2
votes
2answers
47 views

What is the right domain for this Hamiltonian

I want to define a proper domain $D(H) \subset L^2$ for this Hamiltonian ( $\theta$, $\phi$ are the standard angles in spherical coordinates). Furthermore, the wave function is supposed to satisfy ...
0
votes
1answer
87 views

Application of eigenvalueproblems for the wave equation

I'm currently searching for a nice little application of an eigenvalueproblem and found the following for acoustics - but one part doesn't make sense for me. Consider the wave equation to find some ...
0
votes
0answers
16 views

Eigenvalues of the Laplace-Beltrami Operator on Lorentzian manifolds

Are there explicit expressions for the eigenvalues of the Laplace-Beltrami Operator on manifolds with the metric $$ g_{\mu \nu }= \begin{pmatrix} c^2-\|u\|^2 & u^\top \\ u & -I \\ ...
1
vote
0answers
23 views

Spectrum of the operator of differentiation along streamlines

Suppose $\mathbb T^2$ denotes the two-torus and suppose $\psi^0$ is a steady state smooth stream function on $\mathbb T^2$ and define $\mathbf u^0 = (-\partial_y \psi^0,\partial_x \psi^0 )$ be the ...
8
votes
3answers
149 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 ...
3
votes
0answers
101 views

Simultaneous Orthogonal basis for $L^2(\mathbb{R}^n)$ and $H^1(\mathbb{R}^n)$

Given a smooth bounded set $U\subset \mathbb{R}^n$, there is a simultaneous orthogonal basis for $L^2(U)$ and $H^1_0(U)$ by the existence of eigenvectors to the Laplacian in a bounded domain, which ...
6
votes
1answer
62 views

About some properties of the heat kernel

Maybe my questions are stupit, but I'm so unsure about some properties of the heat equation. Suppose we have a compact and smooth manifold M (possibly with boundary), then in a lot of books they ...
5
votes
1answer
177 views

Why is it important to study the eigenvalues of the Laplacian?

Why is it important to study the eigenvalues of the Laplacian acting on regions in $\mathbb R^n$? What information does this give us? What problems does this information help us solve? In particular, ...
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
0answers
94 views

Compactness of advection diffusion operator (elliptic operator)

Consider the ADE defined on a compact domain $\Omega \in \mathbb R^n$, with boundary $\partial \Omega$ Assume $u(x)$ is smooth incompressible vector field, with $u(x).\hat n(x)=0$ for $x\in \partial ...
4
votes
1answer
112 views

Can 0 be an eigenvalue?

Let $-\Delta $ be the positive Laplacian and consider the operator $$ -\Delta + V $$ on $L^2(\mathbb{R}^3)$ with domain the Sobolev space $W^{2,2}(\mathbb{R}^3)$. Here $V:\mathbb{R}^3\to \mathbb{R}$ ...
5
votes
1answer
192 views

How to prove that the spectrum of the Laplacian over $\Omega\subset \mathbb{R}^n$ is negative?

I am looking for a proof of this well known fact and I guess it has to do with integration by parts (Green's identity). Unfortunately, I only know about 1-d integration by parts( I am just 3rd ...
5
votes
1answer
143 views

Self adjointness of an elliptic differential operator

Let $A:D(a)\to L^2(\mathbb{R}^n)$ be an elliptic partial differential operator $$ A(f)=\sum_{i,j=1}^{\infty}\partial_{x_j}(a_{ij}(x)\partial_{x_i}f) $$ where $a_{ij}\in C^{\infty}_b(\mathbb{R}^n)$, ...
2
votes
0answers
87 views

Dirichlet eigenvalue problem on the Hilbert cube

I'm trying to solve the Dirichlet problem for the Helmholtz equation \begin{aligned}-\triangle u & = & \lambda u, & x\in\Omega,\\ u & = & 0, & x\in\partial\Omega, ...
5
votes
2answers
106 views

Looking for a first order perturbation of the Laplacian having 0 in its spectrum

I would like to find an explicit example of a linear elliptic operator having the following form: $$Lu=-\Delta u +b(x)\cdot \nabla u, $$ where $b\colon \mathbb{R}^n \to \mathbb{R}^n$, and such that ...
2
votes
0answers
55 views

What are the connections between spectral expansion and differential operator?

For instance, for a nice function $f$ on the unit circle, we have its Fourier expansion, $$f(x)=\sum_n \hat{f}(n) e^{inx},$$ where the exponentials are eigenfunctions for differential operator ...
3
votes
0answers
86 views

Compute the continuous spectrum of an unbounded operator in $L^2(\mathbb{R}^2)^2$.

In "Béthuel, F. und J. C. Saut: Travelling waves for the Gross-Pitaevskii equation. I. Ann. Inst. H. Poincaré Phys. Théor., 70(2):147–238, 1999." the authors write on page 150, that one can easily ...
9
votes
2answers
226 views

Are there n-th roots of differential operators?

In analogy to a Dirac operator, it seems to me that formally, the equation $$\frac{\partial^n}{\partial x^n}f(x,y)=D_yf(x,y)$$ is solved by $$f(x,y)=\exp{(x \sqrt[n]{D_y})}\ g(y).$$ Is there a ...
1
vote
1answer
141 views

Spectrum Of The Laplacian Question

I'm given the following question (from Davies' book - Spectral theory of differential operators): Use the theorem (*) below to prove that if $\Omega$ is a convex region in $ \mathbb{R}^2 $ , then: $ ...
2
votes
2answers
158 views

Symbol Of Differential Operators

When we are given a differential operator of the form $Lf : = \sum_\alpha a_{\alpha}(x) D^ \alpha f(x) $ , we can define the symbol associated with it to be the function: $a(x,y) := \sum_\alpha ...
5
votes
1answer
322 views

Is it true that the Laplace-Beltrami operator on the sphere has compact resolvents?

We consider the Riemannian structure on the sphere $\mathbb{S}^n$ seen as a submanifold of $\mathbb{R}^{n+1}$ and the Laplace-Beltrami operator defined on $C^\infty(\mathbb{S}^n)$ by the equation ...
4
votes
2answers
301 views

Explicit expression for eigenpairs of Laplace-Beltrami operator

In $R^n$, the Laplace-Beltrami operator is just the Laplacian, and its eigenstructure is well known. There are also explicit expressions for the eigenvalues/eigenvectors of the Laplace-Beltrami ...
1
vote
2answers
531 views

How do the solutions to the wave and heat equations converge in general?

I would like to check my understanding with someone if possible. When we cover the heat and wave equations, for instance, in "methods" courses at university, they normally restrict the initial ...
2
votes
1answer
189 views

Schrödinger operator: where is the generator to be defined?

The theory as I know it Let $\mathcal{H}$ be a Hilbert space and $(A, D(A))$ a self-adjoint operator acting on it. The Spectral Theorem (cfr. Reed & Simon Methods of modern mathematical physics, ...