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Questions tagged [eigenfunctions]

For questions on eigenfunctions, each of a set of independent functions that are the solutions to a given differential equation.

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Sign of the first eigenfunction of the Laplacian

I am trying to prove that the first eigenfunction of the Laplacian operator in an open domain $\Omega$ does not change sign and that the first eigenvalue $\lambda_1$ is simple (with Dirichlet-boundary ...
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How are measurement functions expanded in terms of Koopman eigenfunctions in the Koopman mode decomposition?

I am trying to understand the Koopman mode decomposition as presented in Modern Koopman Theory for Dynamical Systems - Brunton et al. Specifically, in page 16-17, the Koopman mode decomposition where ...
Nikos H.'s user avatar
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Are the eigenvectors of the generalised Laplace operator always periodic?

In $\mathbb{R}$ the eignevectors / eigenfunctions of the Laplace operator yield the fourier series, which, among other things, is made up of exclusively periodic functions. If you have a Riemannian ...
Makogan's user avatar
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Studying stability of pde

I have a problem with studying the stability of this PDE. $$U_t = U_{xx} + f(U).$$ Let $U^{*}(x)$ be a solution for this equation. As conditions we have ${U^{*}}'(x) > 0$ for $x < x_{0}$, ${U^{*}...
Dan's user avatar
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What are the spaces whose linear operators admit eigen"vectors"?

I know that linear operators over $n$-dimensional, $\mathbb{R}-$vector spaces admit $n$ eigenvalues (in the algebraic closure of $\mathbb{R}, \mathbb{C}$) and eigenvectors. The proof of this is just ...
algebroo's user avatar
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Orthogonality of solutions to an eigenproblem

Let $\Omega$ be a connected, closed region in $R^2$, with $\Gamma$ being its boundary. ($\Gamma$ is piecewise smooth and non-self-intersecting but may not be necessarily connected - there may be one ...
G_B's user avatar
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Asymptotics of negative laplacian eigenvalues

I’ve come across in a paper the fact that the eigenvalues of the negative Laplacian $-\Delta v_k = \lambda_k v_k$ (dirichlet BC, bounded domain in $\mathbb{R}^d$) follow the asymptotic relation $\...
fGDu94's user avatar
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Proof of the Orthogonality of Hermite Polynomials

My question is regarding the proof of the orthogonality of Hermite polynomials. Actually, it's not quite the Hermite polynomials: $$ \psi_n(x) = [\dfrac{1}{\sqrt{n} 2^n n!}]^{\frac{1}{2}} e^{-\frac{x^...
Hooman Puyandeh's user avatar
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2 answers
88 views

Eigenvalue problems $y''(x) + λy(x) = 0, 1 < x < 2, y(1) = y'(2) = 0$ [closed]

Consider the eigenvalue problem $y''(x) + λy(x) = 0, 1 < x < 2, y(1) = y'(2) = 0$. Given the fact that its eigenvalues are positive, find all eigenvalues $λ_n$ and the corresponding ...
starry41's user avatar
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Reconstruction of an operator given the eigenfunctions and eigenvalues

I am interested in operator theory, in particular if I know the sequence of eigenvalues $\{\lambda_n\}_{n=1}^\infty \subset \mathbb{R}$ and eigenfunctions $f_n \subset X$ of a self-adjoint ...
mathematurgist's user avatar
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Need help to solve Sturm-Liouville ODE of form $\frac{dy}{dx}\Big(p(x)\frac{dy}{dx}\Big) + (\lambda)w(x)y = 0$

The first bit of information is just to provide context for where the equation I need help with came from, if you are uninterested, you can just skip to equation (2). While solving a Poisson's ...
Researcher R's user avatar
6 votes
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108 views

Eigenvalues of likelihood ratio

Consider the following hypothesis testing problem: Under $H_0$: $(X,Y) \sim N(0,1)\times N(0,1)$, i.e. $X$ and $Y$ are independent standard normal. Under $H_1$: $(X,Y) \sim N\left(\begin{pmatrix} 0\\...
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The Green function of Schrodinger operator is positive for non-negative q(x)

Consider $\left(E_0\right): L y:=-y^{\prime \prime}+q(x) y=0$ with non-negative function $q \in C(\mathbb{R})$. Let $y_1$ be the solution of $\left(E_0\right)$ satisfying $y_1(0)=0, y_1^{\prime}(0)=1$;...
YuerCauchy's user avatar
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Sturm-Liouville boundary value problem with bounded function

$$ y'' + \lambda y = 0, \quad y'(0) = 0 \quad |y(x)| < \infty \quad \text{for all } y \text{ in } (0, \infty),$$ I have tried numerous Sturm Liouville Boundary Value problems, but never done ...
R_Squared's user avatar
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Is this example wrong and are the eigenfunctions orthogonal over any basis?

