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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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Rayleigh-ritz method to estimate eigenvalues

I have a Sturm-Liouville problem as follows: $$u'' + \lambda u = 0 , 0<x<1$$ $$u'(0) = u(1) = 0. $$ It is asked to estimate the first two eigenvalues via Rayleigh-ritz method and compare to ...
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Eigenfunction Expansion and Fourier Series

What is the difference of Eigenfunction Expansion solution and Fourier Series solution in solving Partial Differential Equation? Are they the same? Please to tell me about what it is the eigen value ...
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Spectral Theory of Markov Jump Processes (continuous-time)

I stumbled over the eigenvalue problem while analysing an infinite-dimensional jump process. The state space I am working with is $$ \mathbb{N}^{<\infty}=\bigcup_{k=1}^{\infty}\mathbb{N}^k $$ i.e. ...
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If $f$ is in the span of eigenfunctions, then $|f|$ is also in the span of eigenfunctions.

Let $(X,\mathscr{B},\mu,T)$ be a measure preserving dynamical system. Then $U_T:L^2(X,\mu)\rightarrow L^2(X,\mu)$ defined by $U_Tf=f\circ T$ is an isometry. Let $\mathscr{E}$ be the eigenspace(closed) ...
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Eigenvectors and eigenvalues relational proofs

Let A ∈ R^ n×n How do I prove that "If A has a finite number of distinct eigenvectors then each eigenvector must have a distinct eigenvalue." If A a symmetric matrix in R n×n . A is called positive ...
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1answer
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Eigenfunctions of subgroup of Heisenberg matrices

Consider the subgroup of the Heisenberg matrices given by: $$ \mathcal{S} = \Big\{ \begin{bmatrix} 1 & 0 & z \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} : z \in \mathbb{R}/\...
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Method to find eigenfunctions?

I have a linear operator $L(p(x)) = -p(x+1) + p(x) + \frac6x \int_0^x p(y) dy$ with eigenvalues of it's representative matrix found to be 6, 3, and 2. From this the respective eigen vectors are (1,0,0)...
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How to find the optimal ${\bf Q}$ that maximizes $\left|{\bf I}+{\bf AQA^HB^{-1}}\right|$

I have the following question, which has stumped me for a long time. ${\max}_{\bf Q \succeq 0}~~~~{\rm det}\left({\bf I}+{\bf AQA^HB^{-1}}\right)$ s.t.$~~~{\rm Tr}\left({\bf Q}\right) \leq 1$. ...
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Generalization of Rayleigh quotient

Consider the function $\sigma(x,y) = \frac{x^T A y}{x^T y}$ where $x, y \in \mathbb{R}^{d\times 1}$ and $A\in \mathbb{R}^{d\times d}$. Also, its given that A is invertible with minimum eigenvalue $\...
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An Orthogonality Problem of Eigenfunctions of homogeneous Fredholm equation

Suppose we have a integral equation $$\int_{-1}^1 \frac{\text{sin }c(x-y)}{\pi (x-y)}\psi(y)dy=\lambda \psi(x),\quad|x|\le1.$$ By the Fredholm equation theory, we know that this equation has ...
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Show that for a regular S-L system, if q(t) is increased to q1(t) > q(t), each nth eigenvalue of the new system is larger than that of the old.

Consider the S-L equation $$\frac{d}{d t}\left[p(t) \frac{d u}{d t}\right]+[\lambda r(t)-q(t)] u=0$$ Show that for a regular S-L system, if $q(t)$ is increased to $q_{1}(t)>q(t),$ each $n$ th ...
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1answer
68 views

Eigenvalues and eigenfunctions of an integral operator

Let $T$ be an integral operator with kernel $K(x,y)=e^{|x-y|}$ on $L^2(-1,1)$. How can we find the eigenfunctions and eigenvalues of $T$? Even though I am not sure whether the following arguments are ...
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1answer
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Why is this eigenvalue problem solved by $\phi(r) = J_v(\alpha r)$

I can't seem to see how the bessel function $J_v(ar)$ solves the problem. The eigenvalue problem has an $\alpha^ 2\phi(r)$ term Ive tried writing $x = \alpha r$ in the expression for $J$ but cant ...
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Bound for eigenvalues of $y''-Kxy'$

Given the problem $y''-Kxy'=-\lambda y$, with $y'(-d/2)=y'(d/2)=0$, show that the smallest nonzero eigenvalue is bounded from below by $\sup_{s\in (0,1)}(4s(1-s)\pi^2/d^2+sK)$. The hint I have is ...
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How is the Laplace Transform a Change of basis?

