For questions on eigenfunctions.

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20 views

Same eigenvalue spectrum with different matrices

There are two matrices $D$ and $T$ which provides eigenvalue spectrum (dispersion relation) according to $$ E(K) = \operatorname{eig}(D + T \exp(iK) + T' \exp(-iK)) $$ $$ K = 0:dK:\pi $$ Where K is a ...
2
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0answers
80 views

Spectrum of $Tu=\int^1_{-1} (1-|x-y|)u(y)dy $

Consider the operator $$ Tu(x)=\int^1_{-1} (1-|x-y|)u(y)dy $$ We want to find the spectrum of $T$. The kernel is certainly bounded and so this operator is Hilbert-Schmidt, so $T$ is compact. We ...
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2answers
59 views

Eigenvalues of symmetric elliptic operators

As stated in one of my previous questions already, I have not had so much exposure to theoretical linear algebra. This time, I'm reading a theorem and proof from PDE Evans, 2nd edition, pages 335-356. ...
4
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1answer
36 views

What is the purpose of studying Sturm-Louville eigenvalue problem?

After a cursory read on the SL eigenvalue problem, I did not immediately feel enlightened and failed find much usefulness except for knowing that SL generalizes a broader class of differential ...
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0answers
16 views

Eigenvalues of correlation matrices in the limit of infinite dimensions

Consider a continuous function $f(x,t)$ with $x\in X$ and $t\in[0,1]$, then one may define a series of functions $f_n\in\mathbb{R}^n$ defined naturally as $f_n(x)_i=f(x,i/n)$. Now compare the ...
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0answers
52 views
+50

Block diagonalizing two matrices simultaneously

There are two matrices $A$ and $B$ which can not be diagonalized simultaneously. Is it possible to block diagonalize them? What if the matrices have an special pattern? Physics of the problem is ...
2
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2answers
40 views

Spectrum of $Tu=\int^1_0 (x+y)u(y)dy$

Given the operator $$Tu(x)=\int^1_0 (x+y)u(y)dy$$ on $L^2(0,1)$, find the spectrum of $T$. For all eigenvalues, find their multiplicities and the eigenfunctions. The kernel is Hilbert Schmidt ...
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0answers
24 views

Which casses of matrices contain A and which contain B? Linear Algebra

Am pretty confused about classes, I don't know what it means, so so I can't really do part_A and I need your help with it? For part B, I got all eigen = 1 for matrix A, and 0 for matrix B, Is this ...
1
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1answer
47 views

Fourier-Transformation of Operator

I have an operator $\hat{L}$ which gives $$\hat{L} f(x) = \lambda \cdot f(x)$$ where $\lambda$ is the eigenvalue. Now I Fourier-Transform my function $f(x)$: $$\mathcal{F}(f)(p) = g(p)$$ Question: ...
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2answers
49 views

Eigenvalue and Eigenfunction for a boundary value problem

I'm having trouble understanding some of the concepts related to these problems. Here's an example I'm working on: $$y''+(\lambda+1)y=0 ; y'(0)=0,y'(1)=0$$ The characteristic equation I found was ...
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2answers
472 views

Why integration operator has no eigen values?

Let $V$ be the vector space of all functions from $\mathbb R$ into $\mathbb R$ which are continuous. Let $T$ be the linear operator on $V$ defined by $$(Tf)(x) = \int_0^x f(t) dt$$ Prove that ...
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0answers
23 views

eigen value problem with Robin Boundary Conditions at both ends

This is a problem from the book Partial Differential Equations by Walter.A.Strauss. Consider the eigen value problem with Robin Boundary Conditions at both ends: $-X''=\lambda X$ $X'(0)-a_0X(0)=0$ ...
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1answer
40 views

Bound on number of mutually orthonormal eigenfunctions

Let $E$ be the vector space of real valued continuous functions on an interval $[a,b]$. Let $K = K(x,y)$ be a continuous function of two variables, defined on the square $a \leq x \leq b$ and $a \leq ...
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0answers
15 views

Eigenvalues and fft of sound trying to find similarities

I'm looking at sound resonance patterns using matlab/octave to see if there may be patterns between FFT and Eigenvalues. I can get the frequencies and each of the frequencies amplitude to recreate ...
0
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1answer
88 views

Application of eigenvalueproblems for the wave equation

I'm currently searching for a nice little application of an eigenvalueproblem and found the following for acoustics - but one part doesn't make sense for me. Consider the wave equation to find some ...
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0answers
23 views

Removing extraneous solutions from an eigenvalue equation

I have an eigenvalue problem of the form $\left[ L_1 + \dfrac{L_2}{\Omega} + \dfrac{L_3}{\Omega^2} + \dfrac{\Omega-1}{\Omega+\eta}\right] \phi(x) = 0$ which I am trying to solve for the complex ...
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0answers
23 views

Sturm-Liouville(-like) differential equation

I encountered the following Sturm-Liouville-like (SL-like) partial differential equation in a project, and I decided to solve them numerically ($' = \frac{d}{dx}$): $(x(x+1) y’)’ – l(l+1) y = A ...
1
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1answer
24 views

