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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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What's the name of this “shooting”-like method for numerical solving of PDE eigenproblems?

Consider a PDE of the following form: $$\left(\partial_r^2+\frac5r\partial_r+\frac4{r^2} \hat L\right)\Psi(r,p)+(E- V(p)U(r))\Psi(r,p)=0,\tag1$$ where $\hat L$ is a differential operator ...
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Weird eigensystem of ODE with regular singularity

Consider the eigenvalue problem of the following 2nd-order ODE $$(x/2+a)^2y(x)-xy'(x)-x^2y''(x)=\lambda^2y(x),$$ in which $y\in(-\infty,+\infty)$ and parameter $a>0$. It has a regular singularity $...
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1answer
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Determine the normalised eigenfunctions for the BVP: $y''+λy=0, y(0)=0, y(1)=0$

Solving it I get: $y(x)=c_1 \cos(x \sqrt{\lambda}) + c_2 \sin (x \sqrt{\lambda})$ $y(0)= C1 + 0 = 0, C1=0$ $y(1)=0+C2\sin(\sqrt{\lambda})=0$ So, $(\sqrt{\lambda})=n\pi$, $({\lambda})=(n\pi)^2$ So, ...
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1answer
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Distinct eigenvalues implies $A \in \mathbb{R}^{n \times n}$ is diagonalisable

Theorem: If an $n \times n$ matrix has n distinct eigenvalues then A is diagonalisable. Proof: Let $A \in \mathbb{R}^{n \times n}$. Suppose A is not diagonalisable. Then, by definition, for a ...
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An upper bound of the first eigenvalue of Laplacian on a Riemannian manifold.

I'm reading the Cheng's thesis ""Eigenvalue Comparison Theorems and Its Geometric Applications," and the author obtains an estimate of eigenvalues of the Laplacian based upon his theorem: If $M$ is $...
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124 views

Legendre's Equation, sturm liouville - eigenvalues/eigenfunction

Linear Differential Equation,Legendre's Equation, sturm liouville - eigenvalues/eigenfunction Consider the linear differential operator: $$ L = \frac{1}{4}(1+x^2)\frac{d^2}{dx^2}+\frac{1}{2}x(1+x^...
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1answer
49 views

Legendre Eigenvalue problem

I have the eigenvalue problem, $\frac{d}{dx}\big((1-x^2)\frac{du}{dx}\big)+\lambda u=0$, on $[-1,1]$ subject to single boundary condiction $u(-1) = u(1)$. Assume that there is an eigenfunction of the ...
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2answers
82 views

Eigenvalue of a given operator

If $u_0$ is a positive radial symmetric nontrival solution of $$ -\frac{1}{2}\frac{d^2u}{dx^2}+\lambda u -u^3=0 $$ Then how to show $-3\lambda$ is a eigenvalue of $$ Lu=-\frac{1}{2}\frac{d^2u}{dx^2}+...
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1answer
71 views

Finding eigenfunctions and eigenvalues from a differential equation

Consider the differential equation $$X''(x)+\lambda X=0$$ on $0 \leq x \leq 1$with boundary conditions $$X'(0)+X(0)=0 \ \ \ \ \text{and} \ \ \ \ X(1)=0.$$ I have a few problems here that I think ...
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Addition property of Laplace-Beltrami eigenfunctions in symmetric spaces

Consider the eigenvalue equation for the Laplace-Beltrami operator on a manifold with metric $ds^2=|K|^{-1}[d\chi^2+\sin_K^2\chi(d\theta^2+\sin^2\theta\,d\phi^2)]$, where: $$\sin_K\chi=\left. \begin{...
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85 views

Eigenvalue problem; second order differential equation.