This question is primarily based on the following answer's way of reasoning, https://math.stackexchange.com/a/2156002/525644 If you want to write a new answer to the question; "How is the Laplace ...
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Finding the domain shape, where the fundamental Helmholtz Eigenmode has certain properties at a boundary

The following is a problem about finding the shape of a domain such that the fundamental Helmholtz eigenmode has certain properties at the boundary of the domain. I search for an analytical or ...
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Finding eigenfunction

So i am trying to solve this problem. (solve by the method of eigenfn expansion for y´´=-x with y(0)=0 and y(1)+2y´(1)=0) When i tried to find the eigenvalues amd eigenfn of y´´ +λy=0 with y(0)=0 and ...
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1answer
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Discrete Ergodic spectrum

$\textbf{Problem}$ An ergodic measure preserving transformation $T$ on $(X,B,\mu)$ is called to have ${discrete \ spectrum}$ if there exists an orthonormal basis for $L^2$ which consists of ...
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Show that for any eigenvalue we can find a real-valued eigenfunction (for the laplacian)

Let $\Omega \in \mathbb{R}^d$ be an open bounded set with smooth boundary. Consider $$-\bigtriangleup q(x) = \lambda q(x)$$ with either Dirichlet or Neumann boundary conditions $$q(x)=0, x \in \...
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A Homogeneous Fredholm Equation of Second Kind

in my probability research I encounter the following integral equation for continuous non-negative $f: (0,\pi/4] \to \mathbb R$: $$ f(\varphi) = \int_0^{\pi/4} \frac {4} {\pi} \sin \varphi_0 \cos \...
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1answer
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Eigenfunction expansion to solve non homogenous heat equation

I've been really struggling to figure out how to solve this problem using Eigenfunction expansion, I can solve it using seperation of variables. So this the problem is: $$ \begin{cases} u_t(x,t)=u_{...
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2answers
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How to find Sturm-Liouville problem eigenvalue and function?

So I have the following Sturm-Liouville problem: $$ y'' + \lambda y = 0 $$ Such that $ \lambda > 0 $ and the initial conditions are as follows: $$ y (0) + y'(0) = 0 $$ $$ y(1) + y'(1) = 0 ...
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Eigenvalues of 1D laplacian discretized matrix

I have the matrix resulting form the finite difference discretization and now I should find its eigenvalues. the text of the exercise is Hint: Write out a typical equation of the system $Aw = λw$ ...
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How to Choose Basis Solutions for Eigenvalue Problem

I have a real, fourth order linear operator $L$ and want to solve the eigenvalue problem \begin{equation*} Lv = \lambda v, \end{equation*} where $\lambda \in \mathbb{C}$. I further want to impose ...
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For a compact Riemannian manifold $M$, $L^2(M)$ is spanned by the eigenfunctions of the Laplacian.

In some paper I read the following statement: For a compact Riemannian manifold $M$ and the corresponding Laplace-Beltrami operator $\Delta$ on $M$ we have, that $$L^2(M) = \widehat{\bigoplus_{\...
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Orthogonal Degenerate Eigenfunctions

Regarding Fourier Series, it is easy when talking about non-degenerate eigenfunctions to prove they are orthogonal using Green's identity. However I'd like to know if it is possible to prove that ...
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2answers
70 views

Sturm-Liouville, find normalized eigenfunction

I want to find the eigenvalues and normalized eigenfunctions of the problem $$-y'' = \lambda y, y'(0) = y(1) = 0. $$ By solving $r^2 + \lambda = 0$ I found the general solution $y(x) = c_1\cos(\sqrt{\...
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How to turn ODE into an Eigenvalue Problem

My Question: What is the general formula/method for turning an ODE into an eigenvalue problem? My book turns this equation from an example $$\frac{d^2u}{dx^2}+ u=e^x, u(0)=u(\pi)=0$$ into $$\frac{...
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Solving the BVP $y''+2y'+(\lambda+1)y=0$ for $y(0)=y(\pi)=0$

Convert the differential equation $$y''+2y'+(\lambda+1)y=0$$ to Sturm-Liouville form, and obtain the solutions satisfying the boundary conditions $$y(0)=y(\pi)=0.$$ Using the integrating factor $\mu(...
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Relation eigenfunction of linearized PDE and solution of the original PDE

Consider $$ \frac{\partial^2}{\partial x^2}u+\mu \sin(u) = 0 \\ u(0) = 0 = u(1) $$ The linearized version is for small $u$ $$ \frac{\partial^2}{\partial x^2}u+\mu u = 0 $$ This gives for the general ...
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What functions can be represented as a series of eigenfunctions

Consider the differential equation: $y'' = \lambda y$ with the boundary conditions $y(0) = y(2\pi) = 0$. This equation has eigenfunctions $\mu_n(x) = \sin(\frac{nx}{2})$ with the corresponding ...
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1answer
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Why is a matrix multiplied by an eigenvector not parallel to that eigenvector?