Finding linear transform matrix from characteristic polynomial

I got two similiar very simple question on a notebook. 1)let characteristic polynomial $P_A(x)=x^2+2x-3$ and $T:V\to V$ and DimV=2,S={$\alpha_1,\alpha_2$} is ...
2
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1answer
20 views

Estimation with an orthonormalbasis in some finite dimensional subspace of $L_2(\Omega)$

I'm currently trying to understand a step of a proof of the following estimation: $\displaystyle \sum_{j=2}^{N} \left(\left< \varphi_1-R_N\varphi_1, \varphi_{j,N} \right>_{L_2}\right)^2 \ ...
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0answers
46 views

Galerkin-Approximation of first Eigenfunction

I'm currently trying to understand a certain proof of an error estimate for the first eigenfuntions gained by a Galerkin-Approximation with Finite Elements of the Potential Equation $-\Delta u$ with ...
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1answer
34 views

Eigenpairs of the Potential Equation in 1D

If you consider the Potential Equation in one dimension on some Interval $(0,R)$ and look at the eigenvalue problem: $$-u'' = \lambda \ u \ \ \text{ on } (0,R)$$ with zero-"boundary conditions": ...
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0answers
41 views

Eigenvalues for the double-well potential

I am trying to find eigen-values, $E$, for the following differential operator: $$\left[ -\frac{1}{2}\frac{d^2}{dx^2} +L\left(x^2-a^2\right)^2\right]y(x) = E\,y(x) $$ where $L,a$ are two positive ...
0
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1answer
49 views

I “proved” the first Laplacian eigenvalue has many eigenfunctions. Where's my mistake?

(For simplicity, say our domain $D \subset \Bbb R^2$, $\partial D$ is nice and smooth, and $\overline D$ is compact.) Let $\lambda_1$ be the first eigenvalue of the problem $\Delta u + \lambda u = 0$ ...
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37 views

What does it mean to say a differential equation is an eigenvalue problem?

My text says the following $$ \frac{\mathrm d}{\mathrm dx}\left(x^2 \frac{\mathrm dy}{\mathrm dx}\right) + \lambda y = 0,\;\;\;0\le x\le 1,\; y(1)=c\ge0$$ is an "eigenvalue problem". I don't ...
0
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1answer
30 views

Eigenvalue problem with asymmetric boundary conditions

Consider the unit square $\,\Omega = (0,1) \times (0,1) $ and the normal eigenvalue problem for Laplace's equation $$ -\Delta u = \lambda u $$ with the boundary conditions that on the vertical sides ...
0
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1answer
49 views

Proving subspaces are invariant for different statements [closed]

I would like to know how to prove the next statements regarding invariant subspaces: Statement 1: $f$ and $g$ are endomorphisms from a vector space $V$. If $f$ and $g$ commute, then subspaces ...
2
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0answers
28 views

Eigenfunctions of a 2D fractional Brownian motion covariance

The fractional Brownian motion is a centered Gaussian process with the following covariance function (covariogram): $E[B(t)B(s)]=C(\Vert t \Vert ^{2H}+\Vert s\Vert^{2H}-\Vert t-s\Vert^{2H})$ ...
0
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0answers
23 views

What kind of function do I have to find?

To show that the eigenfunctions of an eigenvalue problem don't form a complete set, I have to show that there is a function that satisfies the boundary conditions of the problem, but it cannot be ...
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2answers
209 views

Sturm Liouville with periodic boundary conditions

Background and motivation: I'm given the boundary value problem: $$y''(x)+2y(x)=-f(x)$$ subject $y(0)=y(2\pi)$ and $y \, '(0)=y \, '(2\pi)$. EDIT: These were not given to be zero !! Maybe this ...
0
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1answer
43 views

General eigenspace & general eigenvector

Consider the following: 1.Suppose $k_i$ is an eigenvalue of $A$ with algebraic multiplicity $n_i$ 2.dim $V_i = m_i$ (geometric multiplicity), $V_i$ is the eigenspace corresponding to $k_i$. 3. $m_i ...
2
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1answer
34 views

Eigenvalue of (1-0) matrix

Assume I have 2 matrices, each of size nxn with only 1 and 0 as entries in both. (n>10) The first matrix (call it A) has each row summing up to 2 (ie: on each row, it has two "1" and n-2 "0"). It is ...
4
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1answer
53 views

Can multidimensional eigenfunction problems be solved to arbitrary precision in constant memory usage?