I have arrived at following differential equation $\psi^{''} + (x^2 - E/x + E^2) \psi =0$, where $E$ is a constant. Is it possible to recast this equation as an eigenvalue problem, that is: $\psi^{'...
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34 views

Neumann Laplace eigenfunctions

Let $u_k, u_m$ be two Neumann Laplace eigenfunctions on a bounded domain $\Omega \subset \mathbb{R}^n$ with smooth boundary $\partial \Omega$, corresponding to eigenvalues $\mu_k, \mu_m$ respectively. ...
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Understanding the product of Normal Random Variable and Eigenfunction

Consider a symmetric function (Mercer Kernel) $K : T \times T \rightarrow \mathbb{R}$ and define an operator $H_k: L^2(T,\nu) \rightarrow L^2(T,\nu)$ where $H_kf(x) = \int_X K(x,y)f(y)d\nu(y)$. ...
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Eigenfunction of Dirichlet Laplacian on smooth domain in $\mathbb{R}^n$

I was reading about eigenfunctions of the Dirichlet Laplacian on bounded domains $\Omega \subset \mathbb{R}^n$. It seems that such eigenfunctions are real analytic in the interior of $\Omega$ and ...
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1answer
112 views

Find eigenvalues and eigenfunctions for integral operator

I'm trying to find the eigenvalues and eigenfunctions for the integral operator $Ku=\displaystyle \int_{-1}^1 (1-|x-y|) \,u(y) \, dy$ Since I want to find $\mu,u$ such that $Ku=\mu u$, we get the ...
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2D Elliptic Eigenproblem

Consider the elliptic eigenproblem \begin{align} \nabla^2\phi&=0 \ \ \ \ \ \ \ \ \text{in $\Omega$}\\ \frac{\partial\phi}{\partial r}&=\lambda\phi \ \ \ \ \ \text{on $\Gamma_1$} \\ \phi&=0\...
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1answer
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Eigenfunctions Laplacian - Bounding the Fourier Coefficients

Let $\Omega \subset \mathbb{R}^{N}$ be an open set with boundary of class $C^{\infty}$ and let $\{\lambda_{k}\}$ and $\{v_{k}\}$ be the eigenvalues and eigenvectors of -$\Delta$ with Dirichlet ...
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2answers
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What is the error I am making on getting general solutions for this Sturm-Liouville problem?

Given this Sturm-Liouville problem: $$X'' + \lambda X = 0$$ There are general solutions (Eigenfunctions) for three cases on $\lambda$: $$\lambda > 0$$ Has the characteristic equation: $r^2+\lambda ...
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Orthogonal polynomials as eigenfunctions of a second-order difference operator

I am Reading the theorem 6.1.3 of this book https://books.google.com.mx/books?id=RusIDAAAQBAJ&pg=PA146&lpg=PA146&dq=up+to+normalization,+the+charlier,+krawtchouk,+meixner,+and+chebyshev%E2%...
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1answer
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SO(n) as a manifold

I cannot find some basic information on $SO(n)$ ($n$ general, not just 3) as a manifold: what is the geodesic distance between two matrices, what are the eigenfunctions and eigenvalues of the Laplace-...
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Eigenvalue/function of Laplace Operator for Exterior of Disk - Reference Request

I'd like to know more about, $$\lambda u = \nabla ^2 u$$ for unbounded domains (particularly the exterior of a disk in $\mathbb{R}^2$ or ball in $\mathbb{R}^3$), but have had a hard time finding ...
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What is the relation between separation of variables and the eigenfunctions and eigenvalues for PDEs?

Studying Fourier Series and its application of solutions for Partial Differential Equations, in particular (historically) for the heat equation, one starts by separating variables. Somehow related to ...
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find the eigen function and eigen value of differential operator

I have an operator, defined in the cylindrical coordinate system with cylindrical symmetry, given by: $\frac{\partial^2}{\partial r^2}+ \frac{\partial}{r\partial r} $ I would like to find the ...
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1answer
59 views

How do I compute the eigenfunctions of an operator that contains another operator?

