For questions on eigenfunctions.

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2
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0answers
21 views

Eigenvectors of matrix equation $AX+XB^\text{T}=\lambda (CX+XD^\text{T})$

We have the matrix eigenvalue problem $$AX+XB^\text{T}=\lambda (CX+XD^\text{T})$$ Where $\lambda$ is the eigenvalue, $X$ is $m\times n$ and plays the role of the eigenvector, and $A$, $B$, $C$, and ...
2
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1answer
30 views

Expansion theorem or Poisson Summation Formula? - Basis of eigenfunctions gives rise to a Fourier series

Does anyone could explain to me why in the Semiclassical's answer on the question Wave kernel for the circle $\mathbb{S}^1$ - Poisson Summation Formula, the basis gives a series of the form ...
2
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0answers
14 views

Set of normalized eigenfunctions - Expand some given function in the eigenfunction basis [on hold]

The spectrum$(\mathbb{S}^1)=\{\lambda_k=k^2\ : k \in \mathbb{N}\}$, and the eigenfunctions $\mu_k(t)$ associated to the eigenvalues $\lambda_k$ are $a_k \cos kt + b_k \sin kt$ under the Laplacian ...
2
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1answer
24 views

Coefficients of the eigenfunction

Related to the question : Eigenvalues of the circle over the Laplacian operator, how is it possible to find $c_1$ and $c_2$ related the explicit function $g(x)=c_1 \cos (\mu x)+ c_2 \sin (\mu x)$? ...
3
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0answers
17 views

Wave trace on $1$-dimensional circle - How about the spectrum of this circle?

I have to find the wave trace for the Laplacian on the $1$-dimensional circle. Generally, the wave trace is defined (see this website) as $$W(t)= \int_{M} K_t(x,y)dy=\sum_j \cos(t \lambda_j)=\Re ...
0
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0answers
40 views

Eigenvalues Eigenvectors and bases of eigenspace

I was given the matrix \begin{pmatrix} 3 & -5 & 4 \\ 2 & 0 & -3 \\ -1 & 2& -1 \end{pmatrix} I'm not sure what to do after I get $-x^3 + 2x^2 - 17x + 9=0$.
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0answers
37 views

Eigenvalue and eigenvector problem

I am looking to get to the answer of Question 3 e). I successfully expand the determinant to -λ^3 + 6λ^2 + 3λ - 13 using the characteristic equation and this is where I get stuck. I tried rational ...
0
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1answer
50 views

Generalized Hermite Function as eigenfunction of a differential operator

I'm going through this paper. The article defines function function $\phi_n^\mu(x)$ that is orthonormal on $L^2$ with measure $dm = dx$: \begin{equation} \phi^\mu_n ...
2
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0answers
4 views

$U(r, \theta) = T_s(\theta) J_s(\sqrt{\lambda}r)$ - Extract the eigenvalues of the eigenfunction $U$?

On a certain problem, I obtain $U(r, \theta) = T_s(\theta) J_s(\sqrt{\lambda}r)$, where $J_s$ is the Bessel's function of order $s$, and we know by the Dirichlet boundary conditions that $U(r_0, ...
4
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1answer
54 views

This linear operator has no eigenvalues

Let $T : L^2(\mathbb R) \to L^2(\mathbb R)$ be a linear operator defined by $$(Tf)(x)=f(x+1).$$ Show that $T$ has no eigenvalues, i.e., there exists no $f \not= 0$ in $L^2(\mathbb R)$ such that ...
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0answers
9 views

Minimal set of functions spanning the same space as a larger set

Apologies if this has been asked already; I'm not sure how to phrase it in a search. I have a set of functions $f_k$ with an inner product $\langle f_i,f_j\rangle$ (which I compute using Monte ...
4
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0answers
56 views

Solving 2D Laplacian eigenvalue problem with non-standard Dirichlet boundary condition

I have to solve the following eigenvalue problem, i.e. find eigenvalues and eigenfunctions (some of you will notice that this is the Schrödinger equation): $$-\frac{\hbar^2}{2m}\left( ...
0
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1answer
7 views

Triangle with the lowest laplacian eigenvalue under the Dirichlet boundary condition

Let us fix the area of the triangle. Which triangle has the lowest Laplacian eigenvalue? The equilateral one?
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0answers
15 views

eigenfunction derivation for 2nd order ODE

Any one help in the attached derivation. I am lost from eqn 4.13 till basically 4.24. Regards PDF file showing the proof steps
0
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1answer
36 views

How do solve this pde problem?

