All Questions
21 questions
2
votes
0
answers
102
views
Semiclassical limit of spectral gap of Schrödinger operators with nonsmooth potential
Let $\Omega$ be a connected compact subset of $\mathbb{R}^d$. It is well known that for a smooth potential $V:\Omega \to \mathbb{R}$ that has a unique nondegenerate minimum $V(0) = 0$, the operator $H ...
- 1,551
2
votes
1
answer
196
views
Estimate for the operator $A A_D^{-1}$
Let $O\subset\mathbb{R}^d$
be a bounded domain of the class $C^{1,1}$
(or $C^2$
for simplicity). Let the operator $A_D$
be formally given by the differential expression $A=-\operatorname{div}g(x)\...
3
votes
1
answer
162
views
On a compact operator in the plane
Let $\Omega \subset \mathbb R^2$ be a bounded domain with a smooth boundary. Let $$\bar{\partial}= \frac{1}{2} ( \partial_{x^1} + i \,\partial_{x^2}),$$
and let $G: L^2(\Omega)\to H^2(\Omega)$ be the ...
- 4,145
4
votes
0
answers
137
views
Eigenvalues of Schrödinger operator with Robin condition on the boundary
Let $(M^2,g)$ be a compact Riemannian surface with boundary and let $L = \Delta_g + q$ be a Schrödinger operator, where $\Delta_g = -\operatorname{div} \nabla$ is the Laplacian with respect to the ...
- 2,095
2
votes
0
answers
67
views
A mapping property for fractional Laplace--Beltrami operator
Suppose that $g$ is a smooth Riemannian metric on $\mathbb R^3$ that is equal to the Euclidean metric outside the unit ball. Let us define the fractional Laplace--Beltarmi operator on $\mathbb R^3$ of ...
- 4,145
4
votes
1
answer
212
views
Existence of eigen basis for elliptic operator on compact manifold
Let $M$ be a compact Riemannian manifold. Let $E$ be a vector bundle over $M$ equipped with a Hermitian (or Euclidean) metric on its fibers. Let $D$ be a linear elliptic differential operator acting ...
- 21.8k
5
votes
1
answer
222
views
Domains with discrete Laplace spectrum
Let $\Omega \subset \mathbb{R}^n$ be a domain. Assume that the Laplacian $-\Delta=-(\partial^2/\partial x_1^2+\cdots+\partial^2/\partial x_n^2)$ has a discrete spectrum on $L^2(\Omega)$ (i.e., we are ...
- 443
2
votes
1
answer
209
views
Kernel for an equation involving the Ornstein-Uhlenbeck operator
Consider the following PDE on $\Omega\subset \mathbb{R}^n$ for $n\geq 2:$
\begin{align}
\Delta u - x\cdot \nabla u &= f(x),\text{ in } \Omega\\
u&=0 \text{ on }\partial \Omega
\end{align}
Are ...
- 537
-1
votes
1
answer
77
views
Applications and motivations of resolvent for elliptic operator
Let $ A(x)=(a_{ij}(x)):\mathbb{R}^d\to\mathbb{R}^{d\times d} $ be a matrix satisfying ellipticity condition, that is
\begin{align}
\mu^{-1}|\xi|^2\geq \sum_{i,j=1}^da_{ij}(x)\xi_i\xi_j\geq\mu|\xi|^2
\...
- 1,219
4
votes
0
answers
82
views
On the convergence of the spectral decomposition of a harmonic function
Let $(M,g)$ be a smooth compact Riemannian manifold of dimension $n\geq 2$ with a smooth boundary. Denote by $0<\lambda_1\leq \lambda_2\leq\ldots$ the Dirichlet eigenvalues of $-\Delta_g$ on $(M,g)$...
- 4,145
2
votes
0
answers
85
views
Proving an eigenvalue bound without resorting to Weyl's law
Suppose $(M,g)$ is a smooth compact Riemannian manifold of dimension $n\geq 2$ with smooth boundary and denote by $\{\phi_k,\lambda_k\}_{k\in \mathbb N}$ its Dirihclet spectral decomposition for the ...
- 4,145
2
votes
0
answers
77
views
Weyl's law and eigenfunction bounds for weighted Laplace-Beltrami operator
I would appreciate any answers or even references for the following problem.
Let $(M,g)$ be a complete smooth Riemannian manifold with an asymptotically Euclidean metric (let's even say that the ...
- 4,145
3
votes
0
answers
67
views
Eigenvalues of an elliptic operator on shrinking domains
This was probably done somewhere 100 times, but I can't find a reference.
Assume that we have a bounded star-shaped domain $\Omega\subset \mathbb{R}^n$ with piece-wise smooth boundary and a general ...
- 445
2
votes
0
answers
100
views
Existence and uniqueness for $\Delta f + \lambda f = g$ on $S^2$ for $\lambda>0$ [closed]
Consider the PDE
$$\Delta f + \lambda f = g$$
on $S^2$, where $\Delta$ is with respect to the round metric, $g \in L^2(S^2)$ and $\lambda>0$. I wish to study the existence and uniqueness of this ...
- 969
1
vote
0
answers
148
views
Spectrum of Laplacian-like operator
Let $\kappa_1, \kappa_2>0$ be fixed.
Consider the unbounded operator $A: D(A) \rightarrow L^2(-1,1)\times\mathbb{R}$ defined by
$$
A\begin{bmatrix} y \\ h \end{bmatrix} = \begin{bmatrix} \...
- 309
21
votes
1
answer
742
views
Non real eigenvalues for elliptic equations
I am looking for an example of a pure second order uniformly elliptic operator
$L=\sum_{i,j=1}^da_{ij}(x)D_{ij}$ in a bounded domain $\Omega$ (with Dirichlet boundary conditions, for example) having a ...
- 6,035
4
votes
1
answer
3k
views
Laplace spectrum of the $2$-Sphere [closed]
The $2$-sphere $S^2$ endowed with usual round metric, we have a Laplacian operator $\Delta_{\mathrm{d}} = \mathrm{d}^*\mathrm{d} + \mathrm{d}\mathrm{d}^*$ acting on functions. The eigenvalues of this ...
- 643
3
votes
0
answers
167
views
Asymptotic behaviour of principal eigenfunctions and large deviations
Dear Math Overflowers,
I am currently interested in a particular problem involving Large Deviations. I am only going to talk about the PDE side of the problem, but I'll be happy to provide more ...
- 5,380
2
votes
0
answers
344
views
Spectrum of Laplacian depending on boundary conditions [closed]
Consider a compact domain $\Omega \subset \mathbb{R}^n$ with smooth boundary for simplicity. Consider the Laplacian operator with zero Dirichlet boundary conditions on $\Omega$. It is well-known that $...
- 1,407
3
votes
1
answer
206
views
Bounded solutions for Schrödinger equation at the edge of the essential spectrum
Let $V:R^d\to R_+$ be with a compact support. The Schrödinger operator $H_a=-\Delta - a V$ acting in $L^2(R^d)$ has then (at most) finitely many negative eigenvalues. Denote the number of negative ...
- 175
3
votes
3
answers
358
views
Limits for eigenvalues for the Dirichlet Laplacian
If $\Omega$ is a bounded domain in $\mathbb{R}^n$, let $\lambda(\Omega)$ be an eigenvalue of the problem
$$
\begin{cases}
-\Delta u=\lambda u & \mbox{in }\Omega\\
u=0 & \mbox{on }\partial\...
- 93