If $\lambda$ is a non-zero eigenvalue with a corresponding eigenvector $v$, then $A v$ is parallel to $v$. This statement is false. Why is that? Would it be parallel to $\lambda v$?
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Proving that the eigenvalues of the Airy problem are positive

I am solving an exercise concerning the Airy eigenvalue problem $$ -y''+xy =\lambda x, \quad y(0)=y(1)=0, \quad (*) $$ which (among other things) asks me to prove that all eigenvalues are positive. I ...
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How can I find the following one dimensional heat conduction solution?

$$\frac{\partial T}{\partial t}=\alpha\frac{\partial^2 T}{\partial^2 t}$$ with an initial condition and boundary conditions $$T(x,0)=T_0$$ $$T(L,t)=T_0$$ $$-k\left.\frac{\partial T}{\partial x}\right|...
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Checking Eigenvalues Of an ODE

When checking the eigenvalues of an ODE that you separate from a PDE like: $\displaystyle \frac{d^2\phi}{dx^2} = -\lambda \phi$ $\phi(0)=0$ $\phi(L)=0$ Why do you separate the problem into cases ...
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Heuristic understanding of the eigenvalue equality between two Neumann quantum graph

Lemma: Let $G$ be a quantum graph (not necessarily connected) with two vertices $v_{1}$ and $v_{2}$ with the Neumann conditions imposed on it. Modifying the graph G by merging the two vertices into ...
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Finding basis of the null space of eigenvector equation

Question I'm trying to do part (ii) which asks for the basis for the null spaces of the eigenvector equation with the two respective eigenvalues. What's the easiest way of doing so? I tried to find ...
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Fredholm Equation with Exponential Sum Kernel

I'm trying to solve the following integral equation to find the function $f(x)$ \begin{equation} f(x) = K(x) - \int_0^\infty K(x-t)f(t)dt \end{equation} where \begin{equation} K(x) = \sum_{i=1}^N ...
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Can this problem be reduced to a Sturm-Liouville form?

From a system of three coupled PDEs \begin{eqnarray} \frac{\partial \theta_h}{\partial x} + \beta_h (\theta_h - \theta_w) &=& 0,\\ \frac{\partial \theta_c}{\partial y} + \beta_c (\theta_c ...
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Help in building final solution after solving the separated Eigenvalue problems

I (with help from a MSE user) used the following substitution to seperate variables in a second order linear PDE $$\theta_w = e^{-\beta_hx}F'(x)e^{-\beta_cy}G'(y)$$ The following two ODEs (...
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How to proceed further in this Eigen Boundary value problem

I have the following eigenvalue BVP $$ \lambda_h F''' - 2 \lambda_h \beta_h F'' + \left( (\lambda_h \beta_h - 1) \beta_h - \mu \right) F' + \beta_h^2 F = 0 $$ wit BC(s) $F(0)=0,\frac{F''(0)}{F'(0)}=\...
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Converting BVP into standard Eigenfunction Eigenvalue form

I have a eigen boundary value problem $$ \lambda_h F''' - 2 \lambda_h \beta_h F'' + \left( (\lambda_h \beta_h - 1) \beta_h - \mu \right) F' + \beta_h^2F=0 $$ $\mu$ is the separation variable or the ...
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30 views

Finding pattern in a Eigenvalue BVP [Analytic possiblity] ??

I have the following two third order linear ODEs which have been arrived at after applying separation of variables to a coupled system of three PDEs. \begin{eqnarray} \lambda_h F''' - 2 \lambda_h \...
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37 views

Eigenvalues of a partial differential equation

Why $\lambda_n=sgn(n)\pi i \sqrt{n^2+\alpha}$? I have this: $\varphi_{xx}-(\alpha+\lambda^2)\varphi=0$ and $\varphi(0)=\varphi(1)=0$ then $\varphi(x)=c\sin(\sqrt{-(\alpha+\lambda^2)}x)+d\cos(\sqrt{-...
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Evaluating Eigen values of [Cubic ODE]

I have a DE (resulting from variable separation applied on a PDE with $\mu$ acting as the separation coefficient, all other terms are constant and $>0$) $\lambda_h F''' - 2 \lambda_h \beta_h F'' + ...
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1answer
161 views

Eigen values of a Third Order Linear Homogenous ODE

I have two third order linear ODE which have been arrived after applying separation of variables to a system of PDEs \begin{eqnarray} \lambda_h F''' - 2 \lambda_h \beta_h F'' + \left( (\lambda_h \...
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28 views

Helmholtz equation with moving boundary in the plane

Let us assume we have a unit disk $D\subset\mathbb{R}^2$ s.t. $\vec{0}\in D$. To obtain the eigenfrequencies and eigenmodes (or eigenvalues and -functions if you like) we must solve $\Delta \psi + \...
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Can this Helmholtz PDE with Robin boundary conditions be solved analytically?

Consider the following Helmholtz problem in the infinite triangle $y>0,\;x>y$ with parameters $Q<0$, $P\ge0$, $P<|Q|$. $$\left\{\begin{align} &\psi^{(2,0)}(x,y)+\psi^{(0,2)}(x,y)+E\...
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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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0answers
47 views

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 $...