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Questions tagged [sturm-liouville-theory]

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0answers
84 views

Limit circle/point of an ODE with finite eigenvalues

Consider the following Sturm–Liouville (SL) eigenvalue problem defined in $(-\infty,0]$ or $[0,\infty)$ or $(-\infty,+\infty)$ $$(py')'-qy=-\lambda^2wy,$$ in which $p(x)=x^2$, $w(x)=1$, and $q(x)=(x/2+...
3
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2answers
138 views

Orthogonal Polynomials and Sturm Liouville operators

Classical Orthogonal polynomials (e.g., Hermite, Legendre) are eigenfunctions of Sturm Liouville operators. For example, define $L[u]=u''-xu'$, then the $n$-th order Hermite polynomial satisfies $...
0
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0answers
61 views

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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0answers
41 views

One class Sturm-Liouville differential equation

Let $\phi_n(x), \psi_n(x)$ be solution Sturm-Liouville differential equation $$p(x) y''(x) - 2n p'(x)y'(x)+2n(2n+1)y(x)=0$$ $$\phi_{n}(0)=0, \hspace{3mm} \phi'_{n}(0)=1;$$ $$\psi_{n}(0)=1, \hspace{...
0
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1answer
150 views

Completeness of the solutions to the Schrödinger Hydrogen Atom

I once did some work on using orthogonal function expansions for fitting 3D distribution functions. To ensure completeness over $L^2$ (which was considered sufficient even though technically a ...
1
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1answer
106 views

A property of a nonlinear ODE under periodic boundary conditions

Let $u_1,u_2 : (0,1)\to \mathbb{R}$, and given that $$u_1(0) = u_1(1),u_1'(0)= u_1'(1)$$ and $$u_2(0) = u_2(1),u_2'(0)=u_2'(1)$$ and also $$(|u_1'|u_1')' = \lambda_1|u_1|u_1$$ and $$(|u_2'|u_2')' = \...
1
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1answer
76 views

How to determine the spectrum from the diagonal Green's function

Let $L: L^2(\mathbb{R}) \supseteq Dom(L) \rightarrow L^2(\mathbb{R})$ be a densely defined closed operator. Assume that the resolvent admits an integral kernel (Greens function) $G$, i.e. for $z\in \...
3
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0answers
64 views

Convergence of Bessel (Sturm-Liouville) Expansions at the End Points

I have asked this question before on MSE but received no answer at all. So I assume that it is proper to ask it here. I am not a mathematician so my language may not be too precise, please correct me ...
3
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1answer
120 views

An indefinite integral containing functions that are solutions to a 2nd order linear ODE

I am trying to evaluate an indefinite integral of the form $\int \frac{dz}{A u_1^2 + Bu_2^2 + Cu_1u_2}$ where $u_1$ and $u_2$ are two independent solutions to the ODE $u'' + F(z)u = 0$ This ...
1
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1answer
176 views

Sturm Liouville problems for non-classical orthogonal polynomials

It is known that for the classical orthogonal-polynomials there exist a set of Sturm Liouville problems. E.g. , the Hermite polynomial of order $n$ is a solution of $$y''(x) -xy'(x)+ny(x)=0 \, .$$ My ...
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0answers
98 views

References for the Sturm oscillation theorem

What is the most general form of the Sturm oscillation theorem? So far I have only seen cases that treat either unbounded intervals or weighted $L^2$ spaces. I would be especially interested in ...
6
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3answers
476 views

Non-self adjoint Sturm-Liouville problem

I'm new to this site, but I felt the need to post when I recently came in to an ordinary differential equation/boundary value problem with this form: $(1)- \frac{d^2 y}{d x^2} + \frac{m(m+1)} {x^2(...
2
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0answers
55 views

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 ...
3
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2answers
250 views

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
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1answer
143 views

