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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4answers
3k views

Proof that Legendre Polynomials are Complete

Can somebody either point me to, or show me a proof, that the Legendre polynomials, or any set of eigenfunctions, are complete?
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1answer
716 views

How to find the eigenvalue and eigenfunction of Laplacian?

Define a bounded domain $\Omega=(0,a)\times(0,b)$ What is the eigenvalue and eigenfunction of the Laplacian with homogeneous boundary condition? my first thought is something like $sin(n\pi x/a)sin(n\...
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1answer
156 views

I don't understand this PDE solution involving Fourier coefficients and orthonormal eigenfunctions.

$u_{xx} + u_{yy} = 0$ with $x \in (0,\pi)$ and $y \in (0, \pi)$ Initial Conditions: $$ u(x,0) = x^2 $$ $$ u(x,\pi) = 0 $$ Boundary conditions: $$ u_{x}(0,y) = 0 = u_{x}(\pi, y) $$ I performed ...
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0answers
969 views

Poisson Equation in a Rectangle

The problem is to solve $$\Delta\phi=\frac \lambda {\varepsilon_0}\delta(x-x',y-y')\quad;\quad \phi(0,y)=\phi(a,y)=\phi(x,0)=\phi(x,b)=0.$$ My idea was to try and represent the RHS as a series of ...
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0answers
92 views

Eigenfunction Expansion for Simple Nonhomogenous PDE

How do I go about finding an eigenfunction expansion for the following equation: $$ u'' = f(x)$$ where: $$ u'(0) = \alpha \quad u'(1) = \beta$$ What about the case when $f(x) = C$ a constant? Edit: ...
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1answer
236 views

Eigenfunction expansion

Use the appropriate engenfunction expansion to represent the best solution. $$u''=f(x), u'(0)=\alpha, u'(1)=\beta$$ I use the function $$\phi''+\lambda\phi=0$$ to get the eigenfunction is $$\phi=A\...
13
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1answer
661 views

Can we construct Sturm Liouville problems from an orthogonal basis of functions?

Given a sequence of functions orthogonal over some interval, which satisfy Dirichlet boundary conditions at that Interval, can we construct a Sturm Liouville problem that gives these as its ...
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0answers
263 views

What are the real world uses of Eigenbasis

The title pretty much says it all, I am wondering what the real world application (especially pertaining to electrical engineering) of an Eigenbasis is. I am also having some trouble understanding ...
4
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1answer
526 views

What can be said about the eigenvalues of the Laplace operator in $H^k(\mathbb{T}^2)$

Consider the Laplace operator $$\Delta: H^{k+2}(\mathbb{T}^2) \to H^k(\mathbb{T}^2)$$ where $\mathbb{T^2}$ is the two-dimensional torus (which is a compact manifold without boundary), so that $$ H^k(\...
2
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0answers
74 views

Given two eigenfunctions and eigenvalues determine existence of eigenvalues between them

Suppose we have two eigenfunctions $f_n(x,y)$ and $f_m(x,y)$ and corresponding eigenvalues $\lambda_n<\lambda_m$ of a differential operator $L$. How can I determine whether there exists another ...
4
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1answer
206 views

Simplest Schrödinger equation with both continuous and residual spectrum

Consider a Schrödinger equation: $$-\frac{\text{d}^2}{\text{d}x^2}f(x)+U(x)f(x)=Ef(x),$$ I need a $U(x)$ satisfying the following: The Schrödinger equation with it must be solvable purely ...
2
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1answer
981 views

Arbitrarily using Sin and Cos as eigenfunctions of a Hamiltonian?

In the context of quantum optics, the rotating wave Hamiltonian can be written: $\hbar\begin{pmatrix} -\Delta & \Omega/2\\ \Omega/2 & 0 \end{pmatrix}$ The eigenvalues can then be calculated ...
0
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1answer
123 views

Completeness of eigenfunctions of higher order differential equation

I have a third order linear differential equation, with a free parameter, and boundary conditions that depend on that parameter. I don't think it is possible to obtain an analytic solution, but I ...
0
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1answer
654 views

Eigenvalues of a second derivative

I have a function f(r) that describes a Gaussian random field. A second derivative can be formed $\nabla_i \nabla_j f(r)$. I am looking at a paper that claims that in finding the extremum, the ...
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0answers
225 views

Eigenfunction of which differential operator?

I suspect this question is rather naive, but here goes. I have a set of basis functions that I suspect are eigenfunctions of an unknown differential operator (due to some results I've seen in some of ...
2
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1answer
2k views

How to find the corresponding eigenfunction after determining the eigenvalues?