So I have this example for solving a PDE in one of my lecture courses and I can't figure out if it's wrong or there's something I'm missing? The question is as follows: PDE: $\quad u_t=a^2 u_{xx} \...
Ahmed Tayee's user avatar
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How do I prove this exact orthogonality of eigenfunctions?

So, I study photonics and there is a theme "Modal decomposition for 1D Phc". It has the following part: Maxwell equation for both polarizations read: $$\eta(z)\frac{d}{dz}\left(\frac{1}{\eta(...
illusive_mikser's user avatar
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Is the Complex Exponential the Only Eigenfunction of LTI Systems?

I've been studying Linear Time-Invariant (LTI) systems and came across the concept of eigenfunctions with respect to these systems. The literature often presents complex exponentials (as well as sines ...
CuriousMind's user avatar
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Eigenvalue problem, BVP with periodic conditions

Good Day, I am struggling to connect 2 parts of the lecture. Consider the Eigenvalue Problem with periodic conditions $$y''+ky = 0, y(0) = y(1), y'(0) = y'(1) $$ $$\text{Solution is in the form:} \\ y ...
SuperMage1's user avatar
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Transformation from Hypergeometric to Whittaker.

I'm working on an eigenvalue problem involving 2 operators, which we will call $\hat{A}$ and $\hat{B}$. They are related for very small values on a coordinate $x \approx 0$, in which $\hat{A}$ ...
MultipleSearchingUnity's user avatar
3 votes
0 answers
153 views

Inverse of kernel integral operator of Gaussian squared exponential kernel

I am doing research related Gaussian processes and Gaussian process regression. What I would like to know is the inverse of the integral operator of the squared exponential kernel in one dimension, $K ...
Alojaco's user avatar
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Is there a function $f$ such that $f(-x)=\sum_{k=0}^{+\infty}{\frac{f(k)}{k!} x^k}$?

You can rewrite this as the requirement that $(-1)^k f^{(k)}(0)=f(k)$ but I do not feel this helps much. I also saw some similarities with Ramanujan's master theorem/interpolation formula but I also ...
user146125's user avatar
3 votes
1 answer
158 views

Sturm-Liouville problem with singular weight

I have the following eigenvalues problem: $ xy'' + \lambda y = 0 $, $0 \leq x \leq 1$, $y(0)=y(1)=0$ that if we rewrite becomes $y''+ \lambda \frac{1}{x} y = 0$, a Sturm-Liouville problem with weight $...
aaa6's user avatar
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1 answer
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Completeness of system of eigenfunctions of Hermitian and Sturm-Liouville operators

My professor in the course "Math for Physicists" mentioned that the eigenfunction of a Hermitian operator or a Sturm-Liouville operator is an orthonormal basis for the function space on ...
R24698's user avatar
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Reference for boundary regularity of Neumann eigenfunctions

Let $\Omega\subseteq\mathbb{R}^n$ be a bounded domain with piecewise $C^{\infty}$ boundary. I have seen it implicitly used in several results that Neumann eigenfunctions of the Laplacian on $\Omega$ ...
Lawford Hatcher's user avatar
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1 answer
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Estimating eigenvalues of second order linear ODE

Suppose an ODE is given by a linear operator $L$ where $$L = P(x)\frac{d^2}{dx^2} + Q(x) \frac{d}{dx}$$ and we would like to find the eigenvalues of this operator so that $$Ly = \lambda y.$$ I am ...
CBBAM's user avatar
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Identifying eigenvalue and eigenfunctions of Sturm Liouville Problem

Here is the problem: $-y''=\lambda y$ with boundary conditions of $y(0)=0$ and $y(\pi)=-y'(\pi)$ My attempt is below: General solution is $$y(x) = a \sin(\sqrt\lambda x)+b\cos(\sqrt\lambda x).$$ ...
raspberry_flapper's user avatar
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Brezis' exercise 8.30.8: how to find the eigenvalues of this self-adjoint compact operator?

