Questions tagged [eigenfunctions]
For questions on eigenfunctions, each of a set of independent functions that are the solutions to a given differential equation.
712
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Uniqueness of radial eigenfunctions
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{ ...
- 511
1
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0
answers
18
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minimization problem for elliptic system
Consider the following minimization problem in $\mathbb{R}^N$:
$$
\lambda_1(R)：=\min_{(\zeta_i)_{i=1}^m \in H_0^1(B_R(0))}
\biggl\{\sum_{i=1}^m \int_{B_R(0)}|\nabla\zeta_i|^2 + \sum_{i,j=1}^m \int_{...
- 801
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69
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+50
Spherical harmonics of order $l$ with smallest $L_\infty$ norm
Let ${\mathbb H}_l$ be a space of spherical harmonics of order $l$, i.e.
$${\mathbb H}_l = \{f: {\mathbb S}^{n-1}\to {\mathbb R} \mid \Delta_{{\mathbb S}^{n-1}}f=-l(l+n-2)f\},$$
where $\Delta_{{\...
- 341
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0
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53
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Bounds on Sturm-Liouville eigenfunctions and its derivatives
Suppose we have the self-adjoint Sturm-Liouville problem
$$
(p(x)y')'+q(x)y=\lambda y \\
Ay(a)+By'(a)=0 \\
Ay(b)+By'(b)=0
$$
whose eigenvalues are $0<|\lambda_1|<|\lambda_2|<\cdots$ and $\{\...
- 587
2
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0
answers
125
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Eigenfunctions of the integral kernel 1/(x^2 + x'^2)
My question seems elementary, yet I could not find the solution after working on and searching for several days...
I'd like to find the eigenfunctions of a simple integral kernel:
\begin{equation}
\...
- 21
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0
answers
37
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How eigenvector evolves when columns of matrix multiplies?
$IX=AX$
So for the above equation, X is the eigenvector corresponding to the eigenvalues of 1.
Let's multiply each column of A with a positive number bigger than one and call the new matrix B where A ...
- 3
1
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1
answer
82
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General solution to $f(cx) = \lambda f(x)$
Which functions satisfy $f(cx) = \lambda f(x)$ where $c, \lambda$ are any constants?
The general solution for $c = \lambda$ is given in Steven Stadnicki's answer to this question: Is the function $\...
- 13
2
votes
1
answer
60
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Derivation of Complex Fourier Series coefficients through inner products (and swapping arugments)
I am trying to derive the complex Fourier series coefficients given by:$$f(x)=\sum_{n=-\infty}^{\infty}{c_n}e^{inx}$$ with coefficients:$$c_n=\frac{1}{2\pi}\int_{-\pi}^{\pi}{f(x)}e^{-inx}dx$$
I am ...
- 109
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0
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16
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$L^{\infty}$ norms of Robin eigenfunctions
Let $\Omega\subseteq \mathbb{R}^2$ be a bounded Lipschitz planar domain. Suppose that $u$ is a Robin eigenfunction of the (negative) Laplacian on $\Omega$: $-\Delta u=\lambda u$ with $\partial_{\nu}u+\...
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1
answer
58
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PDE - separation of variables in a rectangle
I'm trying to solve a PDE problem, and got this S-L problem
$ \nabla^2 u_2(x,y)=0,\,\, x \in (-1, 1),\,\, y\in (0, \pi) \\
u_2(x,0) = \cos(0.5\pi x),\,\, x\in [-1,1] \\
u_2(x,\pi) = 2\sin(\pi x),\,\, ...
- 3
2
votes
0
answers
25
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Solving an eigenvalue problem for 2nd order linear differential operator in a half-infinity square [closed]
I'm trying to solve the eigenvalue problem
$$(1+x^2+y^2)\left[(1+x^2)\frac{\partial^2f}{\partial x^2}+(1+y^2)\frac{\partial^2f}{\partial y^2}+2xy \frac{\partial^2f}{\partial x \partial y}+2x\frac{\...
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0
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22
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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 ...
- 1,213
1
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2
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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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0
answers
13
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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 ...
- 99
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0
answers
31
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Eigenvalues and Eigenfunctions of the Laplace operator an ellipsoid
I am currently trying to find the spectrum of the Laplace operator for ellipsoids in $\mathbb{R}^{3}$ with Dirichlet boundary conditions, i.e., I am looking for solutions to the following PDE,
$$
\...
- 75
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votes
1
answer
70
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Solving real life problem of gravity using Laplace's equation in polar coordinates
Suppose we have an empty space: then the gravitational potential will be equal for all points to 1 constant.
I want to solve the Laplace equation for such space in polar coordinates.
Let the ...
- 115
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1
answer
53
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Determine the values of a for which the eigenvalues are nonnegative for the given regular Sturm-Liouville problem.
I'm seeking assistance in figuring out the values of 'a' that would ensure the eigenvalues of a specific regular Sturm-Liouville problem are nonnegative. The problem is presented as follows:
Given the ...
