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74 votes
2 answers
3k views

Can you hear the shape of a drum by choosing where to drum it?

I find the problem of hearing the shape of a drum fascinating. Specifically, given two connected subsets of $\mathbb R^2$ with piecewise-smooth boundaries (or a suitable generalization to a riemannian ...
Emilio Pisanty's user avatar
15 votes
3 answers
1k views

Version of Banach-Steinhaus theorem

I am wondering about the following version of the Banach-Steinhaus theorem. Let $A$ be a closed convex subset contained in the unit ball of a Banach space $X$ and consider bounded operators $T_n \in \...
Sascha's user avatar
  • 536
11 votes
3 answers
3k views

Dual space of $L^2(\mathbb{R},L^1(0,1))$?

I was wondering what the dual space of $L^2(\mathbb{R},L^1(0,1))$ is? (equipped with Lebesgue measures) Formally, one would suspect that it is just $L^2(\mathbb{R},L^{\infty}(0,1))$. But this may be a ...
Jacob Augstine's user avatar
11 votes
2 answers
1k views

Concentration compactness. Can this concept be stated in a theorem?

I recently attended a talk on NLS which is rather not my main field of interest. Yet, I got interested in a concept called concentration compactness during the talk. When I approached the speaker ...
Zinkin's user avatar
  • 501
10 votes
1 answer
3k views

Trace of integral trace-class operator

I have seen many answers to the converse question (which seems to be difficult in general), but I would like to ask the following: Let $T: L^2 \rightarrow L^2$ be a trace-class operator that is also ...
user avatar
9 votes
2 answers
778 views

Rellich's theorem from compact resolvent

On a compact Riemannian manifold, we know that the Laplacian $\Delta$ has compact resolvent. In proving this, one typical way is to use Rellich's theorem about the compact embedding of $H^1(M)$ into $...
anonymous's user avatar
8 votes
1 answer
359 views

Lax pairs in an abstract formalism

I am reading Integrals of Nonlinear Equations of Evolution and Solitary Waves by Peter Lax and I'm having a hard time. The methods are pioneering, of course, but Lax does not bother much to provide ...
Delio Mugnolo's user avatar
7 votes
2 answers
682 views

Hölder continuity for operators

Let $x,y$ be positive real numbers then $$|\sqrt{x}-\sqrt{y}|=\dfrac{|x-y|}{\sqrt{x}+\sqrt{y}}=\sqrt{|x-y|}\cdot \dfrac{\sqrt{|x-y|}}{\sqrt{x}+\sqrt{y}}\leq 1\cdot |x-y|^{\frac{1}{2}}$$ we obtain $1/...
user avatar
7 votes
2 answers
573 views

Existence of spectral gap

I would like to start by saying that any comment or idea is highly appreciated. Let us observe that for Hilbert-Schmidt operators $H_1,H_2$ on an infinite-dimensional separable complex Hilbert space $...
Dixmier's user avatar
  • 95
7 votes
1 answer
938 views

What is the idea behind interpolation spaces?

I am working through a text on Numerics for SPDEs and there the concept an interpolation (Hilbert-)space associated to an operator is used. To be specific: Definition. Let $H$ be an $\mathbb{R}$-...
alexlo's user avatar
  • 131
7 votes
2 answers
997 views

Uniform continuity of heat semigroup

I would like to illustrate my question with an example: It is well-known that $\Delta$ is the generator of a strongly continuous semigroup $(T(t))$ on $L^2(\mathbb R^n),$ i.e. the heat-semigroup. It ...
Sascha's user avatar
  • 536
6 votes
1 answer
436 views

Definition of the nonlinear part of the drift in a (stochastic) Navier-Stokes equation

Let $T>0$ $d\in\mathbb N$ $\Lambda\subseteq\mathbb R^d$ be bounded and open $\mathcal V:=\left\{v\in C_c^\infty(\Lambda)^d:\nabla\cdot v=0\right\}$, $$V:=\overline{\mathcal V}^{\left\|\;\cdot\;\...
0xbadf00d's user avatar
  • 167
6 votes
0 answers
113 views

