Questions tagged [sturm-liouville-theory]
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Spectrum of this ODE
I noticed something interesting studying this Sturm-Liouville Problem:
$$ \frac{d}{dx}\left(\sqrt{(1-x^{2})}\frac{df}{dx} \right)+\frac{\left(n \alpha x+\alpha^2 x^{2} + \lambda\right)f}{\sqrt{(1-x^{...
0
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1
answer
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Legendre differential equation with additional term
In an application I encountered the ODE
$$ \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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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
$$ \frac{\partial^4\psi}{\...
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A graduate course on Sturm Liouville theory?
I have some general questions on Sturm-Liouville theory. We are planning to introduce a graduate course on Sturm-Liouville theory and every one has been asked to propose topics which might be suitable ...
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Spectral theorem for self-adjoint differential operator on Hilbert space
I need a reference concerning a theorem that shows the following result, stated very roughly:
Given a self-adjoint differential operator densely defined on a Hilbert space, then the given Hilbert ...
4
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1
answer
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Simplicity of eigenvalues of an elliptic operator under a symmetry assumption
A striking difference in the spectral analysis of 2nd order elliptic boundary-value problems between one and several space dimensions is the following. In one space dimension, the eigenvalues are ...
3
votes
5
answers
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Good reference for the construction of a Greens functions fur the Sturm-Liouville
Does anyone know a good reference for the constructions of a Greens functions fur the Sturm-Liouville Boundary Value Problem.
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2
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Lower bound for the eigenvalue
For a given real number $c>0$ define functions $\left(\psi_{k,c}(\cdot)\right)_{k\ge0}$, as an eigenfunctions of the Sturm-Liouville operators $L_c$ defined
$$
L_c(\psi)=(1-x^2)\frac{d^2\psi}{dx^2}...
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0
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$n$-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
0
answers
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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 $ ...
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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 ...
7
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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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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 ...