Let $I$ be the open interval $(0, 1)$. Let $k \in \mathbb R \setminus \{1\}$. We consider the space $$ V := \{v \in H^1 (I) : v(0) = kv(1)\}, $$ and the symmetric bilinear form $a$ defined on $V$ by $$...
Akira's user avatar
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$L^\infty$ bound of an operator on Fourier series or eigenfunction series

Let $f\in C_0^\infty(T^d)$, $T$ is a 1-D torus("$0$" here means that integral of $f$ is $0$). $f=\sum_n a_n\phi_n$, where $\{\phi_n\}$ are $e^{i\vec{n}\cdot\vec{x}}$. $\lambda_n=4\pi |\vec{n}...
MikeMichael_maths's user avatar
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Need help derivation of Fourier Coefficient for inhomogeneous BCs

We are dealing with Poisson's equation with inhomogeneous BCs and (according to the text), any geometry. $$\nabla^2 u = Q(x,y); 0 \le x \le L,0\le y \le H \\ u = \alpha \mid x,y\in\partial\Omega$$ The ...
Researcher R's user avatar
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84 views

Solving a Laplace Equation on a Semi infinite Strip

I am currently working semi-infinite strip question that requires the use of separation of variables. I inserted $q(x,y)=X(x)Y(y)$ into the Laplace equation provided in the image to then get $X''(x)Y(...
Jake Bynum's user avatar
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Functional Analysis - Finding the eigenvalues and corresponding eigenfunctions of a linear differential operator

I've been on this question for ages now, and eventually I came to this solution, however I'm very unsure on if I'm correct or if my approach is right. Question: Let $\hat{H}=\frac{d^2}{dx^2}-2\frac{d}...
cillianlynch_'s user avatar
1 vote
1 answer
88 views

Using method of eigenfunction expansion to solve a second order steady-state PDE. Does what I have look correct so far?

I was following the process in Richard Haberman's Applied Partial Differential Equations, pgs. 388-389, to solve a PDE of the form $\nabla^2 u(x,y) = Q(x,y)$ which was subjected to homogeneous BCs ...
Researcher R's user avatar
1 vote
1 answer
97 views

Eigenfunction Uniqueness Theorem

Consider the following Sturm-Liouville boundary value problem: $\text{Given parameters }c > 0 \text{ and } \beta > 0, \text{ let } y=y(x) \text{ for } 0 \leq x \leq c. \text{ We have }$ $$y''+ \...
RungSoup's user avatar
1 vote
2 answers
90 views

Help Understanding Sturm-Liouville Boundary Value Problem

Consider the following Sturm-Liouville boundary value problem: $\text{Given parameters }c > 0 \text{ and } \beta > 0, \text{ let } y=y(x) \text{ for } 0 \leq x \leq c. \text{ We have }$ $$y''+ \...
RungSoup's user avatar
1 vote
0 answers
26 views

Do analytic eigenvalue branches of the Robin problem converge?

Consider the Robin Laplacian eigenvalue problem on a bounded domain $\Omega\subseteq \mathbb{R}^2$: $-\Delta u=\lambda u$ with $\partial_{\nu}u+\alpha u=0$ on $\partial\Omega$. It is well known that ...
Lawford Hatcher's user avatar
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38 views

Rotational invariance of Laplace-Beltrami eigenvalue problem

I am currently looking at the eigenvalue problems of the Laplace-Beltrami operator. Let $(M,g$) be a smooth oriented Riemann manifold. I am investigating the eigenvalue problem of the Laplace-Beltrami ...
SebastianP's user avatar
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A guess about first eigenfunction and scalar curvature on Riemannian 2-sphere

For any Riemannian 2-sphere $(S^2,g)$, $R$ is its scalar curvature, and $R_A$ is the average value of $R$. $f$ is the first eigenfunction of Laplacian. Then I guess that for $$ \int_{S^2} (R-R_A) f^2 \...
Enhao Lan's user avatar
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2 votes
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118 views

Geodesics and Laplace-Beltrami eigenfunctions

For a smooth, closed Riemannian 2-manifold $M \subset \mathbb{R}^3$, is there a relationship between the geodesics of $M$ and smooth functions $k : M \times M \to \mathbb{R}$ which can be expressed in ...
p__.b's user avatar
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dimension of orthogonal complement of complete orthonormal sets