Schwartz kernel of spectral projection of Laplacian and integrated density of states

I'm reposting here a question I asked on MSE which did not receive an answer. I am considering the Dirichlet Laplacian $\Delta$ on some smooth domain $U$. For now assume that $U$ is bounded, and later ...
GSofer's user avatar
  • 251
6 votes
0 answers
170 views

$L^p$ estimates for solutions of strongly elliptic equations

Let $\Omega\subset \mathbb{R}^d$ be a Lipschitz bounded domain. In $L_2(\Omega;{\mathbb C}^n)$, we consider a strongly elliptic second order operator $A$ with constant coefficients. Namely, it is ...
Suslina  Tatyana's user avatar
5 votes
2 answers
267 views

Positivity for the mild solution of a heat equation on the torus

Consider the following linear parabolic equation in one spatial dimension for $u=u(x,t)$ on the one-dimensional torus $\mathbb{T}^1,$ meaning $x \in \mathbb{T}^1$ and $t \in (0, T]:$ $$ \partial_t u- ...
kumquat's user avatar
  • 185
5 votes
2 answers
459 views

Backward heat equation and forward perturbed heat equation well posed?

I consider the following scenario. Let $I$ be a compact interval in space and $f$ a nice function in the space $C^{\infty}(I)$. In the following we consider a self-adjoint realization of our operators ...
Sascha's user avatar
  • 536
5 votes
1 answer
571 views

Schrödinger operator with Coulomb potential

The free Laplacian $-\Delta$ has absolutely continuous spectrum $[0,\infty).$ The Coulomb Hamiltonian $H=-\Delta-\frac{1}{\vert x\vert}$ on $L^2(\mathbb R^3)$ has absolutely continuous spectrum $[0,\...
Xing Wang's user avatar
  • 119
5 votes
1 answer
171 views

Invariant subspace in infinite dimensions

Let $A(t)$ be a family of skew self-adjoint operator defined on some Hilbert space $H$ with common domain $D(A).$ The dependence on $t$ is in the strongly continuous sense, i.e. for all $x \in D(A)$ ...
Zorgo's user avatar
  • 177
5 votes
1 answer
211 views

Pointwise convergence in functional calculus

Let $A_n$ be a family of (bounded) self-adjoint operator converging pointwise to some (unbounded) self-adjoint operator $A,$ i.e. for all $x$ in the domain of $A$ $$\left\lVert A_n x-Ax \right\rVert \...
Clement G.'s user avatar
5 votes
1 answer
1k views

Trace-norm of integral operator

Let me start by saying that I do appreciate any insight on this. So also if you have a partial result, please share it as a comment or answer. This is somewhat unrelated to what I normally do, so I ...
Jacob Augstine's user avatar
4 votes
2 answers
541 views

What do we know about the semigroup $e^{it\sqrt{-\Delta}}$

I am very interested in the properties of the semigroup $e^{it\sqrt{-\Delta}}$, which may has some fundamental differences (such as the kernel) with the well-known Schrödinger semigroup $e^{it\Delta}$...
user23078's user avatar
  • 1,644
4 votes
1 answer
372 views

Strong positivity of Neumann Laplacian

There are many places in the literature where the positivity of some semigroups is treated. However I did not know anyone which states and proves the strong positivity even for the basic semigroups ...
Bogdan's user avatar
  • 1,759
4 votes
1 answer
221 views

Non-isolated ground state of a Schrödinger operator

Question. Does there exist a dimension $d \in \mathbb{N}$ and a measurable function $V: \mathbb{R}^d \to [0,\infty)$ such that the smallest spectral value $\lambda$ of the Schrödinger operator $-\...
Jochen Glueck's user avatar
4 votes
1 answer
366 views

Dissipative operator on Banach spaces

An operator $A$ is called dissipative if for all $x \in D(A)$ and $\lambda >0$ $$ \left\lVert (A-\lambda)x \right\rVert \ge \lambda \left\lVert x \right\rVert.$$ On a Hilbert space this is ...
Zinkin's user avatar
  • 501
4 votes
1 answer
254 views

