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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13 views

Invariance and Equivalence of extrema under change of variables in infinite-dimensional spaces

In finite dimensionsial spaces, the optimization problem $$ \text{min}_{x \in \mathbb{R}^n} f(x) \quad \text{ s.t. } ||x||_2 = 1 $$ for $f: \mathbb{R}^n \to \mathbb{R}$ is equivalent to $$ \text{min}_{...
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reference request: hormander paper

Did someone know where to find the paper: On the Riesz means of spectral functions and eigenfunction expansions for elliptic differential operators of Lars Hormander?
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What type of kernel has eigenfunctions with orthogonal derivatives (Mercer's theorem)

Suppose $K(x,y)$ is a continuous non-negative definite function, where $x,y\in[0,1]$. By Mercer's theorem, $$K(s,t)=\sum _{j=1}^{\infty }\lambda _{j}\,e_{j}(s)\,e_{j}(t)\,,$$ where the $e_j$'s form an ...
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23 views

Sturm Liouville Problem Transcendental Equation

I am trying to solve the following problem $$ X''(x)+ \lambda X(x)=0$$ $$ X'(0)+2X(0)=0$$ $$ X'(1)=0$$ Show that $$ \tan\left( \sqrt{ \lambda } \right)= -2/ \sqrt{ \lambda } $$ With the ...
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72 views

Which of the statements is not necessarily true?

Let $A$ be a $3\times3$ matrix and $u, v, w$ be linearly independent vectors in $\mathbb{R}^3$ such that: $Au = 2u, Av = 2v, Aw = 0$. Which of the statements are NOT necessarily true? Option 1: $w$ is ...
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33 views

How to find eigenfuctions of shifts operator and dilation operators

I am trying to find eigenfunction of shift operators and dilation operators. Shift operators $S_t$ maps function $f(x)$ to $f(x+t)$. I am trying to find continuous function $f$ such that $(S_tf)(x) = \...
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Poisson equation in a cylinder

I need to solve the problem $\nabla^{2} u(r,\theta,z)=Q(r,\theta,z)$ inside a circular cylinder $(0 < r < a, 0 < \theta < 2\pi, 0 < z < H)$ subject to $u = 0$ on the sides. I'm ...
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28 views

Eigenfunction for solving wave equation

$u(x,t)=\sum_{n=1}^\infty \cos\frac{(2n-1)\pi x}{2L}(c_{1n} cos\frac{(2n-1)\pi t}{2}+c_{2n} sin\frac{(2n-1)\pi t}{2})$, $n=1,2,...$ $x\in[0,L]$ The ICs are $u(x,0)=p(0)x$, $u_t(x,0)=p'(0)x$ Applied ...
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Question in solving wave equation by eigenvalue-eigenfuntion

I am learning to solve PDEs, I found a example the following from a book, $u_{tt}(x,t)$=$u_{xx}(x,t)$ $u_x(0,t)=0$, $u(L,t)=0$ $u_x(x,0)=\sin\frac{3\pi x}{2}$, $u_t(L,t)=\cos\frac{3\pi x}{2}$. Now by ...
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Solving a Fredholm equation

I'm trying to solve this form of Fredholm equation: $$ g(v)=f(v)+\int\limits_{0}^{a} g(v_s)K(v,v_s)\mathrm{d} v_s, $$ where, $f, K$ is a given function $K(v,v_s)=K_1(v-bv_s)+K_2(v+bv_s)$, where $b$ ...
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56 views

Laplace eigenvalues on unit disc intermediate step

If I try for a separable solution $u(r,\theta)=R(r)\Theta(\theta)$ to the equation $\Delta u=-ku$ where $k\geq0$ I run into some problems before getting to the Bessel functions part. After separation ...
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When eigenfunctions equal adjoint eigenfunctions (for self-adjoint operators)

We are told that the eigenfunctions $y_k$ of a linear differential operator $Ly=\lambda y$ are equal to the eigenfunctions $w_k$ of the adjoint problem $L^*w=\lambda w$ when the operator is self-...
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55 views

