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 $W^{1,2}(\mathbb{R})$? $$-\left(\frac{1}{\cosh^2x}\right)y''-\frac{2}{\cosh^4x}y=\lambda y.$$
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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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Techniques for showing non-degeneracy results (PDE)

Motivation: Consider the equation, $$-\Delta u = u^p$$ in $\mathbb{R}^n$ for $n\geq 3$ and $p=2^*-1.$ Then we know that this equation has unique positive solutions given by functions of the form $U_{a,...
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4 answers
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Joint boundedness of solutions of a family of Sturm-Liouville ODE

Let us fix $0 \neq \lambda \in \mathbb{R}$. Let us consider the following ODE, on $[0,\infty)$: $$ y^{\prime \prime} (x) + \frac{r e^{-x}}{(1+e^{-x})^2} y(x) = -\lambda^2 y(x).$$ Here $r \ge 1$ is a ...
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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: \begin{equation}\label{1} (1) \ \ \ \ \ \ \ \ \ \ \ y''(t)+\frac{\theta'(t)}{\...
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2 answers
303 views

Orthogonality of Bessel function $\int_0^bxJ_a(\ell x)J_a(\ell' x)=0$ for $\ell\neq\ell'$

How do I show the above relation with Sturm-Liouville theory (assume the usual boundary conditions for the identity)? Here is what I have tried: if we start with $$ \big(xJ_a'(\ell x) \big)'+\left(\...
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1 vote
1 answer
84 views

Can we drop this annoying integral term to restore a Sturm-Liouville problem?

On $[0,1]$, let $f:[0,1]\to \mathbb{R}$ be positive and continuous, consider the equation: $w''+w+\lambda f\cdot (w-\int_0^1 w)=0$($\lambda$ is an eigenvalue) subject to $w(0)=w(1)=0$. If the integral ...
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Negative eigenvalue for a periodic Sturm-Liouville problem

Let $f \in C^{\infty}([0, 2\pi])$ be a smooth function and consider the following periodic Sturm-Liouville problem: $$\begin{cases} u''(x) + f(x)u(x) = - \lambda u(x) \\ u(0) = u(2\pi) \\ u'(0) = u'(2\...
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References on discrete Sturm-Liouville eigenvectors convergence

Let $ L : u_n \mapsto a_n u_{n + 1} + b_n u_n + a_{n - 1} u_{n -1} = \nabla ( a_n \Delta u_n ) + (b_n + a_n + a_{n - 1}) u_n $ be a discrete Sturm-Liouville operator, with $ \nabla u_n := u_{n + 1} - ...
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1 answer
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Sign of solution to (in)homogeneous linear ODE

Let $N \geq 3$ be a positive integer and $A >0, B \geq 0$ be two constants. Let $y: (0,\infty) \to \mathbf{R}$ be a solution to the following linear, inhomogeneous ODE: $y''(x) + \frac{N-1}{x} y'(x)...
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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 $J=(a,b)$ \begin{equation} \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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1 vote
1 answer
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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 \begin{equation} \bigl(p(x) u'(x; \lambda) \bigr)' + q(x) u(x; \lambda) = -\lambda w(x) u(x; \...
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156 views

Fourier transform of Green function and its derivative

Consider a real Sturm-Liouville operator $L$ on $[0,+\infty)$ and use the following notations : https://www.encyclopediaofmath.org/index.php/Titchmarsh-Weyl_m-function Assume $a = 0$, $\alpha \in [0,\...
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7 votes
1 answer
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Singular Sturm-Liouville problems: criterion for discrete spectrum for zero potential ($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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  • 183
2 votes
1 answer
114 views

Common eigenvalues for two Sturm-Liouville problem

Does exist in literature any results concerning the common eigenvalues for the two eigenvalue problems of the form $$y''(x)=\lambda^2 y(x)+\lambda a(x)y(x), \ x\in(0,1), $$$$z''(x)=\lambda^2 z(x)-\...
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2 answers
152 views

Eigenvalue problem of Schrodinger equation with polynomial growth potential on the real line

