# Questions tagged [sturm-liouville-theory]

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### Boundary conditions for singular Sturm-Liouville problem (boundary behavior of eigenfunctions)

I am not at all an expert in Sturm-Liouville theory, but I ended up on the following Singular Sturm Liouville problem: \label{1} (1) \ \ \ \ \ \ \ \ \ \ \ y''(t)+\frac{\theta'(t)}{\...
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### Sturm-Liouville Problem: When does $w y^2$ vanish at a singular boundary point?

It is well known (e.g. Courant, Hilbert - Methods of Mathematical Physics) that solutions of the Sturm-Liouville problem on an interval $J=(a,b)$ \tag{1} \left(p y' \right)' - qy \; = ...
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### Angular excitations and Schrodinger operators with radial potential in N-dimensions

Can someone please explain the following in mathematical language? "First of all, angular excitations only push the energy up, never down, so it is enough to analyze spherically symmetric s-waves....
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### Fourier mode decomposition and eigenvalues of Schroedinger operators with radial potential in N-dimensions

In the study the stability of minimal hipersurfaces $\Sigma \subset \mathbb{R}^{N+1}$ one is lead to study the Morse index of a Schroedinger operator $J := \Delta_g + |A|^2$ (usually called Jacobi ...
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### The Node Theorem - an argument from physics

The theorem on the number of zeros of a solution to a Sturm-Liouville equation is a well-know result in quantum mechanics. It doesn't seem to have a special name in the mathematics literature, but it ...
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### Set of eigenvalues of the boundary problem

I'm looking for the results about the set of eigenvalues of boundary problem for differential equation \bigl(p(x) u'(x; \lambda) \bigr)' + q(x) u(x; \lambda) = -\lambda w(x) u(x; \...
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### Proof for strict separation of the eigenvalues ​of a Jacobian matrix with its minors

Let's consider a jacobi matrix (or tridiagonal symetric matrix where adjacent diagonals coefficients are strictly positive) : T_n = \begin{bmatrix} a_1 & b_1 & 0 & \...
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### 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 ...
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### 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 ...
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### 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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### 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 ...
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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}...
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 \$ ...