For questions on eigenfunctions.

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2answers
31 views

Solving non-homogenous PDE with forcing function (which diappears!) dependent only on time

Applying the method of eigenfunction expansion to the PDE $$u_t -c^2u_{xx}=F(t)$$ $$0<x<L, t>0$$ $$u(x,0)=f(x)$$ $$u_x(0,t)u_x(L,t)=0$$ for the homogenous part of this equation ($L[v(x,t)]=0$...
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0answers
23 views

Resonance in wave equation

I have solved the non-homogenous equation by the method of eigenfunction expansion $$u_{tt} - c^2 u_{xx}=F(x)\sin(\omega t)$$ $$0<x<L, t>0$$ $$u(x,0)=u_t(x,0)=0$$ $$u(0,t)=u(L,t)=0$$ and got ...
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0answers
9 views

What is the following operator which satisfies:

Question What is the following linear operator as an explicit expression of $s$ given the eigenfunction and the eigenvalue: $$ \hat O a^s = e^a a^s $$ Where $a$ is an arbitrary constant. $$ \hat ...
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1answer
32 views

Laplace equation on a cylinder

For the Laplace equation in 3D $$\nabla^2 u =u_{xx}+u_{yy}+u_{zz}=0$$ in a right cylinder with an arbitrarily shaped base, whose top is $z=H$, bottom is $z=0$, we assume the following boundary ...
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1answer
33 views

Finding all the eigenvalues and eigenfunctions for a BVP with an inequality condition

I am trying to find all the eigenvalues and eigenfunctions for the following boundary value problem \begin{eqnarray} \phi''(z) + \phi'(z) + \lambda \phi(z) &=& 0\\ \phi (0)&=& 0 \\ |\...
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1answer
33 views

Is $\lambda$ the eigenvalue of differential operator

I want to find the eigenvalue of the differential operator $D(f)=f'=λf$. By solving the differential equation $f'=λf$ I get the eigenfunction ${e^{\lambda t}}$ which means $D(e^{\lambda t})=\lambda e^{...
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1answer
128 views

How could I find an orthogonal basis for $H^k$? - HELP!!

Let $P^k=$homogeneous polynomials of degree $k$ in $x$, $y$, $z$, $k=0, 1, 2, \dots, $ i.e. $P^k= \text{span} \{x^{k_x}y^{k_y}z^{k_z} : k_x+k_y+k_z=k\}$ and $H^k= \{f \in P^k : \Delta f = 0\}$, where $...
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1answer
34 views

Eigensolver for Black-box matrix

$\DeclareMathOperator{\diag}{diag}$ Consider the generalized eigenproblem $A\mathbf{x}=\lambda B\mathbf{x}$. When solving PDEs numerically (specifically, I am interested on finding the Dirichlet ...
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0answers
47 views

If $\,\Delta_{\scriptsize\mathbb{R}^n}u = \lambda u\,$ and $\,u(x)=u(x+y)$, then $\,u_{y}(x):=e^{2 \pi i \left\langle x,y\right\rangle}$

Related to Analysis on Manifolds via the Laplacian page $52$, I would like someone explain to me why if we have a function $u$ such that $\,\Delta_{\mathbb{R}^{n\,}}u = \lambda u\,$ and $u(x)=u(x+y)$ ...
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0answers
33 views

Solution to the boundary value problem $y''+\left(\frac{(1-e^{-4x})^2}{2x^3}-\frac{2(1-e^{-4x})}{x}\right)y'-\frac{(1-e^{-4x})^4}{16x^4} y=0$

Solve the following boundary value problem $$y''+\left(\frac{(1-e^{-4x})^2}{2x^3}-\frac{2(1-e^{-4x})}{x}\right)y'-\frac{(1-e^{-4x})^4}{16x^4} y=0, \quad y(0)=y(\tfrac{1}{2})=0.$$ Note: I attempted ...
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0answers
22 views

Showing that ODE is not of Sturm-Liouville form

The PDE $$\frac{\partial u}{\partial t} = D\frac{\partial^2 u}{\partial x^2}-V_0\frac{\partial u}{\partial x}$$ can be separated into two ODEs by the method of separation of variables, and the ODE ...
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0answers
23 views

Eigenfunctions of $x^2M''+xM'+\lambda M=0$ with $M'(1)=0$ and $M'(L)=0$

If we make the substitution of variables by $z=\ln(x)$ in $$x^2M''+xM'+\lambda M=0$$ then we will get $$M''(z)=-\lambda M(z)$$ We can consider different cases for $\lambda$: Case 1: $\lambda>0$ ...
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0answers
79 views

Spherical Harmonics & Beltrami operator

I don't know if I can ask this question here, but there's a question on MO for which I have a good interest. The problem is I don't think I have competencies to do it. On the page The spherical ...
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0answers
23 views

Boundary value problems: eigenvalue and eigenfunction

I'm having trouble in understanding eigenvalues and eigenfunctions in BvP the problem is: $y''$ + $\lambda$$y$ = $0$ $y(0)=0$ $y(2\pi)$ = $0$. Make characteristic polynomial $r^2 + \...
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0answers
31 views

Can “eigenvalues” in eigenfunction expansion be non-scalar?

