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4 votes
1 answer
195 views

Asymptotic spectrum of a complex Sturm-Liouville differential operator

Let $\varepsilon > 0$ and consider the (complex) Sturm–Liouville differential operator on $[0,1]$ given by $$ \mathcal{L}_\varepsilon f(x) = \varepsilon^2 f''(x) + i V(x) f(x), $$ with Neumann ...
Matheus Manzatto's user avatar
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 ...
Lwins's user avatar
  • 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)\...
Yulia Meshkova's user avatar
2 votes
0 answers
94 views

Dimension of Laplacian eigenspaces along a smooth 1-parameter family of metrics

Let $(M^n,g)$ be a closed Riemannian manifold, $n \geq 2$. For a smooth 1-parameter family $g_t$, $t \in (-\varepsilon, \varepsilon)$, of Riemannian metrics on $M$ with $g_0 = g$, let $\lambda_k(t)$, $...
Eduardo Longa's user avatar
5 votes
2 answers
625 views

Reconstruction of second-order elliptic operator from spectrum

Let $M$ be a compact smooth manifold, $(\lambda_n)_{n=1}^{\infty}$ be a square-summable monotonically increasing sequence of non-negative numbers, and let $(f_k)_{k=1}^{\infty}$ be continuous ...
Math_Newbie's user avatar
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 ...
Ali's user avatar
  • 4,135
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 ...
Eduardo Longa's user avatar
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 ...
Ali's user avatar
  • 4,135
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 ...
asv's user avatar
  • 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 ...
mmen's user avatar
  • 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 ...
Student's user avatar
  • 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 \...
Luis Yanka Annalisc's user avatar
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 $(M,g)$...
Ali's user avatar
  • 4,135
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 ...
Ali's user avatar
  • 4,135
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 ...
Ali's user avatar
  • 4,135
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 ...
Ivan's user avatar
  • 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 ...
Laithy's user avatar
  • 969
1 vote
1 answer
75 views

A property for generic pairs of functions and metrics

Let $M$ be a compact smooth manifold with a smooth boundary. Given a smooth Riemannian metric $g$ on $M$, we denote by $\{\phi_k\}_{k=1}^{\infty}$ an $L^2(M)$--orthonormal basis consisting of ...
Ali's user avatar
  • 4,135
5 votes
1 answer
164 views

Spectrum of an elliptic operator in divergence form with a reflecting boundary condition

Let $\Omega$ be a bounded open domain and $v:\Omega\to\mathbb{R}^n$. Consider the following elliptic operator in divergence form, defined on smooth functions $u: \Omega \to \mathbb{R}$ \begin{align} L ...
Mr_Rabbit's user avatar
0 votes
0 answers
161 views

When linear strongly elliptic operators are invertible?

I am studying Pazy's book "Semigroups of Linear Operators and Applications to Partial Differential Equations" and when considering an operator like: A linear differential operator, $$A : W^{...
L.F. Cavenaghi's user avatar
2 votes
0 answers
123 views

Principal eigenvalue of non self-adjoint elliptic operators on closed manifolds

Consider the elliptic operator $Lu = - \Delta u + \langle \nabla u , X \rangle + c \, u $ acting on functions on a closed Riemannian manifold $M$. Here $\Delta$ denotes the Laplace-Beltrami operator, $...
Ramiro Lafuente's user avatar
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} \...
char's user avatar
  • 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 ...
Giorgio Metafune's user avatar
1 vote
0 answers
52 views

Laplacian eigenvalue problem for systems coupled along the boundary

I am looking for references on eigenvalue problems for systems of the following type: Let $\Omega$ be the region enclosed by a right triangle with legs $\Gamma_1$, $\Gamma_2$, and hypotenuse $\...
Justin Erik Katz's user avatar
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 ...
Pierre Dubois's user avatar
2 votes
0 answers
59 views

Zero in the spectrum of an elliptic second order operator

This might be considered as a continuation of my previous question Spectrum of a linear elliptic operator but is independent. I have another question on V. Gribov's paper "Quantization of non-Abelian ...
asv's user avatar
  • 21.8k
2 votes
0 answers
162 views

Spectrum of a linear elliptic operator

In the paper in quantum fields theory by Gribov,V.; (1978) "Quantization of non-Abelian gauge theories". Nuclear Physics B. 139: 1–19; in Section 3 the author makes the following claim from PDE and ...
asv's user avatar
  • 21.8k
5 votes
1 answer
147 views

Stable region of minimal hypersurfaces with finite Morse index

In this Inventiones Mathematicae paper, Fischer-Colbrie proved the following result (Proposition 1): Proposition: Let $ M$ be a complete two-sided minimal surface in a three manifold $N$. Then if $M$...
Onil90's user avatar
  • 823
3 votes
1 answer
182 views

