# 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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Given a smooth radial potential $V$ in the unit ball $B_1\subset\mathbb{R}^n$, consider the eigenvalue problem \begin{equation} \begin{cases} \Delta \varphi + V \varphi = \lambda \varphi & \text{ ...
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### Continuous dependence of eigenvalues for Sturm-Liouville problem

Let's say I have a certain Sturm-Liouville problem with Dirichlet initial conditions of the form $$(p(x) y'(x))'+q_c(x)y(x)=\lambda w(x) y(x), \quad y(a)=y(b)=0$$ where the function $q_c$ is a ...
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### Finding the spectrum of a Sturm-Liouville problem

I have the following Sturm-Liouville problem for $0 \le x\le \pi$ $$y'' + \lambda y=0, \qquad y(0) = 0, \qquad y(\pi)+y'(\pi) = 0$$ How do I find the spectrum of the problem? And how will the ...
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### Compact operator on a orthonormal sequence

Let $\Omega$ a bounded domain in $\mathbb{R}^n$ and let $A$ a compact self-adjoint and positive operator defined from $L^2(\partial\Omega)$ to itself. Let $\lambda_n$ the decreasing sequence of ...
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### Orthogonal function to find for transient diffusion equation in spherical coordinates

I have a question on eigenvalue and eigenfunction in spherical coordinates. Let $0 \leq u \leq 1$, and a function $j_0(u)$ (zero order spherical Bessel first kind): $$j_0(u) = \frac{\sin(u)}{u}$$ ...
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### Reference request: Location of maximum of Laplacian eigenfunction

Background: Let $U\subset \mathbb{R}^n$ be a bounded convex domain. By classical results, the (signed) Laplacian $-\Delta$ on $U$ with zero boundary conditions admits $L^2(U)$-orthonormal ...
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### Proving continuity of eigenfunction

I have a matrix that looks like a discrete Laplacian. Now I can plot the entries of the eigenvector by their index on the x axis and value on the y-axis. Now increasing the size of the matrix makes ...
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### Differential equation - Laplace with Dirichlet condition on a rectangle

In a rectangular domain $R$ with sides $a$ and $b$, with $b^2/a^2$ irrational, we look at the differential equation $- \Delta u = \lambda u$ with Dirichlet boundary conditions. How to show that the ...
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### Weight for orthogonal eigenfunctions

Is there an easy way to find the weight to make eigenfunctions orthogonal? (by hand and with an "easy" logic process, no numerical method type stuff) Let's move on to an example taken from ...
### How to compute $\partial^2_{n,n} u(x_0)$ for $x_0\in\partial\Omega$?
Here is the setting: Let $\Omega\subset\mathbb{R}^2$ is a bounded smooth strictly convex domain, $\mu_1(\Omega)$ is the first positive Neumann eigenvalue and $u$ is the corresponding eigenfunction. ...