Given the operator $A = (X\frac{d}{dx}+2)$, where $X$ is a linear operator, how can I find the eigenfunction of $A$ corresponding to a zero eigenvalue? In general, this is just a matter of solving ...
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1answer
56 views

Need Help Interpreting the Sturm-Liouville Operator

I am given the following "Sturm-Liouville Problem with Operator $\mathcal{L}$ ": $$\mathcal{L}_{SL}=-\frac{1}{x}\left[\frac{d}{dx}\left(x\frac{d}{dx}\right)-\frac{1}{x}\right]$$ which is defined on ...
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Finding the eigenfunctions of an operator with multiple derivatives

The differential operator is given by $L= x\cdot \frac{d^2}{dx^2} + \frac{d}{dx} -\frac{a}{x}$ Any advice or strategies to find the eigenfunctions of this operator would be greatly appreciated. I ...
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Eigenfunctions of PDE

I came across the following eigen-function problem $$ \begin{align} (a(x,t)\nabla + b(t)\Delta)f =& \lambda f\\ f(x)=&0 (\forall x \in \partial \Omega),\\ f(x)>&0 (\forall x \in \Omega),...
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PDE Problem on Eigenvalues and the Heat Equation

Let $a$ be any fixed real number. Consider the eigenvalue system: $$ \begin{cases} X'' + \lambda X = 0, & 0 ≤ x ≤ 1\\[0.1cm] aX(0) = X(1)\\[0.1cm] aX'(0) = -X'(1) \end{cases}. $$ Prove that if $...
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2answers
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Expanding the function $f(x) = 1, 0 < x < 1$ in a series of the eigenfunctions?

I have a problem that asks me to expand the function $ f(x) = 1, 0 < x < 1$ in a series of the eigenfunctions of the given problem. For example, one given problem is $y'' + 2y' + ( \lambda + 1)y ...
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1answer
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Since $\phi_n$ and $\phi_m$ are eigenfunctions, they must satisfy the ODE …

I came across this: Theorem 1 The eigenfunctions of Sturm-Liouville BVP above satisfy the integral relationship: $$\int_a^b r(x)\phi_n(x) \phi_m(x) \ dx = 0$$ if $m \not= n$, where $\...
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3answers
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Finding eigenfunction to this operator

I have the operator $$ A : -e^{-2ax} \frac{\partial}{\partial x} \left(e^{2ax}\frac{\partial}{\partial x}\right)\\ D_A = \left( v \in C^2[0,L] \quad | \quad v(0) = v(L) = 0 \right) $$ ...
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how to make use of t' in tx = Ax vs general case

I ran into this problem on my differential equations homework set. Previously, the questions were in the form x' = Ax, and they were relatively straight forward and easy to complete. I was gifted a ...
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4answers
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Find eigenvalues & eigenvectors for an integral.

Can anyone please explain me how to solve it? Find the nonzero eigenvalues and the corresponding eigenvectors: $T:[-1,1]\rightarrow[-1,1]$ $$T((f(x))=\int_{-1}^1(x^2 y + y^2 x) f(y) \, dy$$
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1answer
103 views

How to find eigenvalues and eigenfunctions of simple-looking differential operator

What process can be used to solve for the eigenvalues and eigenfunctions of the following differential operator? $$H=A\left(\frac{\partial^2}{\partial x^2}+\frac{\partial^2}{\partial y^2}\right)+\cos(...
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Fixed points of cubic transformation

Let $f: \mathbb{R}^n \rightarrow \mathbb{R}^n$ be a Lipschitz continuous operator and let fix$(f)$ denote the set of fixed points of $f$. Define the operator $g = (1-a) f + a(1-b) f^2 + a b f^3$, for ...
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Operator acting on a product of functions

Let $f \in \mathcal{H}$ be a function $f: \mathbb{R} \rightarrow \mathbb{C}$ in a Hilbert Space $\mathcal{H}$. Now suppose that for some linear operator $T$ that acts on $\mathcal{H}$ and for some $f \...
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Solve this integral equation.

