For questions on eigenfunctions.

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17 views

Sturm Liouville with periodic boundary conditions

Background and motivation: I'm given the boundary value problem: $$y''(x)+2y(x)=-f(x)$$ subject $y(0)=y(2\pi)=0$ and $y \, '(0)=y \, '(2\pi)=0$. The text (Nagle Saff and Snider, end of Chapter 11 ...
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1answer
28 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 ...
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1answer
28 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 ...
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0answers
10 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 ...
3
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1answer
35 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
30 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
191 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 ...
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0answers
25 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
32 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 $1$. ...
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1answer
44 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 ...
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1answer
38 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 ...
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0answers
19 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
25 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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3answers
59 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
112 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
38 views

Completeness of eigenvectors of Hermitian Matrix.

How do you show that eigenvectors of a Hermitian matrix form a complete set of basis?
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0answers
44 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 ...
0
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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. ...
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0answers
36 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
39 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 ...
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1answer
102 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.
2
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1answer
41 views

Is Helmholtz equation in arbitrary regular polygon solvable in closed form?

A Helmholtz equation $\Delta f=-\lambda^2 f$ with Dirichlet boundary conditions can easily be solved in a square and also not too hard to solve in equilaterial triangle. In both cases the solution is ...
4
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1answer
52 views

can the first eigenfunction of the Dirichlet Laplacian have any saddle points

Let $\Omega$ be a connected, bounded region of $\mathbb{R}^2$. The Laplacian $\Delta$ has a discrete spectrum of functions satisfying $$\Delta f = \lambda f$$ on $\Omega$ with $f=0$ on the boundary ...
1
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1answer
53 views

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

What is the analytical solution for the following diffusion partial differential equation (initial value problem)? $$\frac{\partial f}{\partial t} = (ax^2+b)\frac{\partial f}{\partial ...
0
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0answers
42 views

Regular Sturm-Liouville Boundary Value Problem

Let $L[y]:=y''''$. Let the domain of $L$ be the set of functions that have four continuous derivatives on $[0,π]$ and satisfy $y(0)=y'(0)=0$ and $y(π)=y'(π)=0$ a) Show that $L$ is self adjoint b) ...
1
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1answer
56 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 ...
1
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1answer
80 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
94 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
28 views

Testing if a n-dimensional ellipsoid lies completely in an n-dimensional cuboid

I have a n-dimensional ellipsoid A and a n-dimensional cuboid B. I need to know if A lies completely in B. How can I test whether this condition holds? Even better would be a solution telling what ...
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0answers
47 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
97 views

On max-min representation for the principal eigenvalue of second order elliptic operator

(Just to be upfront about things, this is a homework problem.) I'm asked to show that the principal eigenvalue, $\lambda_1$ of an uniformly elliptic operator can be represented by \begin{equation} ...
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1answer
93 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 ...
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0answers
93 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
36 views

Do eigenfunctions from symmetric ODEs really make up a base for $L^2$?

I'm confused about a theorem in some hand-out material on spectral theory for ODEs. The BVP is stated in the form $$ \begin{cases} L u = f\\ R u = 0 \end{cases} $$ over $I = [a,b]$. And ...
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0answers
76 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
129 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 $$ ...
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0answers
44 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 ...
2
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0answers
72 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
159 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 ...
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1answer
65 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 ...
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1answer
96 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
136 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
148 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} ...
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1answer
357 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
80 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 ...
3
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1answer
241 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 ...
0
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2answers
222 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 $\lambda$ ...
0
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1answer
106 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 ...
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1answer
72 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.