Eigenfunction expansion of Gaussian kernel over a closed interval

$\newcommand{\Xc}{\mathcal{X}}$ Let $\Xc=[-1,1]$ and consider a Gaussian kernel $k(x,t)\propto \exp(-(x-t)^2/2\sigma^2)$ for some $\sigma>0$ on $\Xc$. I am looking for an eigenfunction expansion of ...
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Eigenfunctions for $K(x,t)=\log (1-\cos(x-t))$

I've done all parts except for the last one where I need to find $\lambda_n$ and $\phi_n(x)$. It is easy to see that for $n=0$ the kernel is just $K(x,t)=-\log2$ and then we get $$y(x)=\lambda \int_0^{...
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43 views

$K(x,t)=-2 \log \left( \sin \left( \frac{1}{2} \left( x-t \right) \right) \right)$ eigenvalues and eigenfunctions

Find the eigenvalues and eigenfunctions for the following kernel: $$K(x,t)=-2 \log \left( \sin \left( \frac{1}{2} \left( x-t \right) \right) \right) \in \mathcal{L}_2([0,2 \pi ]^2).$$ What I have: $$...
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36 views

Finding eigenvalues and eigenfunctions of differential equation

I need to write the following d.e. in Sturm-Liouville form and find the eigenvalues and eigenfunctions. $$\frac {\mathrm d ^2 y}{\mathrm d x^2} + 7 \frac {\mathrm d y} {\mathrm d x} + (e^{3x} + \...
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24 views

Find function for multiplication operator such that there are at least 2 eigenvalues

Consider $V=C([a,b])$ and let $A:V\to V: f(t)\mapsto u(t)f(t)$ be the multiplication operator with fixed $u(t)\in V$. Let $E$ denote the set of eigenvalues of operator $A$. Let $S$ denote the ...
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Eigenvalues of the operator $A = -v'' + B(x) v$

How can I prove that for the eigenvalues of the operator $$A := -v'' + B(x) v$$ on $(0,L)$ with zero Dirichlet boundary conditions the following holds $$ \left| \lambda_n - \frac{\pi^2n^2}{L^2}\right| ...
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Localization property of spherical harmonics on incomplete spheres

This question follows from another question I asked here. I am currently trying to read this paper and I am having difficulty in understanding the interpretations of eq. 7 which are given in lines ...
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Laplace eigenfunction on a polygonal domain symmetric about an axis

Consider a polygon $\Omega \subseteq \mathbb{R}^2$, and let us consider the usual Laplacian operator $\Delta = \partial_x^2 + \partial_y^2$, with Dirichlet boundary conditions. My question comes from ...
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eigenfunctions of $ -\epsilon u''(x) + u'(x) = \lambda u(x)\text{ with } u(x=0)=1,u(x=1)=0$

How do I find the eigenfuntions of the following boundary value problem $$ -\epsilon u''(x) + u'(x) = \lambda u(x)\text{ with } u(x=0)=1,u(x=1)=0$$ I tried it with the substitution of $u(x)=ce^{\mu x}$...
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Proving self-adjointness of operator $\mathcal{L}$ regarding eigenvalue problems.

Question: Define $X = \{y \in C^2\left( [a,b] \right) : y(a) = y(b) = 0\}$ Consider the operator $\mathcal{L}$ defined on X by: $\mathcal{L} := P(x)y'' + Q(x)y' + R(x)y \quad \forall y \in X$ Where $P(...
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Sturm Liouville problem - Showing there exists an infinite number of eigenvalues

I've got the equation $\phi''+2\phi'+\lambda \phi=0$ where $0<\phi<\pi$ with boundary conditions $\phi(0)=0$ and $\phi'(\pi)+2\phi(\pi)=0$ I've shown its a S-L problem and written the equation ...
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62 views

Spherical Harmonics on incomplete sphere [closed]

