For questions on eigenfunctions.

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1answer
10 views

Given a set of non orthogonal functions. Find another set of functions that are orthogonal to the first set.

Let's say I have a set of complex functions $\{\phi_1,\phi_2,...\}$ defined on $0\leq x \leq1$, and they are not orthogonal i.e. $\int_0^1\phi_m^*(x) \phi_n(x)dx \neq \delta_{mn}$. Is it possible to ...
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0answers
35 views

Find eigenfunctions of the integral operator with kernel $\sum\limits_{n=1}^\infty \frac{1}{n^2} \sin((n+1)x)\sin(ny)$

Find the eigenvalue and eigenfunctions of the integral operator $Ku=\int_0^\pi k(x,y)u(y)dy$. $k( x,y) = \sum\limits_{n=1}^\infty \frac{1}{n^2} \sin\big((n+1)x\big)\sin(ny)$. This is how I ...
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0answers
10 views

Applying different boundary conditions in a quasi Helmholtz problem.

I have tried to solve this exercise from Applied Partial Differential Equations-Richard Haberman, Consider the two-dimensional eigenvalue problem $$ \nabla^2\phi+\lambda\sigma(x,y)\phi=0 $$ ...
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0answers
16 views

Sobolev space for Mixed Dirichlet - Neumann boundary condition

Consider the subset $\Omega\subset \mathbb{R}^N$ with boundary $\partial\Omega$ sufficiently regular and let $\Gamma\subset\partial\Omega$ be a $(N-1)$- dimensional submanifold of $\partial\Omega$. ...
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0answers
15 views

Integro-differential eigenvalue problem

In my research I encounter an eigenvalue integro-differential equation of the form: $$f_n(x,y)=\lambda_n\iint_D\frac{\nabla'\cdot\big\lbrace ...
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0answers
27 views

Question about the spectrum of linear (unbounded) operator

I'm not much confident with functional analysis, but I found in my lecture note a statement that doesn't convince me. For a linear (possibly unbounded) operator $T$ in a Banach space the following ...
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0answers
6 views

Implied meaning of “existence” of inner products

I read somewhere that the inner products of the eigenfunctions of an operator with a discrete eigenvalue spectrum are guaranteed to exist. Does existence here just mean that they can be defined or ...
2
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1answer
36 views

Eigenfunctions of an operator using Laguerre Polynomials

I am trying to find the eigenfunctions of the following operator: $$\mathcal{L}f=(-\gamma x+\frac{\mu}{x})f_x+\mu f_{xx}$$ I know that I must somehow use Laguerre polynomials, the solutions to the ...
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0answers
27 views

Finding an equation satisfied by the negative Eigen values

I have a system $$\frac{d^2y}{dx^2}+\lambda{y} =0$$ subject to $2y(0)+y'(0)=0$ and $y(1)=0$ Im trying to find the equation satisfied by negative eigenvalues. Heres what ive tried: let ...
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0answers
21 views

Eigenfunctions of the Fourier transform on locally compact abelian groups

The eigenfunction theory of the Fourier transform on $\Bbb R$ is well-understood. For example, the Hermite-Gauss functions are eigenfunctions with eigenvalues $i^n$; in fact, this comprises the ...
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0answers
35 views

show linear transformation bijective

Can you please help me prove this? Let $T:\mathbb{R}^7\to\mathbb{R}^7$ be a linear transformation such that 9 is an eigenvalue of $T$ and $dim(E_9)=6$ Prove that either T-4I or T-5I is a bijection ...
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0answers
17 views

Sturm-Liouville eigenfunction expansion converges to 1-x instead of -1

I am presented with a Sturm-Liouville problem $y''+\lambda y = 0$, $y(0)=0$, $y'(1)=0$. I solve this to obtain eigenfunctions of the form $y = \sin{(n-\frac{1}{2})\pi x}$. Now I wish to find a ...
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0answers
40 views

Finding normalized eigenfunctions for $y'' + \lambda y = 0$

Find the normalized eignefunctions for $$y'' + \lambda y = 0$$ $$y(0)=0, y(\pi)-2y'(\pi)=0.$$ My teacher gives me this hints: Consider$$(py')'+qy+\lambda ry=0$$ where $p, p', q, r$ are ...
-1
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2answers
43 views

Finding the eigenvalues and eigenvectors with each eigenvalue, solving the general solution with initial conditions.