I was reading this page (http://www.jirka.org/diffyqs/htmlver/diffyqsse25.html) example 4.1.4, which says: Again $A$ cannot be zero if $\lambda$ is to be an eigenvalue, and $sin(\sqrt {\lambda} \pi)...
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1answer
2k views

Finding eigenvalues and eigenfunctions for a BVP

Find the eigenvalues and eigenfunctions for $$y'' + \lambda y = 0, y(0) = 0, y'(\pi/2) = 0$$ According to my book we must check 3 cases: $\lambda < 0$, $\lambda = 0$, $\lambda > 0$. I started ...
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0answers
133 views

Understanding orthogonality in two-scale asymptotic expansion (cf. G. Allaire)

This question is about equation (2.16) of Lecture 2 on Homogenization in Porous Media_ by Allaire page 28. There are two spacial scales: $x$ being macroscopic and $y=\dfrac{x}{\varepsilon}$ being ...
6
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1answer
1k views

Forced vibrations in an annulus / annular membrane.

I am trying to find out how to solve the following problem: $$ \frac{ \partial^2 u }{\partial t^2 } = c^2 \nabla^2 + Q(x,y,t) , $$ in which we have the initial conditions $u(x,y,0) = f(x,y)$ and $\...
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2answers
573 views

Problem related with boundary value problem and eigenvalue, eigenfunctions

I was looking at previous year exam papers and was stuck on the following problem: For the boundary value problem, $\,\,y''+\lambda y=0; y(0)=0,y(1)=0, \,\,\exists$ an eigenvalue $\lambda$ ...
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1answer
179 views

Are these functions orthonormal?

Are the following set of functions orthonormal over the interval $0$ to $1$? $$Y_r(x) = \sin{\beta_r x}-\sinh{\beta_r x}-\frac{\sin\beta_r-\sinh\beta_r}{\cos\beta_r-\cosh\beta}\left(\cos\beta_r x-\...
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1answer
91 views

Show that eigenvalues are negative

I have to consider the eigenvalue problem: $$ L[u] := \frac{d^2 u}{dx^2}= λu,x \in (0,1)\quad u(0)-\frac{du}{dx}(0)=0, u(1)=0.$$ I need to show that the eigenvalues are negative.
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2answers
158 views

Upper bound on the difference between two elements of an eigenvector

Let $W$ be the non-negative, symmetric adjacency/affinity matrix for some connected graph. If $W_{ij}$ is large, then vertex $i$ and vertex $j$ have a heavily weighted edge between them. If $W_{ij} = ...
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1answer
83 views

What subjects should I study to learn about eigenfunctions? What good textbooks would you recommend for learning the subject?

I googled eigenfunction and look it up in wikipedia, but still I do not know where I should start to learn the subject. I have two questions, and allow me to repeat the title of this question. What ...
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0answers
350 views

Completeness of eigenfunctions

In my computations I have obtained a sequence of eigenvalues $\lambda_k, \; k\in \mathbb{N}$ of double multiplicity. Thus, the basis for the eigenspace of $\lambda_k$ is given by $\psi_k(x) = \left\{e^...
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0answers
52 views

First eigenvalue of the given linear operator

I have the following question: Let us denote $H_2^N: = \{u\in (H^2(0,1))^2: u'(0) = u'(1) = 0\}$. Let an operator $L:H_2^N \to (L^2(0,1))^2$ be given by $Lu = -Du'' + Cu$, where $D$ is a positive ...
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1answer
282 views

Sturm-Liouville problem_positive eigenfunctions

I would like to solve next problem: Show that all the eigenfunctions of the Sturm-Liouville problem are positive: $$u''+(\lambda-x^2)u=0$$ $$0<x<∞$$ $$u'(0)=\lim_{x \to \infty}u(x)=0$$ Any ...
5
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1answer
2k views

Eigenvectors and Principal component

What is the difference between eigenvectors and principal component. I got confused about this point because some researches reported that the principal components are the same eigenvectors of ...
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1answer
214 views

Can any PDE of certain form be solved via separation of variables?

I know there are some examples how a PDE can be solved by separation of variables even when it doesn't have some obviously useful symmetry - e.g. Laplace equation in ellipse can be solved in elliptic ...
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1answer
318 views

How do you find the (complex) eigenvalue and each eigenspace over C

My book only has eigenvalue and eigenspace and does not say anything about complex eigenvalue and eigenspace. $$A = \begin{pmatrix} 0&4\\-1 & 0 \end{pmatrix}\hspace{10pt}B =\begin{pmatrix} -1&...
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2answers
871 views

Evaluating using BRA KET Notation?

Evaluate $\langle 0 \mid x^3 \mid 1\rangle$, assuming that all the wave functions you encounter are normalized eigenfunctions of the harmonic oscillator Hamiltonian, without Mathematica, Maple, or ...
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0answers
115 views

The value interpretation of eigenvectors.