- 68
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1
answer
53
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Helmholtz equation eigenvalue problem; without separation of variables?
Consider the eigenvalue problem Δφ=λφ
with the Dirichlet boundary condition
on the rectangle Ω=[0,1]×[0,1]
.
By using separation of variable method { φ(x,y)=f(x)g(y) }, I found that φ(x,y)=sin(mπx)...
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1
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What does it mean if there arent any eigenvalues and eigenfunctions?
I have been trying to solve this problem of eigenvalues:
$X''(x)+\lambda X(x)=0 , X(1)=0,X'(1)=0$ however I cannot find any eigenvalues or eigenfunctions.
What does this mean for $X(x)$?
- 115
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0
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37
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How to find the eigenfunctions of this PDE?
Consider the differential operator $$ \mathcal{L} = -A \partial_x f_1(x,y) - B \partial_y f_2(x,y) + \frac{A}{2} \partial_x^2 f_1(x,y) + \frac{B}{2} \partial_y^2 f_2(x,y)$$ where $f_1 = (x+1)(M-x-y)$, ...
- 123
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1
answer
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Why if $u$ satisfies $-\Delta u+tu=f$ and $\int_{X}\nabla u\cdot\nabla v=\lambda\int_{X}uv$ has non-trivial solutions $u$ then $-\lambda\leq c$
Let $X\subset\mathbb{R}^2$ be a bounded open set with either Dirichlet $u = 0$ or Neumann boundary $\frac{\partial u}{\partial \nu} = 0$ conditions on the boundary of $\Omega$. Let $V$ be either $H^...
- 725
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0
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59
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Finding the Green's function of a given PDE
I'm currently studying this article because I'm interested in the technique for finding the Green's function $G$ of the following PDE:
$$\frac{\partial B}{\partial t}+\frac{\sigma^2}{2}\frac{\partial^...
- 131
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0
answers
15
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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):
$$ j_0(u) = \frac{\sin(u)}{u} $$
...
- 127
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0
answers
30
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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 ...
- 560
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0
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102
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This linear elliptic PDE has no solutions
Let $(M^n,g)$ be a compact Riemannian manifold with boundary, with $\operatorname{dim} M = n \geq 3$. Denote by $\Delta$ the Laplacian operator acting on functions as $\Delta f = -\operatorname{div} \...
- 5,135
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Can the eigenvalue of an differential operator be a function itself?
My issue is this:
Let's say I have the operator $\hat{X} = \frac{d}{dx}$ and I want to find complex-valued functions $f:\mathbb{C}\rightarrow\mathbb{C}$ which satisfy $\hat{X} f = \lambda f,$ with $\...
- 93
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35
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Looking for the eigen-function of a system
How could I find the eigen-function of the following system.
Consider the following ODE system
$$\begin{bmatrix}\dot{x}\\\dot{y}\end{bmatrix} = \begin{bmatrix}0&1\\1&0\end{bmatrix}\begin{...
- 46
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votes
0
answers
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Must solutions to the time-independent Schrodinger equation that have discrete spectra or negative eigenvalues be square-integrable?
Consider the following version of the time-independent Schrodinger equation:
$$
\left( -\frac{d^2}{dx^2} + V(x) \right) \psi(x) = \lambda\ \psi(x)
$$
(where we have absorbed some unimportant physical ...
- 5,919
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Two types of eigenpairs in eigenproblem of PDE
I am trying to solve the following PDE using separation of variables:
$$\begin{cases}
u_{tt}=c^2u_{xx} & -l<x<l & t>0\\
u(-l, t)=u(l, t)=0 & t>0\\
u(x, 0)=\phi(x),\quad u_t(x, ...
- 1,069
1
vote
1
answer
101
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Green function of a forced undamped oscillator using two different methods don't match!
The differential equation for a forced undamped oscillator has the form $$\mathcal{L}x\equiv \frac{d^2x}{dt^2}+\omega_0^2x=f(t),$$ and the Green function $G(t,t')$ defined as $$\mathcal{L}G(t,t')\...
- 641
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0
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Solution to ODE in terms of eigenfunctions of differential operators
Can anyone please recommend a textbook or publication, where the theory of linear ordinary differential equations is explained from operator-theoretical perspective? That is, solutions to the ODEs are ...
- 11
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0
answers
73
views
Eigenfunctions of Laplacian on genus 0 Riemann surface
Consider a closed compact genus 0 Riemann surface $\Sigma$ on which we install a metric $g$. The eigenvalue equation for the scalar Laplacian is
$$
- \nabla^2 \psi_n(x) = \lambda_n \psi_n(x) , \qquad \...
- 600
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0
answers
78
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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 ...
- 198
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0
answers
66
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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 ...
- 41
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How does the constant in $\|u\|_{L^p} \leq C \|\Delta u \| _{L^p}$ depend on the smallest eigenvalue of the Laplacian?
