For questions on eigenfunctions.

learn more… | top users | synonyms

1
vote
0answers
75 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 ...
0
votes
0answers
30 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 ...
0
votes
1answer
28 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
vote
1answer
36 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 $u\in L^2(U)=1$ ...
0
votes
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{\...
0
votes
0answers
17 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 ...
0
votes
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
votes
1answer
33 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 ...
0
votes
0answers
24 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 (...
1
vote
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
votes
1answer
35 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}...
1
vote
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
votes
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 ...
0
votes
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",...
0
votes
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=(\...
0
votes
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
votes
1answer
103 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
votes
1answer
42 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 ...
0
votes
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
votes
0answers
35 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 $...
1
vote
1answer
33 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 $\...
1
vote
0answers
14 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 ...
0
votes
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 $g(x)=c_1 \cos (\mu x)+ c_2 \sin (\mu x)$? ...
1
vote
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 \...
0
votes
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 $-x^3 + 2x^2 - 17x + 9=0$.
0
votes
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
votes
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)}{\...
2
votes
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
votes
1answer
56 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)(...
0
votes
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
votes
0answers
58 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
votes
1answer
9 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?
0
votes
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
votes
1answer
38 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
votes
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 ...
0
votes
1answer
82 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
votes
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
48 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
votes
0answers
26 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
90 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
34 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
25 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
34 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
vote
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
votes
1answer
39 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 $$\phi_n(x)=\sqrt{2}\sin{(n\pi x)}?$$
0
votes
0answers
29 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 ...