Let $h(x)$ be a known well-behaved function, I have to solve for $\sigma(t)$: $$ \phi(x) = \int_a^b\log\left[\left(x-t\right)^2 + \left(h(x) - h(t)\right)^2\right]\sigma(t)dt $$ Where, $b>a>0$, ...
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Cylindrical Operator

Maybe you can help me solve this (simple?) problem I'm too stupid to tackle :-( I want to find the eigenfunctions to the Operator $$ \widehat{O} = -\partial_z^2 - \frac{1}{r}\partial_r r\partial_r $$ ...
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1answer
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Eigenvalues of second order ordinary differential equation

The question is find all the eigenvalue of the following equation $-\frac{d^2y}{dx^2}+x^2y=\lambda y$ I have found the first function which is $y=e^{\frac{-x^2}{2}}$, however I have no clue on how ...
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Eigenvectors of discrete Laplace matrix for 2D unit square under Neumann boundary condition

Eigenvectors of discrete Laplace matrix for 2D unit square with free boundary is simply $$ \phi(x,y)= \cos(\frac{\pi}{n} kx) \cos(\frac{\pi}{m} ly) $$ It is easy to see that its 2nd order derivative ...
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Scale-dependent isoperimetric inequalities (e.g. with heat kernel?)

Suppose $\Omega\subseteq\mathbb R^2$ is a compact, connected planar region with a smooth connected boundary $\partial\Omega$. Take $\lambda_0,\lambda_1,\lambda_2,\ldots$ to be the Laplacian spectrum ...
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Eigenvalue solution of a 2nd order ODE of a geophysical fluid dynamic problem

community. I am working on a project with a professor of mine and he suggested me to numerically solve a geophysical fluid dynamic instability ODE equation of a paper of Boccaleti et al. (2007) (...
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Eigenvalues of the Black-Scholes operator

The Black-Scholes operator is given by $$L_{BS}u(x) = \frac{1}{2}\sigma^2x^2\frac{\partial^2}{\partial x^2}u(x) + rx\frac{\partial}{\partial x}u(x) - ru(x)$$ I want to prove that this operator has ...
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1answer
31 views

How to find the eigenvalues of this operator?

The Question: Consider the system \begin{align} & Ly(x) \equiv y''(x)+y(x)=f(x),\qquad 0<x<1 \\ & y(0)=1, \qquad y(1)=0 \end{align} (i) Find the eigenvalues and eigenfunctions of $L$ ...
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1answer
101 views

Solution Space of Schrodinger's Equation

We are given Hermitian operator of the form $H(x) =(-\hbar^2/2m) \partial^2/\partial x^2 + V(x)$ (where $\hbar$ and $m$ are real constants) which has orthogonal eigenfunctions corresponding to a ...
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The Proper Form of Eigenfunctions and Linear Operators

So I'm learning about eigenfunctions and eigenvalues and there appear to be 2 main forms 1) $L[y]=\lambda y$: This is intuitive to me as a direct extension of what I learned in Linear Algebra. 2) $L[...
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87 views

Trick to prove orthogonality for eigenfunctions?

Is there a magic trick to see that $$\int_0^LX_m(x)X_n(x)dx=0$$ for $$X_n(x) = c_n\Big[\big[\sin(\omega_n L) - \sinh(\omega_n L)\big]\big[\cos(\omega_n x) - \cosh(\omega_n x)\big] \\ - \big[\cos(\...
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1answer
89 views

Self-Adjoint Operator on Even Periodic Functions

I am attempting to show the following operator, which acts on even $2 \pi$-periodic functions, is self-adjoint and find its eigenfunctions and eigenvalues. $Ly=\frac{d^2y}{dx^2}, \:\: -\pi\leq x \leq ...
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1answer
62 views

How and why the eigenfunctions were computed like that?

In this question Understanding solution of PDE using method: separation of variables. (in which I put a bounty..) how and why the eigenfunctions were computed like that? The origin of the $T_n$ ...
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1answer
83 views

Find the eigenvalues and eigenfunctions of the operator

Find the eigenvalues and eigenfunctions of the operator $$Ly=\frac{d^2y}{dx^2},-\pi\le x \le \pi,$$ which operates on even-2$\pi$ periodic functions. I am unsure of where to start. Any help would ...