Consider the system $x'_1 = x_1 + 2x_2$ and $x'_2 = 3x_1 + 2x_2$ If we write in matrix from as $X' = AX$, then a) $X =$ b) $X' =$ c) $A =$ d) Find the eigenvalues of A. e) Find eigenvectors ...
1
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1answer
19 views

Not understanding why equation 2 and equation 3 has to be multiplied with $v(x)$

The discrete differential operator $L_h$ is defined as: $(L_h v)(x_j)=-\frac{v(x_{j+1})-2v(x_j)+v(x_{j-1})}{\Delta x^2}$ (Equation 1). The contineous problem had solution $v(x_j)=sin(\beta x_j)$. It ...
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3answers
30 views

Find eigenvalues for $T(f) = \int_{-\infty}^x tf(t)dt$

Let $V$ be the linear space of all functions continuous on $(-\infty, \infty)$ and such that that the integral $\int_{-\infty}^x tf(t)\,dt$ exists. If $f \in V$, let $g=T(f)$ be defined as $g(x) = ...
0
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0answers
35 views

largest eigenvalue

D is a $n \times n$ diagonal matrix whose diagonal entries lies in $(0,1]$. B is any $n \times n$ n.n.d. matrix. What will be the sharpest upper bound on the largest eigenvalue of: ...
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0answers
25 views

Sturm Liouville eigenvalue problem on infinite domain

Consider a Sturm Liouville problem: $$\frac{d}{dx} \left( p(x) \frac{dy}{dx}\right) + q(x) y = f(x)$$ on $(a,b)$ with some boundary conditions. Denote by the Sturm Liouville operator $L:= ...
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0answers
24 views

How to find Eigenvectors of a non symmetric matrix using QR decomposition?

The QR method efficiently calculates the eigenvalues of a matrix. If I try to find eigenvectors alongside the eigenvalues($Q_1Q_2.....Q_n$), the results seem to be correct only for symmetric matrices. ...
0
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1answer
23 views

Order of solving Eigenvalue equation

When solving eigenvalue equation, we usually have three scenarios. The first one is the simplest: i.e. it's diagonalised, so the eigenvectors can be found by inspection. The second one is that you ...
0
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1answer
43 views

Sturm-Liouville expansion

How would I solve this : $-(xu')'=\frac{1}{x} \ln{x} \enspace 1<x<e$ $u(1)=0 \enspace, u'(e)=0$ I want to expand this with: $u(x) = \sum_{n=1} ^{\infty} b_n X_n(x)$ Where $X_n(x)$ are the ...
2
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0answers
84 views

Eigenvalues for $y''+2y'=\lambda y$

I must find the eigenvalues and eigenfunction for $$y''+2y'=\lambda y$$ with initial conditions $y(0)=0$, $y'(1)=0$. I have found the non-trivial case, and made an attempt to solve for $\lambda$, but ...
2
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0answers
38 views

Eigenvalue Function of Laplace Equation discretizes by nine-point stencil

I'm trying to plot the eigenvalue function of the Laplace equation $$-u_{xx}-u_{yy}=0,\;(x,y)\in (0,1)^2$$ with $$u(x,y)=0$$ on the boundary of the unit square. I have the nine-point stencil ...
0
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0answers
13 views

Closed form of chained Stationary Functions

Consider the following operator $$D_{(a-1)x,x}\left[ f \right] = \frac{f(ax) - f(x)}{(a-1)x} $$ It's pretty easy to see that a nontrival function $g(x)$ such that $$D_{(a-1)x,x}[g] = g$$ Is given ...
0
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1answer
69 views

Eigenvalues and eigenfunctions of fourth order ODE

Find the eigenvalues and eigenfunctions of the problem $$y^{(4)} − λy = 0$$ with the boundary conditions (i) $\quad y(0) = y'' (0) = y(β) = y'' (β) = 0$ (ii) $\quad y(0) = y' (0) = y'' (β) = y''' ...
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0answers
15 views

Does the magnus convergence test not hold for the factorization of second order differential operators?

