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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 ...
2 votes
0 answers
180 views

Approximating $L^p$ functions by eigenfunctions of Laplacian

I'm reading a paper https://www.sciencedirect.com/science/article/pii/S0022039608004932. In this paper, the authors assume that $\mathcal{O}$ is a bounded domain of $\mathbb{R}^N$ with $C^m$ boundary ...
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 ...
3 votes
0 answers
211 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 ...
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:[...
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 $x \in D(A)$ ...
1 vote
1 answer
737 views

$L^2$ function in Schwartz space?

Let $f:\mathbb R^n \rightarrow \mathbb R$ be a smooth function whose derivatives are all polynomially bounded and $f \in L^{\infty}.$ Such a function has the property that when multiplied with any ...