Consider the operator $L=-\frac{d^2}{dx^2}+q(x)$, where $q(x)$ is the potential with polynomial-type growth, say $|x|^s,s>1$. The eigenvalue problem $$L\phi=\lambda\phi$$ where $\phi$ is in $L^2(\...
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Equivalence of solutions to Sturm-Liouville problem after translation of boundary conditions

My doubt is related to the equivalence between solutions of the following Sturm-Liouville problem: \begin{equation} r^{2}f''(r) + 2rf'(r) + \{\omega^{2}r^{2} - [j(j+1)-|q|^{2})]\}f(r)=0\,,\label{SL1} \...
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2 answers
293 views

Sturm Liouville differential equation and hypergeometric functions

I'm trying to understand how to solve this differential equation: $ [z^2(1-z)\dfrac{d^2}{dz} - z^2 \dfrac{d}{dz} - \lambda] f(z) = 0 $ I know the solution is related to the hypergeometric function ...
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1 vote
0 answers
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How to solve or analyse the smallest eigenvalue of 2 coupled 1st-order linear ODEs?

I am trying to solve the eigensystem of a 1st-order linear ODE system in the region $(-\infty,\infty)$ and with Dirichlet boundary condition at the infinities \begin{align} -\mathrm{i} u'(x) +f^*(x) ...
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1 vote
0 answers
52 views

Sturm-Liouville-like Eigenproblem

Consider the piecewise-deterministic Markov process on $\mathbf{R}$ which moves according to the vector field $\phi (x) = 1$, experiences events at rate $\lambda(x) = 1$, and at events, jumps ...
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Showing a differential operator is positive semidefinite

Let $R>\lambda>\chi$ be positive real constants and $\alpha$ be a real number. The following differential operator \begin{multline} \mathcal{L}g = -\frac{d}{d\xi}\left[(1-\xi^2)\frac{dg}{d\xi}\...
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7 votes
1 answer
342 views

Monotonicity of Schrödinger Eigenvalues

Let us consider the Schrödinger operator $$ H_hf(x)=-\frac{d^2}{dx^2}f(x)+h(h\sin^2(x)-\cos(x))f(x) $$ on $L^2[-\pi,\pi]$ with Neumann boundary conditions $f^\prime(\pm\pi)=0$. Here, $h\geq 0$ is a ...
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2 votes
0 answers
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1D Schrödinger Equation with Measure-Valued Coefficients

I've been looking at one of the simplest systems I can think of: a one-dimensional infinite square well on $[0,1]$ with Hamiltonian given by the following: $$\hat{H}=-\frac{\hbar^2}{2m}\frac{d^2}{dx^2}...
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2 votes
1 answer
195 views

Eigenvalues Sturm-Liouville Operator

Is the eigenvalue decomposition of the Sturm-Liouville operator $$ Lf(x)=-f''(x)+h\sin(x)f'(x),\quad h>0, $$ with Neumann boundary conditions $f'(-\pi)=f'(\pi)=0$ on the Hilbert space $L^2([-\pi,\...
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4 votes
1 answer
126 views

Monotonicity/Scaling of Sturm-Liouville Eigenvalues

Consider the regular Sturm-Liouville eigenvalue equation $$ \frac{d}{dx}(p_t(x)f^\prime(x))=\lambda_t f(x) $$ for $p_t\in\mathcal{C}^\infty([0,1])$ with Dirichlet boundary conditions on $[0,1]$. Here $...
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2 votes
0 answers
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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+...
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4 votes
2 answers
410 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 $...
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0 answers
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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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2 votes
0 answers
65 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{...
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1 vote
1 answer
331 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 ...
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1 vote
1 answer
113 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')' = \...
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1 answer
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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 \...
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3 votes
0 answers
87 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 ...
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3 votes
1 answer
137 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 ...
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2 votes
1 answer
299 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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0 answers
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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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6 votes
3 answers
679 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(...
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2 votes
0 answers
68 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 ...
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3 votes
2 answers
354 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 ...
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1 vote
1 answer
312 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(...
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5 votes
1 answer
446 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^{...
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0 votes
1 answer
323 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\,...
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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}{\...
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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
3k 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 ...
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4 votes
1 answer
371 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 ...
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