Quasinilpotent vectors of Newton potential vanish

Suppose $\Omega$ is a smooth bounded domain in $\mathbb{R}^3$. Consider the Newton potential \begin{equation} T [\phi](x) = \int_{\Omega} \frac{1}{|x-y|} \phi(y)dy. \end{equation} It is well know ...
Bowen's user avatar
  • 31
8 votes
1 answer
399 views

Spectrum of a first-order elliptic differential operator

Suppose that I have a first-order elliptic differential operator $A: \mathrm{dom}(A) \subset L^2(E) \to L^2(E)$, where $(E,h^E) \to M$ is a hermitian vector bundle and $M$ is a compact manifold. I ...
Lashi Bandara's user avatar
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 ...
leo monsaingeon's user avatar
4 votes
0 answers
410 views

Spectral Gap of Elliptic Operator

Under what conditions on $a(x)$ and domain $D$, the spectral gap of the elliptic operator $ \nabla \cdot(a(x)\cdot \nabla)$ defined on $D$, can be controlled? The boundary condition is that the ...
Elliott's user avatar
  • 325
2 votes
0 answers
343 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 $...
SMS's user avatar
  • 1,407
1 vote
0 answers
259 views

An estimate for the solution of an elliptic PDE depending on a parameter

Let $\Omega\subset\mathbb R^n$ be a bounded domain with a sufficiently smooth boundary $\partial\Omega$. We assume $\lambda_1\in\mathbb R$ is the principle eigenvalue of the operator $$ -\Delta:\ H^...
CooLee's user avatar
  • 375
1 vote
0 answers
89 views

Eignfunctions of an elliptic operator

I am looking for a reference which makes it possible to say that one can devellop $f$ in the form of the sentence underlined by the yellow. Thank you in advance.
Fadil Kikawi's user avatar
1 vote
1 answer
178 views

Parabolic PDE Long Time Asymptotics and Elliptic Operator Spectrum II

This is a follow-up on a previous question. Now the parabolic PDE of $P(t,x,v)$ has two spatial dimensions. $$ \partial_t P = L^* P \tag1 $$ $$L^*P = \frac12\left(\kappa^2\frac{\partial^2}{\partial v^...
Hans's user avatar
  • 2,239
2 votes
1 answer
315 views

Parabolic PDE Long Time Asymptotics and Elliptic Operator Spectrum

How does one show directly that the solution following parabolic partial differential equation (PDE) of $p(t,v)$ approaches its stationary solution which is a solution of an elliptic partial ...
Hans's user avatar
  • 2,239
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 ...
poupy's user avatar
  • 175
2 votes
0 answers
218 views

Weyl's law for minimal surfaces

I wanted to know if there was some equivalent of Weyl law for the spectrum of the Jacobi operator of a minimal surface in the non-compact case. If the minimal surface is not closed, for example in $\...
Paul-Benjamin's user avatar
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\...
de Araujo's user avatar
2 votes
0 answers
71 views

Localized eigenfunctions of drift Laplacians

I am looking for literature which discusses localization of eigenfunctions of drift Laplacians, i.e. $L\underline u=-\Delta \underline u+\underline{v}.\nabla \underline u$ in 2D/3D domains with ...
user69741's user avatar
5 votes
2 answers
216 views

significance of the Fučík spectrum

The Fučík spectrum seems to gain momentum among people working on spectral theory, with almost 300 articles published on this topic over the last 5 years, according to Google scholar. There exist ...
Delio Mugnolo's user avatar
3 votes
1 answer
775 views

Lp estimate for resolvent of Laplace operator

Consider for $1<p<\infty$ operator $A_p:L_p(0,1)\to L_p(0,1), \ D(A_p)=\{u\in W^2_p(0,1): u'(0)=u'(1)=0\}, \ A_pu=u''$, i.e. $L_p$-realisation of the Laplace operator with Neumann bcd on the ...
Marcin Malogrosz's user avatar
4 votes
1 answer
413 views

Simplicity of eigenvalues of an elliptic operator under a symmetry assumption

A striking difference in the spectral analysis of 2nd order elliptic boundary-value problems between one and several space dimensions is the following. In one space dimension, the eigenvalues are ...
Denis Serre's user avatar
  • 52.3k
6 votes
3 answers
252 views

Stability of the spectrum for perturbations of the boundary

Consider the Laplace operator on a smooth bounded open set with Dirichlet boundary conditions. I need some result of the following type: if one perturbs the boundary in a suitable sense to be ...
Piero D'Ancona's user avatar
3 votes
1 answer
588 views

orthonormal basis of eigenvectors for laplacian on a concave polygon

I am interested in the Laplace operator $\Delta$ on a concave polygon. When the polygon is convex, it is known that $\Delta: H^2(\Omega) \rightarrow L^2(\Omega)$ is boundedly invertible. In addition, ...
localizer's user avatar