Let me start by saying that I am starting to fall in love with the spherical harmonics and analysis of functions defined on a sphere. I am a physicist studying Cosmology, so you can imagine I get to ...
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41 views

eigenfunction expansion to represent the best solution

Use the appropriate eigenfunction expansion (if it exists) to represent the best solution of the following problems. $$u''+ u =f(x)$$ with boundary condition $$u(0) = u(2\pi), u'(0) = u'(2\pi)$$ What ...
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2answers
45 views

Eigenvalues of an almost diagonal matrix [duplicate]

I know that the eigenvalue of a diagonal matrix is simply the values in the diagonal. However, if I have a matrix of the following form: $$ \begin{bmatrix} a & b & 0 & 0 \\ b & c & ...
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Expansion coefficient

I'm working through some derivations for a hydrodynamic stability textbook and I cannot follow the following derivation. The system is the following initial value problem $$\left[(\frac{\partial }{\...
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22 views

function must change sign between two successive zeros

$(\lambda_n-\lambda_m)\int^{x_2}_{x_1}\rho y_n y_mdx=[y_n\rho y'_m-y_m\rho y'_n]^{x_2}_{x_1}$ Deduce that if $\lambda_n > \lambda_m$ then $y_n(x)$ must change sign between two successive zeros of $...
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How to prove that eigenfunction of translationally invariant continuous operator $ K(t-t') $ is $ \exp(iwt) $?

I was studying a book in computational neuroscience where I came across the following equations: $$ \int{W(t,t')e(t')dt'}=\lambda e(t) $$ and read that if $ W(t,t')=K(t-t') $ then the eigenfunction $ ...
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32 views

How do I find the number of clients with a matrices application?

Four competing companies (A, B, C, and D) are competing in a finite market of 5000 clients. Each month, the clients can either renew their contract with the company they were with or switch companies, ...
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27 views

For matrix $A$ , find $m$, $n$ , $r$ in case that $mA^2 + nA + rI=0$

I have a matrix $A= \begin{pmatrix} 5 & 3 \\ 2 & 1 \end{pmatrix} $ and I should find $m$, $n$, $r$ in case that $𝑚A^2+nA+rI=0$ ($I$ is Identity matrix) . and after that find $A^{-1}$ with ...
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40 views

Sturm-Liouville problem with mixed BCs and even-odd eigenfunctions

My problem is a relatively straightforward form of Sturm-Liouville problem, in which the general solution is expressed as $$ y(\lambda,t) = A_1 y_1(\lambda,t) + A_2 y_2(\lambda,t), $$ where $y_1(\...
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How do I numerically solve for the spectrum of a second order differential operator with periodic coefficients?

I have a second order linear differential equation of the form $u'' + k^2\epsilon(x)u = k^2q^2u$. Here, k is a known parameter, $\epsilon(x)$ is a known periodic function and $q^2$ is basically the ...
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Find eigenvalues and eigenfunctions for integral operator in detail

Find eigenvalues and eigenfunctions for integral operator I was trying to do the same problem as this one. I have checked many times, but my first derivative ends it up with $𝜇𝑢′(𝑥)=x*u(-1)+x*u(1)$,...
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54 views

Find the eigenfunction of a differential operator

I'm very familiar with solving differential equations. I think I'm just struggling with the setup here because I've never done it with operators. I've always just been given the differential equation ...
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30 views

Finding eigenvalues and eigenfunctions

Find the eigenvalues and eigenfunctions of the problem $$ \begin{aligned} \phi^{\prime \prime}+\lambda^{2} \phi=0, & 0<x<a \\ \phi(0)-\phi^{\prime}(0)=0, & \phi^{\prime}(a)=0 \end{...
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How to solve the eigendifferential integral for a continuous eigenvalue spectrum?

I don't understand how to solve the following integral in order to obtain the reported solution found in the book "Fundamentals of atomic mechanics" written by Enrico Persico. The following ...
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1answer
33 views

Analogue of eigenvalues for matrix of polynomials?