Given the operator \begin{align} H = V(x)-\partial_x^2 \end{align} and given an eigenfunction $\phi_0(x)$ such that $H\phi_0=0$ with a zero eigenvalue, I can factor $H$ into \begin{align} H = h_+h_- ...
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0answers
16 views

Eigenelements of Neumann Laplacian satisfy $\sum_{k=1}^\infty |(u,\varphi_k)_{L^2}|^2 \lambda_k^{-\frac 12} < \infty?$

Let the eigenvalues of Neumann Laplacian on a bounded open domain be given by $0 = \lambda_0 \leq \lambda_1 \leq \lambda_2 ...$ associated to eigenfunctions $\varphi_0, \varphi_1, ...$. Let $u \in ...
0
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1answer
54 views

Space of continuous functions linear operator eigenvalues

Let $V$ be the vector space of continuous functions from $\mathbb R$ to $\mathbb R$. Let $T$ be the linear operator on $V$ defined as $$(Tf)(x)=\int_0^x f(t)\,dt$$ Prove that $T$ doesn't have ...
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0answers
24 views

Questions about eigen vectors calculation in eignfaces

I am calculating the eigen vectors from a set of M grayscale faces images and I am using two methods. I would expect to get the same results but calculation gave me the different outcomes. Suppose I ...
0
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2answers
45 views

Why does the set of an hermitian operator's eigenfunctions spans the functions space

During a discussion about linear hermitian operators, my professor claimed that if a linear operator $M$ is hermitian under a certian set of conditions, then genrally any function that fulfills these ...
0
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1answer
38 views

Solving 2D Laplace eigenfunction equation

I want to solve the equation $$\nabla^{2}\,{\rm P}(x, y) = \frac{k}{c^{2}}{\rm P}(x, y)$$ or $$\frac{\partial^{2}{\rm P}}{\partial x^{2}}+\frac{\partial^{2}{\rm P}}{\partial y^{2}}=\frac{k}{c^{2}}{\rm ...
0
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1answer
37 views

Can Eigen vectors be different with the same normalization proceduce?

I calculated Eigen vectors of two badly-conditioned symmetric matrices of $K$ and $M$ ($M$ is positive definite). I employed two algorithms, the 1st algorithm is ...
0
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1answer
60 views

Can you help with the Method Of Eigenfunction Expansion of a Non-Homogeneous PDE problem?

Here is the Problem: Solve $\frac{\partial T(x,t)}{\partial t} = \frac{\partial^{2} T(x,t)}{\partial x^{2}} +2xe^{-t} $ with the following boundary conditions $T(0,t)=10, and \frac{\partial ...
1
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1answer
73 views

Eigenvalue problem $y'' + \lambda y = 0,$ $y'(0) = 0$, $y(1) = 0$

Find the eigenvalues of $$y'' + \lambda y = 0, \; y'(0) = 0, y(1) = 0$$ For $\lambda >0$, $$y(x) = c_1 \cos(\sqrt{\lambda} x) + c_2 \sin(\sqrt{\lambda}x)$$ We get that $y'(0) = 0 \implies ...
0
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1answer
59 views

How to change property of Eigen vectors

I calculated the Eigen values & vectors of two $K$ and $M$ matrices with LAPACK DGGEVX routine. I need to make Eigen vectors orthonormal like this: $\Phi^T M \Phi = I$ . How can I do that?
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0answers
48 views

Neumann eigenvalue problem for the Laplacian

Let $\Omega$ be a bounded, connected open subset of $\mathbb R^n$. Consider the Neumann eigenvalue problem $$ \Delta u =\lambda u \, \text{at} \, \Omega \\ \frac{\partial u}{\partial \vec{\nu}}=0 ...
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3answers
53 views

EigenFunction for $\frac{\partial f}{\partial t}+f\frac{\partial f}{\partial x} =\frac{2f^2}{x}$