Let $(M,g)$ be a Riemannian manifold, $\Delta$ be the (positive semi-definite) Laplacian on it, and $\lambda_1>0$ its smallest non-zero eigenvalue.
There exists $C>0$ such that
$$\|u\|_{L^p} \...
- 608
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1
answer
35
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How does a sine function solve an Euler equation (rSLP)?
I'm currently trying to solve an rSLP which looks like this (Euler equation):
$$x^2f''(x)+xf'(x)+cf(x)=0,\ f(1)=f'(2)=0,\ x\in [1,2] \text{ and }c\neq0.$$
It's simple that $f(x)=x^{\pm\sqrt{-c}}$ ...
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0
answers
41
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How to use Slater Type Orbitals as a basis functions in matrix method correctly?
This question is a continuation of my previous series of questions about basis functions.
I would like to find the minimum energy of Coulomb potential motion using matrix method.
$H=-\frac{1}{2}\Delta-...
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0
answers
26
views
Why does using different basis functions in the matrix method give different ground state energies?
I would like to find the ground state energy of the following Hamiltonian:
$$H=-\frac{1}{2}\Delta -\frac{1}{r}-\frac{1}{2r}e^{-1.5r}$$where $-\frac{1}{r}-\frac{1}{2r}e^{-1.5r}$ is a Potential energy.
...
- 121
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0
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Eigenfunction of $\Delta$ in $L^2(\mathbb{R}^d)$
I had this weird thought that should be not too difficult, but I can't seem to resolve. So, we know that $-\Delta$ is positive and self-adjoint on $H^2(\mathbb{R}^d)$, which implies that all of its ...
- 75
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votes
1
answer
87
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PDE with mixed boundary conditions
I am trying to solve this PDE with the following boundary and intial conditions: $$n_t(x,t) - an_{xx} = -bn(x,t) $$
$$n_x(0,t) = 0; n(L,t)=0$$
$$n(x,0)=n_o$$
where $0<x<L$ and $a,b,n_0, L$ all ...
- 33
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votes
1
answer
79
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Why are the Dirichlet eigenfunctions smooth
I am reading Spectral Theory (Chapter 6.3) by David Borthwick and having a question about how to argue the smoothness of eigenfunctions.
Consider the Dirichlet problem on a bounded open set $\Omega \...
- 597
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3
answers
121
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Physical interpretation of $-\Delta u = \lambda u$ on $\Omega$ with boundary conditions $u|_{\partial \Omega} = 0$
Let $\Omega$ be a (regular) domain in $\mathbb{R}^d$, let $\lambda \in \mathbb{R}$ and let $u \colon \Omega \to \mathbb{R}$ be a non-null (regular) function such that
$$-\Delta u = \lambda u \quad \...
- 5,503
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votes
0
answers
69
views
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 ...
- 211
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votes
0
answers
27
views
Evolution of the heat equation at different times
I have some questions about the following problem:
We have a heat function in a $1$D rode on the domain $[0,L]$ with a temperature field $u(x,t)$ and a diffusivity $\alpha$. Boundary conditions are: $...
- 211
1
vote
0
answers
49
views
Eigenfunction cannot have vanishing gradient along the boundary
Let $\Sigma$ be a compact Riemannian surface with boundary and let $f \in C^{\infty}(\Sigma)$ be an eigenfunction of the Laplacian which vanishes on $\partial \Sigma$. How do I show that $\nabla f$ ...
- 5,135
4
votes
1
answer
140
views
Having trouble understanding the solution of $\frac{d^{2}y}{dx^{2}}+2\frac{dy}{dx}+\lambda y=0,$ $y(0)=y(1)=1$
Good day.
I was working on this problem from my lecture notes.
"Find eigenvalues and corresponding eigenfunctions
for the BVP
$\frac{d^{2}y}{dx^{2}}+2\frac{dy}{dx}+\lambda y=0,$
$y(0)=y(1)=1$
and ...
- 296
4
votes
1
answer
303
views
Solving a nonlinear ODE
$$
y''-i(\sin(x)y)'-i\omega y-\lvert y \rvert^2y'=0 , \quad y\rvert_{x=0}=0 \quad y'\rvert_{x=0}=0
$$
Hello, im looking for advice on how to solve this equation, im intrested in knowing what possible ...
- 509
-2
votes
1
answer
118
views
Why does the n-th Sturm-Liouville eigenfunction have n-1 zeros?
I know this question is a duplicate of Proof the n-th Sturm-Liouville eigenfunction has n-1 zeros. however, the link in that question's comments is broken. Also, having a proper answer to this ...
- 13
0
votes
0
answers
58
views
Laplace - Beltrami eigen problem in 1D Riemannian manifold
Consider a 1D Riemannian manifold defined on the unit segment $[0,1]$ and equipped with an unknown metric $g=g(x)$ which is positive for all $x$.
In this case the Laplace-Beltrami operator $\Delta f= ...
- 173
2
votes
1
answer
107
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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. ...
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