EDIT: I know somehow, we end up with an equation relating the derivative of some coefficients to the rest of the stuff. I'm not sure where this equation, or even the constant that we use to get it, ...
0
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1answer
40 views

eigenvaue of Sturm Liouville problem

Let the limit probem $$ \begin{cases} (P(x) y')' + q(x) y' + \lambda r(x) y=0\\ \alpha_0 y(0)+ \alpha_1 y'(0)\\ \beta_0 y(l) + \beta_1 y'(l) \end{cases} $$ with $\alpha_0^2 + \alpha_1^2 >0$ and ...
0
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0answers
35 views

Eigenvalue of Integral Operator and Gamma Function

$''$ Prove that the following integral operator $ Ku(x) = \int_{0}^{ \infty } \ e ^{-xy} u(y) dy $ has as eigenfunction the $ φ_α(x) = \sqrt {Γ(α)} x^{-α} + \sqrt {Γ(1-α)} x^{α-1} $ for $ ...
2
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2answers
35 views

Spectrum of an unbounded operator

Consider a densely defined unbounded operator $A_0:D(A_0)(\subset{H})\to H$ which has the following properties: 1- Symmetric, $\langle A_0x,y\rangle=\langle x,A_0y \rangle$ 2- Positive, $\langle ...
0
votes
2answers
69 views

Find the eigenvalues and eigenfunctions for $y''+\lambda y=0$ where $y'(1)=0$ and $y'(2)=0$

As stated in the title: Find the eigenvalues and eigenfunctions for $y''+\lambda y=0$, where $y'(1)=0$ and $y'(2)=0$. So I have already eliminated the cases for $\lambda=0$ and $\lambda<0$ and I'm ...
0
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0answers
20 views

Computing eigenfunctions, difference between beta and beta prime.

I am trying to implement the method described in the following paper. I am just kind of confused as to the difference between beta and beta prime. I am much more a computer programmer than ...
1
vote
1answer
46 views

Why can we assume that $||w||^2 = 1$?

The context: We are looking at orthogonality and general Fourier series. Given $\lambda$ an eigenvalue, and $w$ the corresponding eigenfunction, we are studying the eigenvalue problem: $w'' = -\lambda ...
1
vote
1answer
27 views

Having trouble with an Eigenvalue Differential Equation

Here is the problem: $$ x^2y''-xy+\lambda y = 0,\quad y(1)=0,\quad y(L)=0,\quad L>0 $$ I am asked to find the Eigenvalues and Eigenfunction. I can't figure out how to get a general equation for ...
0
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0answers
22 views

Orthogonality of eigenfunctions' derivatives

I asked a similar question very recently on here but realised it was very poorly phrased so I will try again with a simpler version. Apologies. We know that modes of vibration of an Euler-Bernoulli ...
5
votes
2answers
82 views

Proving that the eigenfunctions of the Laplacian form a basis of $L^2(\Omega)$ (and of $H_0^1(\Omega)$)

I am studying the eigenfunctions and eigenvalues of the Laplacian on an open, bounded domain $\Omega \subset \mathbb{R}^n$ with homogeneous Dirichlet boundary conditions. I have read about the the ...
0
votes
1answer
29 views

Laplacian eigenvalue with inhomogeneous boundary condition

Let $\Omega$ be some closed, bounded subset of $\mathbb{R}^2$ and $\Delta = \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2}$ be the Laplacian operator. Standard texts on PDEs always ...
0
votes
1answer
24 views

