# Questions tagged [sturm-liouville-theory]

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### Identity principle of solutions of SL-problems with matching values on open set

Situation (cut short): Corresponding solutions (by eigenvalue) of two given regular Sturm-Liouville problems with homogeneous Neumann BC, same spectrum but possibly distinct coefficient functions, &...
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### Eigenvalues of Sturm–Liouville operator

Can we calculate the eigenvalues and eigenfunctions of the following operator in $W^{1,2}(\mathbb{R})$? $$-\left(\frac{1}{\cosh^2x}\right)y''-\frac{2}{\cosh^4x}y=\lambda y.$$
1 vote
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### How does boundary perturbation affect the eigenvalues of differential equations?

There is a well-known procedure (at least to me) to compute how a small perturbation will affect the eigenvalues of a differential equation. However, the method deals only with perturbing the ...
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1 vote
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### Singular Sturm-Liouville problems: criterion for discrete spectrum for zero potential ($q=0$) and Hermite Polynomials

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 \...
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1 vote
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### Sturm Liouville problem on entire line, substitution

Observe Sturm-Liouville problem on entire line $$-(p(x)y'(x))' + l(x)y(x)= \lambda r(x)y(x), \hspace{3mm} -\infty<x<\infty \tag{1} \label{1}$$ where $p(x)$ and $r(x)$ are positive on $\mathbb{R}$...
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1 vote
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### Error bounds for eigenvalue expansion of the Mathieu equation

The Mathieu equation is an important eigenvalue problem in Mathematical Physics that is completely understood in its properties, although there is no "direct way" of expressing eigenvalues and ...
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### Schrödinger operators on a sphere

if you have a Schrödinger operator on a sphere ( $\mathbb{S}^2$) $-\Delta_{\theta,\phi} \psi(\theta,\phi) + V(\theta) \psi(\theta,\phi) = E\psi(\theta,\phi),$ where the potential does not depend on ... 1 vote
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### A graduate course on Sturm Liouville theory?

I have some general questions on Sturm-Liouville theory. We are planning to introduce a graduate course on Sturm-Liouville theory and every one has been asked to propose topics which might be suitable ...