I'm thinking about the following problem and get stuck. I consider a complete orthonormal set $\{\psi_i\}^{\infty}_{i=0}$ defined on $\mathbb{R}$, and the inner product is defined by $$(\psi_i,\psi_j)=...
xfireskyx's user avatar
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0 answers
113 views

Eigenvalue problem- (Fourier Collocation Method)

I have an eigenvalue problem $$ \mathbf{L} \Psi=\lambda \Psi, $$ where $$ \mathbf{L}=i\left(\begin{array}{cc} G_0 & \nabla^2+G_1 \\ \nabla^2+G_2 & -G_0 \end{array}\right), \quad \Psi=\left(\...
egc's user avatar
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2 votes
1 answer
83 views

If a matrix contains a differential operator, can it have eigenvalues? If yes, how can I determine them?

If a matrix contains a differential operator, can it have eigenvalues? If yes, how can I determine them? I will assume that the answer is yes and try to solve the following problem. Consider the ...
FriendlyNeighborhoodEngineer's user avatar
1 vote
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43 views

Computing Spectrum of Linear Differential Operator non-constant Coefficients

I am trying to solve directly (numerically if direct methods are not possible) for the spectrum of the differential operator \begin{align*} & \mathscr{L}v \equiv (D\partial_{\xi\xi} -c \...
vlovero's user avatar
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Applying WKB Method for a Fourth Order Schrodinger Like Equation

I was trying to apply the WKB method to analyze the approximate eigenvalue condition for the Schrodinger like equation \begin{equation} \varepsilon^{4} y^{(4)} = (E-V(x))y, \quad y(\pm \infty) = 0, V(...
theo's user avatar
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1 vote
1 answer
79 views

Uniqueness of radially symmetric eigenfunctions

Given a smooth radially symmetric potential $V = V(|x|)$ in the unit ball $B_1\subset\mathbb{R}^n$, consider the eigenvalue problem \begin{equation} \begin{cases} \Delta \varphi + V \varphi = \lambda \...
Ignatius's user avatar
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0 answers
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minimization problem for elliptic system

Consider the following minimization problem in $\mathbb{R}^N$: $$ \lambda_1(R):=\min_{(\zeta_i)_{i=1}^m \in H_0^1(B_R(0))} \biggl\{\sum_{i=1}^m \int_{B_R(0)}|\nabla\zeta_i|^2 + \sum_{i,j=1}^m \int_{...
Stephen's user avatar
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249 views

Spherical harmonics of order $l$ with smallest $L_\infty$ norm

Let ${\mathbb H}_l$ be a space of spherical harmonics of order $l$, i.e. $${\mathbb H}_l = \{f: {\mathbb S}^{n-1}\to {\mathbb R} \mid \Delta_{{\mathbb S}^{n-1}}f=-l(l+n-2)f\},$$ where $\Delta_{{\...
qwerty43's user avatar
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70 views

Bounds on Sturm-Liouville eigenfunctions and its derivatives

Suppose we have the self-adjoint Sturm-Liouville problem $$ (p(x)y')'+q(x)y=\lambda y \\ Ay(a)+By'(a)=0 \\ Ay(b)+By'(b)=0 $$ whose eigenvalues are $0<|\lambda_1|<|\lambda_2|<\cdots$ and $\{\...
Lazward's user avatar
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2 votes
0 answers
154 views

Eigenfunctions of the integral kernel 1/(x^2 + x'^2)

My question seems elementary, yet I could not find the solution after working on and searching for several days... I'd like to find the eigenfunctions of a simple integral kernel: \begin{equation} \...
Yuli Nazarov's user avatar
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39 views

How eigenvector evolves when columns of matrix multiplies?

$IX=AX$ So for the above equation, X is the eigenvector corresponding to the eigenvalues of 1. Let's multiply each column of A with a positive number bigger than one and call the new matrix B where A ...
Ali K's user avatar
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1 vote
1 answer
108 views

General solution to $f(cx) = \lambda f(x)$

Which functions satisfy $f(cx) = \lambda f(x)$ where $c, \lambda$ are any constants? The general solution for $c = \lambda$ is given in Steven Stadnicki's answer to this question: Is the function $\...
psychgiraffe's user avatar

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