Strongly continuous semigroups and symbols of pseudo differential operators

I am considering the Cauchy IVP for the evolution equation $$u_t + \Psi u =0$$ where $\Psi$ is a linear pseudo differential operator with symbol $\widehat{\Psi}\left(\underline{\xi}\right)$. The ...
Frubiclé's user avatar
  • 155
4 votes
1 answer
279 views

Schroedinger operator in 2 dimensions with singular potential

Consider the Schroedinger operator $$H = -\Delta + \frac{c}{\vert x \vert^2}$$ in two dimensions with $c >0$ This operator has a self-adjoint realization, since it is a positive symmetric operator ...
António Borges Santos's user avatar
4 votes
1 answer
201 views

Spectrum Cauchy-Euler operator

A Cauchy-Euler operator is an operator that leaves homogeneous polynomial of a certain degree invariant, named after the Cauchy-Euler differential equations We consider the operator $$(Lf)(x) = \...
Sascha's user avatar
  • 536
4 votes
1 answer
213 views

Mapping properties of backward and forward heat equation

In a previous question on mathoverflow, I asked about the following: Let $\Delta$ be the Laplacian on some compact interval $I$ of the real line with let's say Dirichlet boundary conditions. The ...
Sascha's user avatar
  • 536
4 votes
1 answer
655 views

Generator of Laplace operator as analytic semigroup on $L^1(\mathbb{R}^n)$

The Laplace operator is the generator of an analytic semigroup on $L^p(\mathbb R^n)$ for $1 < p < \infty$. Is the same true for $L^1(\mathbb R^n)$? If it is, could someone give a reference? The ...
Martin's user avatar
  • 271
4 votes
0 answers
135 views

Zygmund class, Schwartz class and Littlewood-Paley projection operators

I'm studying Littlewood-Paley theory in harmonic analysis, where I encountered the following problem related to the Zygmund class of functions: Consider the Zygmund class of functions defined as ...
pureorapplied's user avatar
4 votes
0 answers
169 views

Inverse Laplacian and convolution in Albeverio's “Solvable Models in quantum mechanics”

I asked this question on math.stackexchange.com two weeks ago but got no answers so far and I got no clues from literature, so maybe someone here knows a reference. I hope it is ok to ask this ...
MrMatzetoni's user avatar
3 votes
2 answers
947 views

Can one hear the shape of a drum for operators?

M. Kac in his famous paper "Can one hear the shape of a drum?" asked whether one can "hear" the area of the ambient domain by looking at the spectral picture. Although he was not the first who came up ...
BigM's user avatar
  • 1,583
3 votes
1 answer
224 views

Extension of Sobolev function defined on unit cube

Im wondering about theorems concerning extending Sobolev functions defined on the $d$-dimensional unit cube to all of $\mathbb{R}^d$. More precisely, given $f:[0,1]^d \to \mathbb{R}$ with $f\in H^k([0,...
Jjj's user avatar
  • 93
3 votes
2 answers
717 views

Existence for ODE in Banach space (accretive operators and Crandall-Liggett)

There is a theory of mild solutions $u \in C^0(0,T;X)$ where $X$ is a Banach space for equations of the form $$\frac{du}{dt} + Au = f$$ where $A$ is an accretive nonlinear operator under some ...
TheBook's user avatar
  • 155
3 votes
1 answer
517 views

A closed extension of the Laplace operator with respect to the supremum norm

Let $X$ be a bounded connected open subset of the $n$-dimensional real euclidean space. Consider the Laplace operator defined on the space of infinitely differentiable functions with compact support ...
user36746's user avatar
3 votes
1 answer
255 views

Closure of tensor product /tensor product semigroup

In this reference the following claim is made in Remark 2 Let $A,B$ be closable operators on Banach spaces $X,Y$, then $A \otimes 1$ and $1 \otimes B$ are closable operators on the Banach space $X \...
user avatar
3 votes
1 answer
876 views

Is Quantum Mechanics (norm)-consistent?

