Questions tagged [sturm-liouville-theory]

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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^{...
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1 answer
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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\,...
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3 votes
2 answers
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}{\...
Moritz Reinhard's user avatar
5 votes
5 answers
2k 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 ...
15 votes
6 answers
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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 ...
Stephane's user avatar
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4 votes
1 answer
398 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 ...
Denis Serre's user avatar
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3 votes
5 answers
489 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.
warsaga's user avatar
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1 vote
2 answers
574 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}...
David's user avatar
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0 answers
141 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}...
David's user avatar
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2 votes
0 answers
225 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 $ ...
user19179's user avatar
136 votes
9 answers
19k 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 $S(f) = \frac{f'''}{f'} - \frac{3}{2} \Big(\frac{f''}{f'}\Big)^2$ Here is a somewhat more ...
Paul Siegel's user avatar
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7 votes
2 answers
592 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 ...
skupers's user avatar
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3 votes
0 answers
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 ...
Ryan Thorngren's user avatar

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