1
vote
2answers
173 views

Eigenfunctions of the Laplacian with imaginary eigenvalue

What are all the $\pm i$ eigenfunctions of the Laplacian on $\mathbb{R}^2$ (or on some domain in $\mathbb{R}^2$)? I know of a few: things like $e^{e^{i \frac{\pi}{4}}x} + e^{e^{i \frac{\pi}{4}}y}$ or ...
2
votes
0answers
21 views

eigen value problem with Robin Boundary Conditions at both ends

This is a problem from the book Partial Differential Equations by Walter.A.Strauss. Consider the eigen value problem with Robin Boundary Conditions at both ends: $-X''=\lambda X$ $X'(0)-a_0X(0)=0$ ...
8
votes
3answers
151 views

What is spectrum for Laplacian in $\mathbb{R}^n$?

I know very well that Laplacian in bounded domain has a discrete spectrum. How about Laplacian in $\mathbb{R}^n$?(not in some fancy-shaped unbounded domain, but the whole domain) Where can I find ...
2
votes
1answer
118 views

Second Derivative of Eigenvalue

Consider the potential $V_t=tx$ and the corresponding Schrödinger operator $H_t=- \frac{\partial ^2}{\partial x^2}+V_t$ on $L^2([0,R])$ with Neumann or Dirichlet boundary conditions. Let ...
2
votes
0answers
40 views

Eigenvalue problem from the PDE point of view

I study the eigenvalue problem for the Laplacian $-\Delta$ on a bounded domain $\Omega\subset \mathbb{R}^n$ with Dirichlet boundary conditions, but unfortunately my knowledgue about solving a boundary ...
0
votes
0answers
33 views

A domain in which the dirichlet laplacian has eigen values of all orders

I am trying to come up with an example to the following. Construct a domain $\Omega$ in all dimensions $n \in \mathbb{N}$, such that the spectrum of the Dirichlet Laplacian on such a domain (i.e., ...
1
vote
1answer
161 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} ...
1
vote
1answer
63 views

compare of eigenvalues $\lambda_1(D_a)$ and $\lambda_1(D_c)$.

Let $f(x)$ be a smooth function on $[-1,1]$, such that $f(x)>0$ for all $x\in(-1,1)$,$f(-1)=f(1)=0$. consider $\gamma\subset\Bbb{R}^2$ the graph of the $f(x)$. Let $T_a$ the symmetry with respect ...
1
vote
1answer
153 views

the first eigenfunction of Dirichlet problem

Let $\Omega$ be a bounded planar domain which has a axis of symmetry and $T:\Bbb{R}^2\longrightarrow\Bbb{R}^2$ symmetry with respect to this axis. Let $u_{1}(x)$ be the first eigenfunction of ...
3
votes
1answer
156 views

Inequality between Neumann and Dirichlet eigenvalues

Let $\Omega$ be a fixed, smooth and bounded domain in $\mathbb{R}^n$. If we denote with $\{\lambda_n\}_{n \ge 1}$ the nondecreasing sequence of eigenvalues of the Dirichlet problem $$\left\{ ...
2
votes
0answers
153 views

Biharmonic operator

Consider the problem: $$ \Delta^2 u = f$$ on the square domain $U=(0,1)\times(0,1)$ with boundary conditions: $$ u(x,y)=\Delta u(x,y) = 0$$ for $(x,y) \in \partial U.$ I try to solve it with the ...
1
vote
1answer
97 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 ...
4
votes
1answer
498 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}) ...
1
vote
1answer
44 views

Inequality from eigenvalue relations

This arises from the Saddle Point theorem as found in Rabinowitz "Minimax Methods in Critical Point Theory", but I'll just provide the relevant information to what I'm trying to understand. We ...
3
votes
1answer
100 views

Numerical Computation of Eigenvalues

I am trying to find the first few eigenvalues of an operator defined by the following PDE: $$ \begin{cases} -\Delta u +(1-\varphi)u=\lambda u, & \text{ on }\Omega = [0,1]^2 \\ u=0 & \text{ ...
3
votes
3answers
305 views

Stuck on Laplace's Equation

I am trying to solve the following: $-\theta_{yy}-\theta_{xx}=0$ $\theta(0,y)=-1$ $\theta(1,y)=1$ $\theta_y(x,0)=1$ $\theta_y(x,1)=0$ I can separate this into two different ...
5
votes
1answer
258 views

Physical interpretation: weighted eigenvalues of the Laplacian with a potential

I've already posted this question on Physics.SE, but I thougth it could be useful to ask also here. No problem if moderators will ask me to cancel this thread... But, please, have mercy! :-D Let ...
2
votes
0answers
84 views

First Weighted Eigenvalue of the Laplacian

Let $\Omega$ be a ball centered in the origin and let $\lambda_1(\Omega)$ be the first (or lowest) eigenvalue of the Dirichlet Laplacian in $\Omega$: $$\lambda_1 (\Omega) =\min_{u\in H_0^1 (\Omega),\ ...
1
vote
1answer
71 views

modular group differential equation solutions

given the PDE Eigenvalue problem $ y^{2}( \partial _{x}^{2}f(x,y) +\partial _{y}^{2}f(x,y))= E_{n}f(x,y) $ (1) if we are on the poincare disc so i impose the conditions $ x'=x+1$ invariance $ ...
1
vote
0answers
223 views

About solving an homogeneous PDE and its eigenfunctions

I'm not a mathematician, I may be inaccurate ; sorry for that ! I have two questions. First, I need some help to clarify the method about solving and homogeneous PDE. I have read somewhere (and my ...
2
votes
0answers
115 views

Principal eigenvalue

How is the principal eigenvalue of elliptic differential operator defined? Is it just a spectral radius?
4
votes
2answers
685 views

How to prove Weyl’s asymptotic law for the eigenvalues of the Dirichlet Laplacian?

The following comes from Springer Online Reference Works: Consider a bounded domain $\Omega\subset\mathbb{R}^n$ with a piecewise smooth boundary $\partial\Omega$. $\lambda$ is a Dirichlet eigenvalue ...
10
votes
1answer
1k views

How to compute the first eigenvalue of Laplace operator in an ellipse?

Let $\mathcal{E}$ be an ellipse in the $\mathbb{R}^2$ plane with center in $o=(0,0)$, given focal distance $c\geq 0$ and given area $A>0$. It is a fact that the eigenvalue problem for the Laplace ...