Find the Eigenvalues of $xy''+y'+λy=0, y(1)=5, y(e)=2$

Hi everybody I have to Find the Eigenvalues of $xy''+y'+λy=0,$ in $y(1)=5, y(e)=2$ I think it has to be in the Stourm-Liouville form: $d/dx(xy')+λy/x=0$ but Im not sure about this
0
votes
1answer
32 views

Help on proof the Sturm-Liouville eigenfunctions are all real

Now, there is a guiding question as follows Let $\lambda$ and $\phi$ be an eigenvalue and a corresponding eigenfunction respectively. Let $\phi(x)=U(x)+iV(x)$ and show that $U,V$ are ...
1
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1answer
21 views

eigenfunction of Schrodinger equation

I am trying to derive the eigenfunctions from $$-\frac{1}{2}\partial_x^2\phi_k(x)+\frac{1}{2}x^2\phi_k(x)=\lambda_k\phi_k(x).$$ I got stuck here. I don't know which method I need to use. Actually, I ...
0
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1answer
37 views

How to find the eigenfunctions of a differential operator.

Consider a linear differential operator $$L=\frac{d^2}{dx^2}.$$ How would one determine that the normalised eigenfunctions of $L$ are $$\phi_n(x)=\sqrt{2}\sin{(n\pi x)}?$$
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0answers
27 views

How to calculate the dominant eigenvalue of the Hessian matrix?

Now, I'm using L-BFGS method to solve the minimization problem of $\phi = \phi_d + \phi_m$. The problem is ill posed and large scale, so it is necessary to use the approximation of the Hessian matrix ...
1
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0answers
37 views

Eigenfunction and eigenvalues of Laplacian

I'm wondering about some definitions of the eigenvalues and eigenfunctions of the laplacian operator and I would be really glad if you can help me on these definitions. Let's make things simple. If ...
2
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2answers
43 views

Solve eigenvalue problem $(\frac{u'}{x})'+\frac{\lambda}{x}u=0$

Consider the eigenvalue $(\frac{u'}{x})'+\frac{\lambda}{x}u=0$,$x\in (1,2)$. And $u(1)=u(2)=0$. I want to determine the sign of the eigenvalues first. But since it is not a standard eigenvalue ...
1
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0answers
14 views

Change of basis to a reduced set when function has known values

Let $\gamma_1\left(x\right) = \sum_{n=1}^{N} \alpha_n f_n\left(x\right)$ where $f_n\left(x\right)$ are orthonormal under an inner product over a range $\left[x_1, x_2\right]$. I already know all of ...
0
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0answers
48 views

Eigenvalues of $-\Delta u -u\ln u=\lambda u$

$\Omega$ is a bounded open subset of $R^n$ $$ -\Delta u -u\ln u=\lambda u \\ u|_{\partial\Omega}=0 $$ What should I read about this eigenvalue question ? I mean some reference or book.I want to ...
0
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0answers
22 views

First eigenfunction of laplacian has minimum in compact set of $\mathbb{R}^n$

I encountered a problem which leads to prove $\frac{1}{\left | \varphi _{1}(x) \right |}$ is a bounded function, $ \varphi _{1}$ is the first eigenfunction of laplacian in an open, bounded, smooth ...
3
votes
3answers
93 views

Can $e^{ax}$ be said to be the eigenfunction of the operator $\frac{d^{(n)}}{dx}$?

I'm gradually getting familiar with operators (as they are used in QM) and the terminology surrounding them, and I was wondering whether all the (to me) well-known operators have straight-forward, ...
1
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0answers
36 views

Eigenfunction expansion in several dimensions

Im trying to solve the following problem \begin{aligned} \psi_xx+\psi_zz &= 0, -h(x,t) < z < \eta(x,t) \\ \psi_z + h_x \psi_x + h_t &= 0, z = -h(x,t) \\ \psi_z - \eta_x \psi_x - \eta_t ...
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0answers
22 views

Prove that the special Hermite functions are eigenfunctions of $R_{x,y}$?