When studying a computer vision problem I end up with a function $f(x,t)$ that satisfying $\frac{\partial f}{\partial t}+f\frac{\partial f}{\partial x} =\frac{2f^2}{x}$. My question includes two ...
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2answers
63 views

Finding Fourier cosine series of sine function

I am trying to find Fourier cosine series of following function, but think that I am messing up somewhere. $$ f(x)=\sin \bigg ( \frac{\pi x}{l} \bigg ) $$ Fourier cosine series can be written as $$ ...
2
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1answer
68 views

Finding eigenvalue and eigenfunction of the boundary value problem

How to find eigenvalues and eigenfunctions of this boundary value problem? $$ y'' + \lambda y = 0 \\ y'(0)=0, y(\pi/2)=0 $$ I want to find only positive eigenvalues. I proceed like this: $$ y=C_1 ...
0
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1answer
60 views

How to find eigenvalues and eigenfunctions of this boundary value problem?

I want to find eigenvalue and eigenfunction of this problem: $$ y''+ \lambda y=0, 0<x<l \\ y(0)=0, ly'(l)+ky(l)=0 $$ And $y'$ stands for $\frac{dy}{dx}$ and similar for $y''$. I get the ...
1
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1answer
26 views

How to find solution of this eigenfuction?

The eigen function boundary value problem is ($y'=\frac{dy}{dx}$ and similar for $y''$) $$ y'' - \lambda y = 0,\\ y'(0) = y(2)=0 $$ I think the solution is like this Case 1: $\lambda <0 $ putting ...
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0answers
33 views

Imaginary Eigenvectors and Systems

NOTE: This is a homework problem (not graded). I want to learn how to do this math problem, not just be given an answer. Thank you for any help you can give! So, in this problem I am dealing with ...
0
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1answer
65 views

Karhunen-Loève Expansion (Finding the Eigenfunctions)

A WSS random process $X(\mu,t)$ has $\eta_X(t) = 0$ and $R_X(\tau) = \sum_{k=1}^K\cos(\frac\pi k\tau)$ for a given positive integer $K$ in the interval $|t|<1$. Find the Karhunen-Loève expansion of ...
5
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0answers
50 views

Accelerated Eigenfunction Expansions of Random Functions

I am interested in eigenfunction expansions of random functions. We know that the autocorrelation of brownian motion, $\{ B_t \}_{t \geq 0}$, is given by $$ E[B_t B_s] = \min\{s,t \}, $$ which can ...
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0answers
30 views

Eigenfunctions of the exponential of the derivative

In the space of smooth functions of one variable is there a way to tell what are the eigenfunctions of the operator $\exp(\partial_x)$, i.e. what is the solution of the eigenvalue problem ...
2
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1answer
62 views

Linear Algebra question relating to eigenvectors

Let A be an m x m positive definite symmetric matrix with eigenvalue-eigenvector pairs $(\lambda_1,e_1),....,(\lambda_m,e_m).$ The eigenvectors are orthonormal. Let $C = e_1e_1'+....+e_me_m'$. ...
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0answers
34 views

Conceptual difficulty about eigenfunction expansions

I'm having a fairly large conceptual block on my understanding of eigenfunctions - I've tried out two different methods that yield embarrassingly different answers, and I would appreciate if someone ...
0
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1answer
81 views

Poisson's Equation in a Disk

How can we solve Poisson's equation in a disk in plane polar coordinates?: $$ \nabla^2 \phi = u_{rr} + \frac{1}{r}u_r + \frac{1}{r^2} u_{\theta \theta} = f(r, \theta)$$ (My attempt): We know that ...
2
votes
1answer
68 views

Why does an eigenvalue expansion 'work' for PDEs?

I understand the logic and rationale behind using a series of eigenfunctions to represent general solutions to simple partial differential equations with prescribed boundary values, such as the ...
2
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1answer
35 views

eigenfunctions on covering spaces of graphs

I am reading about lifts of graphs in relation to covering spaces. Before I pose my question I will explain some of the terminology. Let $G$ and $H$ be two graphs. We say that a function $f: V(H) ...