Tagged Questions

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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 ...
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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: $-X''=\lambda X$ $X'(0)-a_0X(0)=0$ ...
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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 ...
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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 ...
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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 ...
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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., ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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),\ ...