How to prove that the special Hermite functions are eigenfunctions of the rotation operators $$R_{x,y} = x\frac{\partial}{\partial y}- y\frac{\partial}{\partial x}.$$ Where the special Hermite ...
3
votes
2answers
60 views

About Second-order Linear Homogenous ODE

One way to solve second-order linear homogeneous ode with constant coefficients is to do the following things: $$a\left(\frac{\mathrm d^2}{\mathrm dx^2}\right)f+b\left(\frac{\mathrm d}{\mathrm ...
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1answer
55 views

Separable solution to a nonlinear parabolic PDE

I seek a separable solution to the nonlinear parabolic partial differential equation, $\frac{\partial u}{\partial t} = u \frac{\partial u}{\partial x^2} + u^2.$ The physics of the problem allow ...
3
votes
0answers
28 views

Heat equation with mixed boundary conditions

I am trying to solve the following problem $$\left\{\begin{matrix}\frac{\partial u}{\partial t}=\frac{\partial^2 u}{\partial x^2},& 0<x<1,t>0,\\ u(0,t)=\frac{\partial u}{\partial ...
1
vote
1answer
78 views

rayleigh quotient of eigenvalue problem (sturm liouville theory and partial differential equations)

I am reading "A First Course in Partial Differential Equations with Complex Variables and Transform Methods" (Weinberger, p. 168). if we have the eigenvalue problem $$ (pu')'- qu + \lambda \rho u = 0 ...
0
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0answers
24 views

Neumann Series for integral equation with inhomogeneous term zero

Consider the method described in the following article: http://mathworld.wolfram.com/IntegralEquationNeumannSeries.html In this notation, what happens when $ f(x)=0 $? All the terms seem to be zero ...
0
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1answer
47 views

Can the zero vector be within the eigenspace

I have a matrix that looks like this: $$ \begin{pmatrix} 1 & 2 & 4 \\ 2 & 4 & 2 \\ 4 & 1 & 1 \end{pmatrix} $$ Now the calculated eigenvalues are: $-3$, $2$ and ...
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0answers
49 views

Eigenfunctions of integral operator

I am faced with the problem of calculating the eigenfunctions for an operator of the form: $(Kf)(x) = \int_{-\infty}^{\infty} K(x-\alpha y) f(y)dy $ Does anyone know for which functions (or types of ...
0
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0answers
30 views

For what operators are $sin(ax)$ and $cos(bx)$ eigenfunctions?

So it is clear that the operator $\frac{d^{2}}{dx^{2}}$ is an eigenfunction of $sin(ax)$ and $cos(ax)$. For what other operators are sins and cosines eigenfunctions?
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0answers
30 views

How to identify a process via its Karhunen-Loeve expansion?

Suppose that you are given the following Karhunen-Loève expansion of a real-valued continuous Gaussian stochastic process, $x$. $$x(t) = \sum_{k=1}^{\infty}z_{k}\cdot \frac{\sqrt{2}\sin((k-0.5)\pi ...
2
votes
1answer
89 views

Nodal Lines of the Eigenvalue problem $\Delta u=\lambda u$

I have really enjoyed performing the method of separation of variables to identify the eigenfunctions and nodal lines (the set of points for which each eigenfunctions vanishes) of the 2-D wave ...
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0answers
18 views

Bounds on eigenfunctions of integraloperator

Let $K: [0,1]\times[0,1] \to \mathbb{R}$ be a symmetric positive definit and continuous function. It is known my Mercer's theorem that $$ [T_K \varphi](x) =\int_0^1 K(x,s) \varphi(s)\, ds $$ is ...
1
vote
1answer
42 views

Solving Laplace equation in a square with one insulated border

I keep getting stuck on this problem, so if someone could point out where my method is flawed and how I should approach this problem, that would be extremely useful. We're considering the square ...