This question is a bit nebulous but I don't have a particular example in mind... In general under certain assumptions, one can use eigenfunction expansion to represent an operator. For instance, for ...
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1answer
31 views

Solving $\sin(\sqrt{\lambda}L) + \beta \cos(\sqrt{\lambda}L)\sqrt{\lambda} = 0$

I'm working with the ODE $$-\frac{d^2u}{dx^2}=\lambda u$$ and trying to find eigenvalues and eigenfunctions corresponding the boundary conditions $$u(0)=0, u(L)+\beta \frac{du}{dx}(L)=0$$ Assuming ...
1
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1answer
37 views

variational principle for the principal eigenvalue

I am reading the proof of theorem 2 in chapter 6 Evan PDE. I have difficulty verifying the following part of the proof, i.e. 3 questions here. 1) The assumptions $u\in H_0^1(U)$ and $u\in L^2(U)=1$ ...
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0answers
19 views

Constructing the general solution of a PDE using eigenfunctions

As I am learning how to solve PDE, I came into a rather bothersome problem. Let's say that I have to solve the Laplacian of a field $u(x,y)$, where I have $$\nabla^{2}u=0,$$ where $$\nabla^{2}=\frac{\...
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0answers
20 views

Linear system of advection diffusion equations

I am trying to find the eigenvalues and eigenfunctions of the coupled PDE system $$ \partial_t \vec{u} = - \stackrel{\leftrightarrow}{A} \partial_x \vec{u} + \stackrel{\leftrightarrow}{D} \partial_x^2 ...
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1answer
29 views

If we know Spec($M_1$) and Spec($M_2$), what could we say about Spec($M_1 \cup M_2$)?

Let two domains $M_1$ and $M_2$ (Dirichlet conditions). If we know the spectrum of the Laplacian on $M_1$ and $M_2$, what could we say about Spect($M_1 \cup M_2$)? Is there a theorem that might give ...
0
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1answer
35 views

The bigger the domain, the smaller the first eigenvalue - $\lambda(M_2) \leq \lambda(M_1)$ on the Laplacian

I know it is probably a silly question, but is there anyone could help me to complete of the corollary $3.1$ of that document? I pass a lot of time to try understanding the problem, but I can't ...
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0answers
25 views

Nullspace of strange operator

I have the following equation: $0=\frac{\partial}{\partial y}(e^{-\beta U(x,y)}\frac{\partial}{\partial y}(P(x,y,t)e^{\beta U(x,y)}))$ and would like to study its solvability (Fredholm) conditions (...
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1answer
47 views

Weyl's asymptotic law for eigenvalue on the rectangle $D = \{0 < x < a, 0 < y < b \}$ - $N(\lambda) \geq \frac{\lambda ab}{4 \pi} - C \sqrt{\lambda}$

I have a few difficulties understanding the example on the rectangle in the book Strauss W.A. Partial differential equations - an introduction (Wiley, $2008$, $2$nd Ed.) page $326$. I've managed to ...
2
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1answer
38 views

Weyl's asymptotic law for eigenvalues - Rectangle $D = \{0 < x < a, 0 < y < b \}$

Let the domain $D = \{0 < x < a, 0 < y < b \}$ in the plane. We now that $$\lambda_{n,m} = \frac{n^2 \pi^2}{a^2}+\frac{m^2 \pi^2}{b^2}$$ with the eigenfunction $$u_{n,m}= \sin(\frac{nπ}{a}...
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0answers
42 views

Dirichlet conditions - Explanation of the proof of theorem $4$

I have a few difficulties understanding the first part (Dirichlet conditions) of the proof of theorem $4$ in the book Strauss W.A. Partial differential equations - an introduction (Wiley, $2008$, $2$...
2
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0answers
38 views

Max-Min Principle with an example on intervals

I have a few difficulties to understand the max-min principle (Intuitive understanding of Maximin Principle). Is there anyone could explain this theorem in using $[a,b] \subset [a',b']$? I know that ...
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1answer
24 views

Any additional constraint will increase the value of the maximin

In the book Strauss W.A. Partial differential equations - an introduction (Wiley, $2008$, $2$nd Ed.) page $325$, there is the comment "any additional constraint will increase the value of the maximin",...
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3answers
37 views

Explication on how obtaining $\int \langle \nabla w, \nabla w\rangle = \lambda \int \langle w, w\rangle$

Could anyone is able to explain to me how to obtain $\int \langle \nabla w, \nabla w\rangle = \lambda \int \langle w, w\rangle$ related to user7530's comment in the question : Rayleigh quotient $Q=(\...
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1answer
20 views

Finishing off this Sturm-Liouville BVP

I'm looking at the Sturm-Liouville BVP $$\begin{cases} y'' + \lambda y = 0\\ y(0) + y'(0) = 0, y(1) = (0) \end{cases}.$$ I can do the problem but I can't finish it off at the very end (it's probably ...
4
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2answers
135 views

Can I integrate an asymptotic expression?

Suppose that $y(x; \epsilon)$ is a real-valued function of $x \in [a,b] \subset\mathbb{R}$ depending on a real parameter $\epsilon$, and that \begin{align} \int_a^b dx \ y(x; \epsilon) =& 1 &&...
4
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1answer
43 views

Intuitive understanding of Maximin Principle

From the the book page $324$, does someone could explain to me the Theorem $2$. Maximin principle? I have a bit of difficulties to well understand how works this theorem. A simple example would be ...
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0answers
11 views

Problem on Principal Component Analysis (P.C.A.)

Let $X \; = \; (X_1, X_2, \ldots, X_m)^T$ and $Y \; = \; (Y_1, Y_2, \ldots, Y_n)^T$. Let, $S$ = pooled variance-covariance matrix obtained from $X$ and $Y$. Let, $\alpha$ = principal component ...
2
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0answers
39 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 $...
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1answer
36 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 $\...
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0answers
15 views

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

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 ...
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1answer
25 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 $g(x)=c_1 \cos (\mu x)+ c_2 \sin (\mu x)$? ...
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0answers
18 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 \...
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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 =\left(\frac{\gamma_\mu(n)}{\...
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0answers
5 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
57 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 $(Tf)(...
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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 Carlo)....
4
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0answers
61 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( \frac{\partial^...
0
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1answer
11 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
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1answer
39 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
votes
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
44 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
votes
2answers
39 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 ...