Zeroes of Sturm-Liouville solutions as a function of the (complex) eigenvalue

Given the Sturm-Liouville type (time independent Schroedinger) equation \begin{equation} \frac{d^2 y}{d x^2} - \left(\mu + V(x)\right) y = \lambda \, y,\quad x \in \mathbb{R} \end{equation} where $V(...
5
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1answer
315 views

Spectrum of this ODE

I noticed something interesting studying this Sturm-Liouville Problem: $$ \frac{d}{dx}\left(\sqrt{(1-x^{2})}\frac{df}{dx} \right)+\frac{\left(n \alpha x+\alpha^2 x^{2} + \lambda\right)f}{\sqrt{(1-x^{...
0
votes
1answer
265 views

Legendre differential equation with additional term

In an application I encountered the ODE $$ \left( {x}^{2}-1 \right) {\frac {{\rm d}^{2}}{{\rm d}{x}^{2}}}f \left( x \right) +x \left( {\frac {\rm d}{{\rm d}x}}f \left( x \right) \right) \left( 8\,...
3
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2answers
2k views

Eigenfunctions of fourth-order differential operator

This a question where I have thought quite long about: The eigenfunctions (or also normal modes) of an dry Euler beam subject to free-free boundary conditions are given by $$ \frac{\partial^4\psi}{\...
5
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5answers
1k views

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 ...
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6answers
2k views

Spectral theorem for self-adjoint differential operator on Hilbert space

I need a reference concerning a theorem that shows the following result, stated very roughly: Given a self-adjoint differential operator densely defined on a Hilbert space, then the given Hilbert ...
4
votes
1answer
298 views

Simplicity of eigenvalues of an elliptic operator under a symmetry assumption

A striking difference in the spectral analysis of 2nd order elliptic boundary-value problems between one and several space dimensions is the following. In one space dimension, the eigenvalues are ...
3
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5answers
310 views

Good reference for the construction of a Greens functions fur the Sturm-Liouville

Does anyone know a good reference for the constructions of a Greens functions fur the Sturm-Liouville Boundary Value Problem.
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2answers
472 views

Lower bound for the eigenvalue

For a given real number $c>0$ define functions $\left(\psi_{k,c}(\cdot)\right)_{k\ge0}$, as an eigenfunctions of the Sturm-Liouville operators $L_c$ defined $$ L_c(\psi)=(1-x^2)\frac{d^2\psi}{dx^2}...
0
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0answers
112 views

$n$-th derivative of the prolate spheroidal function

For a given real number $c>0$ define functions $\left(\psi_{k,c}(\cdot)\right)_{k\ge0}$, as an eigenfunctions of the Sturm-Liouville operators $L_c$ defined $$ L_c(\psi)=(1-x^2)\frac{d^2\psi}{dx^2}...
2
votes
0answers
205 views

probability interpretation of Sturm-Liouville eigenvalue problem

For a diffusion process , Feymann Kac formula connects the second order elliptic equation. Look at the eigenvalue problem $u''-xu'=-\lambda u $, $ u(0)=0,u(1)=0 $. for some discrete eigenvalues of $ ...
103
votes
9answers
12k views

Is there an underlying explanation for the magical powers of the Schwarzian derivative?

Given a function $f(z)$ on the complex plane, define the Schwarzian derivative $S(f)$ to be the function $S(f) = \frac{f'''}{f'} - \frac{3}{2} (\frac{f''}{f'})^2$ Here is a somewhat more conceptual ...
7
votes
2answers
477 views

Does there exist a potential which realizes this strange quantum mechanical system?

I have done some courses on quantum mechanics and statistical mechanics in the past. Since I also do math, I wonder about converge issues which are usually not such a problem in physics. One of those ...
3
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0answers
1k views

Eigenvalue Problems with Linear Constraints

The motivation for this problem comes from trying to develop a simple way to decompose domains into non-overlapping subdomains to solve for the eigenvalues of some differential operator. The idea is ...