Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

It is well known that the fundamental solution $\Gamma_1$ in $\mathbb{R}^n$ of the Schrödinger operator $-\Delta + 1$ decays exponentially fast, viz. $|\Gamma_1(x)| \leq C_1 \mathrm{e}^{-C_2|x|}$ as $|x| \to +\infty$. I have read some comments in several papers about a different behavior of the fundamental solution $\Gamma_s$, $0<2<1$, of the fractional Schrödinger operator $(-\Delta)^s +1$; it should decay like $1/|x|^{n+2s}$. Unluckily, I cannot find a good reference for this fact, and I wonder if somebody knows a paper or has some hint for the proof.

share|improve this question
I think it is considered not appropriate to cross-post here and simultaneously, verbatim, on Math Overflow. –  paul garrett Aug 10 '12 at 14:05
I could not decide whether my question is more appropriate for this site or for the sister site. It is a research-level question, but also of probable interest for people who study mathematics. I've just deleted my question on Math Overflow. But tell me: do you have any answer or suggestion about the question? –  Siminore Aug 10 '12 at 15:12
I would look at the equation in Fourier space. –  Fabian Aug 10 '12 at 15:52
@Fabian Maybe. I am a bit suspicious, since the problem for the laplacian operator is usually faced by using radially symmetric solutions, for which explicit estimates are easier. But the fractional laplacian is non-local, and radial symmetry does not reduce everything to an ODE. –  Siminore Aug 10 '12 at 16:30

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.