My question is may be strange but I wanna lie it any way. The direction of an eigenvector is the most important as we normalize it. This view is right but what about the value of this eigenvector in ...
2
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3answers
393 views

Spectrum of eigenvalues and eigenfunctions

Our O.D.Es professor had the "amazing" idea of heavily introducing advanced linear algebra material (which is not an official prerequisite for the course) along with boundary value problems. Not being ...
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1answer
471 views

Eigenvalues and Eigenvectors, difference between integer results and absolute results

i am developing an application where i need to calculate the eigenvalues and it's corresponding eigenvectors,i understand how to calculate it from this links: http://people.revoledu.com/kardi/tutorial/...
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1answer
187 views

Which one is the right definition of eigenvalues for a differential operator?

This question might be trivial, but I have problems understanding the definition of eigenvalues for the Laplacian \begin{equation} \Delta : C^2(U) \to C(U). \end{equation} on some open, bounded domain ...
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0answers
128 views

show that the function satisfies condition of the lemma

Let $(f_n)_{n\geq 0}$ be an eigenfunctions of the compact integral operator $F$, defined on $L^2([-1,1])$ by $$ F(f)(x)=\int_{-1}^1\frac{\sin a(x-y)}{x-y}\,\psi(y)\,\mathrm{d}y, \quad \mbox{here ...
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1answer
2k views

degenerate eigenvalues

I have a problem in understanding the exact meaning of degenerate eigenvalue. I have some database and I calculate the covariance matrix among it. the obtained eigenvalues are same ( all of them =5000)...
4
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1answer
2k views

Rayleigh-Ritz Theorem

Let $U$ be an $n$-dimensional subspace of $L:=L_2([-1,1])$. Let $F$ be an acting on $L$, given at $f \in L$ $$ (Ff)(x):=\int_{-1}^1 \frac{\sin a(x-y)}{(x-y)}f(y) dy, \quad x \in [-1,1], \quad a>0. $...
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2answers
4k views

Show that the function is an eigenfunction of the equation

I'm not sure how to use the bbcode so I've taken a screenshot instead: Came up on a past exam paper that I'm working towards and I'm not sure how to answer it. I assumed that EQN . EIGENFUNCTION = ...
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1answer
2k views

How to find an orthonormal basis for $L^2(\mathbb{R},\mathbb{C})$?

Consider the Hilbert space $X:=L^2(\mathbb{R},\mathbb{C})$ Now consider the operator that takes the second derivative, i.e. $A := \partial_{x}^2$, i.e. $A: H^2(\mathbb{R},\mathbb{C}) \subset X ...
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1answer
566 views

Eigenfunctions of special kernel

Suppose $K:[0,1]^2 \times [0,1]^2 \to \mathbb{R}$ is continuous and positive semi-definite, and define the corresponding Hilbert-Schmidt integral operator $$ [Cu](x) = \int_{[0,1]^2} K(x,y) u(y) dy.$$...
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1answer
987 views

physical meaning of laplace-beltrami eigenfunctions?

The eigenfunctions of Laplace-Beltrami operator are often used as the basis of functions defined on some manifolds. It seems that there is some kind of connection between eigen analysis of Laplace-...
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1answer
834 views

1-periodic function; fourier; eigenvalues

Let $k\colon\mathbb{R}\to\mathbb{C}$ be a 1-periodic function with $k|[0,1]\in L^2([0,1])$. Define the convolution operator $T$ as $f\mapsto\int\limits_{[0,1]}k(s-t)f(t)\, dt$. Develop $k$ in a ...
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0answers
112 views

Partial Differential Equation Eigenvalue of zero question

In the event that I'm solving a partial differential equation through separation of variables, if I end up with an eigenvalue of zero, what do I do with the corresponding eigenfunction? That is to ...
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1answer
93 views

What (if anything) am I doing wrong in this eigenvalue boundary value problem?

So If I'm looking at a boundary value problem with boundary values of $U_x(0,t) = 0 = U(2\pi,t)$. I get to $\frac{T'}{kT} = \frac{X"}{x} = \lambda$ and try $\lambda = -\alpha^2$. This leaves me ...
4
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1answer
7k views

What is the difference between eigenfunctions and eigenvectors of an operator?

What is the difference between the eigenfunctions and eigenvectors of an operator, for example Laplace-Beltrami operator?
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1answer
145 views

Eigenstate and quantum mechanics position opperator

Quantum mechanics math question: Suppose that there is eigenstate $|q \rangle$ where $q$ is position observable . The question is, 1) What is eigenstate? How is this different from eigenvector? ...
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0answers
552 views

Finding a proper solution of a given functional

It's my first post here, but I worked very hard to find solution and I failed. Hereinafter, I skip physical background and directly proceed to my mathematical problem. No matter how, you know the ...