Suppose we have a differential operator like a quantum mechanical Hamiltonian: $$\hat H=-\nabla^2+U$$ with zero Dirichlet boundary conditions. In one dimension its eigenvalues can be easily found ...
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1answer
42 views

Eigenvalue problem-Expand the function to the eigenfunctions of the problem

Having solved the eigenvalue problem $$y''+ λ y=0, 0 \leq x \leq L$$ $$y(0)=y(L)=0$$ which solution is: $$\text{The eigenvalues are: } λ_n=(\frac{n \pi}{L})^2$$ $$\text{ and the eigenfuctions are: ...
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0answers
40 views

Laplacian eigenvalue problem

I'm working through a PDE problem and we are given the eigenvalue problem $-\Delta u = \lambda u$ with $\frac{\partial u}{\partial n} = 0$ along the boundary given by the rectangle $\Omega = (0, \pi)$ ...
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2answers
210 views

Positivity of principal eigenvalue for $L\phi=-\triangle \phi + \nabla \cdot( u \phi )$

EDIT: This question is still unresolved as of April 18. The two answers provide useful work in the right direction, but neither resolves the question. A counterexample should have $u \in ...
0
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0answers
27 views

Deriving the particular integral for this ODE using eigenfunction expansion?? :S

In my past paper I'm asked to derive the particular integral using an eigenfunction expansion for this ODE: The mark scheme says this is the answer but shows no working: I've already ...
0
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2answers
36 views

Eigenvalue of a linear transformation

Let $V$ be the linear space of all real polynomial $p(x)$ of degree $\leq n$.If $p \epsilon V$, define $q=T(p)$ to mean that $q(t)=p(t+1)$ for all real $t$. Prove that $T$ has only the eigenvalue $1$. ...
0
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1answer
67 views

Another diffusion partial differential equation, or Sturm-Liouville eigenvalue ODE

What is the solution for the following diffusion partial differential equation (initial value problem)? $$\frac{\partial f}{\partial t} = \pm\frac{\partial f}{\partial x}+(ax+b)^2\frac{\partial^2 ...
2
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1answer
46 views

the solution of matrix polynomials

In order to get the eigenvalues of \begin{equation} P=\left[ \begin{array}{cc} 0_{n\times n} & I_{n\times n} \\ -A & -B% \end{array} \right], \end{equation} where $A$ and $B$ are both $n\times ...
0
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0answers
30 views

eigenfunctions in a Sturm-Liouville problem

I've found that the eigenfunctions in a certain Sturm-Liouville problem satisfy a differential equation whose general solution is $\phi(x)= x^{a}[C_1M(a,2a+2,x)+C_2U(a,2a+2,x)]$, $x\ge0$, where $M$ is ...
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0answers
37 views

Sturm-Liouville boundary value problem with two different eigenfunctions

I am trying to express a function $f(x)$ in terms of a complete set of eigenfunctions found from a Sturm-Liouville boundary value problem: $$2y''(x)+4y'(x)+\lambda y(x)=0$$ $$y(0)=0, y'(2)=0$$ For ...
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4answers
84 views

Stuck in finding Eigen values

The given matrix A is $$ \left[\begin{matrix} 2 & 1 & -2 \\ 0 & 1 & 4 \\ 0 & 0 & 3 \\ \end{matrix}\right] $$ I know that the Eigen values are the diagonals (2, 1, 3) as it is ...
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2answers
32 views

Which function satisfies the conditions?

I'm solving a problem and, in order to run test case, I need a function $ b(x,y) $ that satisfies: $$ \int_0^L \int_0^H b(x,y) \, dx \, dy = 0 $$ and $$ \int_0^L \int_0^H b(x,y) \cos \left(\frac{n ...
2
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0answers
188 views

Eigenvalues of self-adjoint eigenvalue problem

I am stack with the following problem: Consider the following eigenvalue problem $$ u \in H_B(0,1), \; \langle Lu, Lv\rangle = \lambda (\alpha \langle u, v\rangle + \langle u', v'\rangle) \; \forall ...
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1answer
58 views

Completeness of eigenvectors of Hermitian Matrix.

How do you show that eigenvectors of a Hermitian matrix form a complete set of basis?
2
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0answers
47 views

Eigenfunctions.

I have the following ODE: $$y''-2xy'+2\alpha y=0$$ whose solution $y(x)$ may be recursively represented as: $$a_{n+2} = \frac{a_n(2n-2\alpha)}{(n+2)(n+1)}$$ I have found the eigenvalues to be ...
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2answers
71 views

Eigenvalues of $\frac{d^2}{dx^2}$ in $C^2(\mathbb{R})$

Consider the eigenvalue problem \begin{equation} \left\{ \begin{array}{l} \Phi \in C^{2}(\mathbb{R}) \ \text{and bounded }\\ -\Phi^{''}(x)=\lambda\Phi(x), \ x\in \mathbb{R}. \end{array} \right. ...
0
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0answers
38 views

Convert an eigenvalue equation to ODE/s

For example define: $K=-i\frac{d}{dx}$ (non-discrete spectrum), so: $$Kf(x)=-i\frac{df}{dx}=kf(x)$$ Define $g(x,k)=kf(x)$, so: $$\frac{-i}{k}\frac{\partial{g}}{\partial{x}}=g(x,k)$$ ...
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0answers
40 views

“Fixed Domains” of a Linear Transformation

Given a linear transformation $T$, I need to find the set of all domains $D$ such that $T:D\mapsto D$. Equivalently, I need to find the set of all domains $D$ that are symmetric under $T$. Aside from ...
1
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1answer
157 views

Eigenvalue problem for the Laplacian on the unit ball [closed]

I want to find out what are the eigenvalues and eigenfunctions of the eigenvalue problem for the Laplacian on the unit ball in $\mathbb R^3$, with the Dirichlet boundary conditions.