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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28 views

Graph Fourier transform: the adjoint notation for the eigenbasis matrix

It is well-known that for a real symmetric matrix $L$ (here, graph Laplacian) one can write the eigenvalue decomposition as $$ L = U \Lambda U^{\mathsf T}, $$ where $U$ is a real eigenvector ...
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44 views

How to analyse the smallest eigenvalue of this linear ODE?

I am trying to solve the eigensystem of a 1st-order linear ODE system in the region $(-\infty,\infty)$ and with Dirichlet boundary condition at the infinities \begin{align} -\mathrm{i} u'(x) +f^*(x) ...
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17 views

The eigenfunctions of the Sturm-Liouville problem can be chosen real [duplicate]

I need to show that the eigenfunctions of the Sturm-Liouville problem can be chosen to be real. I have already proven that all the eigenvalues are simple. Does it follow directly by saying that if the ...
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1answer
34 views

Eigenfunctions of Sturm-Liouville are linearly independent [closed]

I need to show that the Wronskian of $n$ eigenfunction of the Sturm-Liouville equation $Ψ_{𝑥𝑥}+𝑢(𝑥)Ψ=𝜆Ψ$ is non-zero. I tried to show that they are linearly independent and then it follows, but I ...
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41 views

Why does separation of variables give all solutions to a PDE in $L^2(\Omega)$?

Let $A$ be the set of solutions. I think that if we can show these solutions are dense in $L^2(\Omega)$ then this would mean that we have found every solution. If we show that $A$ vanishes nowhere ...
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51 views

Sturm-Liouville problem with regular-singular point in the interval

I am confronted with an eigenvalue problem on $\mathbb{R}$ whose differential equation has a regular-singular point at $x=0$. For that reason, I am interested to know if there are known results or ...
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2answers
38 views

Solve for the eigenvalues and eigenfunctions for the equation $y''+\lambda y =0$

The boundary conditions are $$y(-L)=0=y(L)$$ where $L>0$. I know how to solve these kind of eigenvalue problems on the interval from 0 to L, but I don't know how to approach this set of boundary ...
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32 views

How to find the structure of ${\bf Q}$ that diagonalizes the matrix ${\bf F}={\bf U^HAQA^HU}$?

Define ${\bf F}={\bf U^HAQA^HU}$, where ${\bf U}$ is a unitary matrix, ${\bf A}$ is an arbitrary matrix, and ${\bf Q}$ is a positive semi-definite matrix. Assume all the complex matrices are $N$-by-$N$...
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32 views

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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23 views

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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28 views

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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2answers
202 views

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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1answer
22 views

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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36 views

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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1answer
135 views

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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15 views

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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1answer
105 views

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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17 views

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
75 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
23 views

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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20 views

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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89 views

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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11 views

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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25 views

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
37 views

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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28 views

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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110 views

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
78 views

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
77 views

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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16 views

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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23 views

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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57 views

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
89 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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2answers
49 views

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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1answer
122 views

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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3answers
160 views

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
66 views

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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1answer
38 views

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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28 views

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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32 views

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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6 views

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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30 views

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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43 views

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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50 views

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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53 views

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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51 views

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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0answers
37 views

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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0answers
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 \...