotics of the error functions correctly.)

I obtained this by rewriting the SL equation as a canonical system $Jy'=-\lambda Hy$, and the coefficient matrix $H$ we obtain here is given by $$ H = w\begin{pmatrix} 1 & v \\ v & v^2 \end{pmatrix} . $$ There is a precise criterion for purely discrete spectrum for (general) canonical systems, and this gave me (1).

This will probably all sound quite cryptic if you haven't seen these things before, and there seem to be too many small details to explain them all here. You could take a look at my papers 1, 2 on my homepage for full background information.

Also, if I hadn't had that available, I would have tried to analyze the Prufer equation $$ \varphi' = \frac{1}{p}\cos^2\varphi + (\lambda w+q)\sin^2\varphi $$ for the phase of the solution vector $(y,py')$. Purely discrete spectrum is equivalent to $\varphi(x)$ staying bounded in $x$ for all $\lambda$. This might also be good enough to get (1).