I edited a few small comments to the question in order to make it perhaps more comprehensible. Today I came across the following question in quantum mechanics. In Quantum mechanics it is common to ...
Hilbertspace's user avatar
3 votes
1 answer
212 views

A question on the Frechet derivative

Suppose the derivative of a functional is given by \begin{equation*} \int_{\Omega}(\vec{v}.\nabla u)|\nabla u|^{p-2} \phi=\int_{\Omega}\nabla.(u\vec{v})|\nabla u|^{p-2} \phi,~\phi\in W_0^{1,p}(\...
Alexander's user avatar
  • 157
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
  • 4,145
3 votes
1 answer
107 views

Finiteness of Schatten $p$-norm of truncated free resolvent

Consider the resolvent operator $ R(z) := (-\Delta - z)^{-1}$ of the Laplace operator on $L^2(\mathbb R^d)$, where $z\in \rho(-\Delta) = \mathbb C \setminus \mathopen [0, \infty)$. For $p \geq 1$, let ...
user271621's user avatar
3 votes
1 answer
274 views

Function square-integrable

Let $f$ be an arbitrary function in $L^2(0,\infty)$ and consider the function $$(g_f)(y) = \frac{1}{y-x_0} \int_{0}^{\infty} f(x) \left(\frac{xy}{(x^2+y^2+1)}\right)^2 \ dx$$ where $x_0$ is an ...
Andrea Tauber's user avatar
3 votes
0 answers
108 views

A question on essentially self-adjoint differential operators of the type $\Delta=D^{\ast}D$

Let $(M,g)$ be a smooth (connected, complete, oriented) Riemannian manifold and let $D:C^{\infty}(M)\to C^{\infty}(M)$ be a linear partial differential operator, which I view as an operator in $L^{2}(...
B.Hueber's user avatar
  • 1,171
3 votes
0 answers
203 views

On the spectrum of Fokker–Planck with linear drift

The paper by Liberzon and Brockett, "Spectral analysis of Fokker–Planck and related operators arising from linear stochastic differential equations." SIAM Journal on Control and ...
mathamphetamine's user avatar
3 votes
0 answers
322 views

Heat equation damps backward heat equation?

In a previous question on mathoverflow, I was wondering about the following: Let $\Delta$ be the Laplacian on some compact interval $I$ of the real line with let's say Dirichlet boundary conditions. ...
Sascha's user avatar
  • 536
3 votes
0 answers
210 views

Singular integral operators and PDEs

What is the relation between the notion of singular integral operators and partial differential equations? I know, for example, that there is a relation between the Cauchy transform (Riesz transforms ...
XIII's user avatar
  • 747
3 votes
0 answers
126 views

Exponentially weighted spaces: Effect on spectrum

My question is somewhat broad, but I do not know how to precisely state the issue. I am investigating stability of certain class of scalar PDE on $\mathbb{R}$. Previous work in this topic has ...
mystupid_acct's user avatar
2 votes
1 answer
314 views

Generation of strict contraction semigroups

Let $T(t)$ be a $C_0$-semigroup on Banach space $X$, and $A$ its generator. By Lumer-Philipps theorem we know that if $A$ is densely defined and m-dissipative operator then it generates a $C_0$-...
S. Maths's user avatar
  • 571
2 votes
3 answers
303 views

Uniqueness of solution depending on constant?

I am a physicist and I am aware that this forum is for professional mathematical questions, but please be not too hard on my notation. I encountered the following integral equation for functions $f:[...
Andrea Tauber's user avatar
2 votes
1 answer
644 views

Reference request: inverse of differential operators

I have asked a similar question on MSE but I did not receive any replies, so I am reposting here in case it is more appropriate (though I have slightly generalized the question). As an example ...
CBBAM's user avatar
  • 721
2 votes
1 answer
93 views

Lipschitz bound on semigroups

Let $T$ be a self-adjoint operator (possibly unbounded) and $S$ a bounded self-adjoint operator. Then one can study the unitary groups $R_T(t):=e^{itT}$ and $R_S(t):=e^{itS}.$ Now if you think about ...
Oliver Seifert's user avatar