Let $A(x)$ be a matrix with entries that are polynomials in say $\mathbb{Z}$. Suppose furthermore that $A$ is invertible. For any fixed $x$ we can find an eigenvalue of $A$ (in $\mathbb{C}$) and this ...
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44 views

Linear algebra with a transformation i think

Let $V$ be a finite-dimensional real vector space and let $P:V→V$ be a linear map such that $P^2 = P$. Which of the following must be true? $P$ is invertible $P$ is diagonalizable $P$ is either the ...
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1answer
166 views

Solving the heat equation in spherical polars with nonhomogeneous boundary conditions

Trying to find the series solution of $r\rho''+2\rho'-\lambda r \rho = f$ with certain ICs & BCs. Question: what is wrong with solution strategy (numerical check does not confirm result). My ...
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94 views

Eigenvalues of Schrodinger equation in cylindrical coordinates

I have Schrodinger equation of the form: $$ \left[\frac{\partial^{2}}{\partial{r}^{2}}+\frac{1}{r} \frac{\partial^{}}{\partial{r}^{}}+\frac{1}{r^{2}}\left(\frac{\partial^{}}{\partial{\varphi}^{}}+i\...
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1answer
52 views

Solving for Eigenvalues of Bessel like differential equations

How to solve for $\lambda$ and $R(r)$ in this Bessel like differential equation when $R(r) = 0 \ \forall \ r\geq r_{0}$. $$\frac{\partial^{2}R}{\partial r^{2}} + \frac{1}{r}\frac{\partial R}{\partial ...
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73 views

Simultaneous eigenstates and commuting operators

It is stated that commuting Hermitian (linear + self-adjoint) operators have 'simultaneous eigenstates'? Which of the following is most correct, and why? 2 commuting operators share AN eigenstate 2 ...
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31 views

$i^{th}$ eigenvalue of $\text{exp}A$

Suppose we have an $N \times N$ matrix $D$ which is a diagonal matrix, and suppose we raise it exponentially, i.e. $e^D$. Now, I know that to find the $i^{th}$ eigenvalue of this new matrix, I have ...
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41 views

Orthogonal Eigenbasis Proof Help

Let $D = d^2/dx^2$ and $V$ be the set of functions that are infinitely differentiable (and continuous), real, and $2\pi$-periodic. I've proven that $\langle Df, g \rangle = \langle f, Dg\rangle$, for ...
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2answers
92 views

How Different is an Eigenvalue Problem from an Ordinary Differential Equation

I have been thinking about how different an eigenvalue problem such as that of the Sturm-Liouville Equation (SL) with that of a second-order linear differential equation. It doesn't seem to be the ...
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1answer
91 views

Help: Orthogonal Eigenbasis of Differential operators related to Fourier series convergence theorem

Let $D = \frac{d^2}{dx^2}$ and $V$ be the set of functions that are infinitely differentiable, real, and 2$\pi$-periodic. I've found the following about $V$ and $D$: $D$ is symmetric: for any two ...
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1answer
56 views

Rephrasing statements in terms of Eigenfunctions, Eigenvalues and Eigenspaces of Differential Operator

Let $V$ be a set of functions that are infinitely differentiable, real, $2\pi$-periodic functions. Also, let $D = \frac{d^2}{dx^2}$ be the second derivative function that is a linear transformation. ...
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1answer
39 views

Why does the completeness of a set imply that functions can be built out of linear combinations of its elements?

(Most of my training is in physical sciences, so forgive my question if its trivial) I read in my course on partial differential equations that the set of eigenfunctions of a regular Sturm-Liouville ...
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38 views

Do we have eigenfunctions for Laplacian in $\Bbb{R}^n$ [duplicate]

Does there exist $u\in H^2(\Bbb{R}^n)$ so that $-\Delta u=\lambda u$. Of course if such $u$ exists, it must be a smooth function by elliptic estimate and bootstrap strategy. So we need to find $u$ so ...

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