# Barry Simon's decay of eigenfunctions for pseudo differential operators

In his celebrated paper on Schrödinger semigroups, Barry Simon proves the following result.

Let $V_{-} \in K_\nu$, $V_{+} \in K_\nu^{\mathrm{loc}}$ and suppose that $Hu=Eu$ where $E$ is the discrete spectrum, i.e. $E$ is an isolated eigenvalue of finite multiplicity. Then for some $\delta>0$ and $C$ $$|u(x)| \leq C \mathrm{e}^{-\delta |x|}.$$

Here $H=-\frac{1}{2}\Delta + V$ is a Schrödinger operator with potential $V$, and $K_\nu$ denotes the usual Kato class.

My question is: does there exist a generalisation of Simon's theorem to pseudo differential operators like $H=(-\Delta)^s+V$ or $H=(-\Delta+m^2)^s-m^{2s}+V$? It is very important that the exponential decay should hold for any eigenvalue of finite multiplicity, since I need this kind of result to prove the exponential decay of solutions to nonlinear equations governed by a pseudo differential operator.

Edit: the decay of solutions to equations governed by $(-\Delta)^s$ is known to be of rational type. However, it is known to be exponentially fast for other fractional operators, but only under more restrictive conditions on the point spectrum. My question is whether exponential decay does hold true without further assumptions, whenever it is known under restrictive assumptions on $V$ and on the spectrum.

• Can you add a link to Barry's paper ? – Rene Schipperus May 17 '18 at 12:26
• @ReneSchipperus Done! – Siminore May 17 '18 at 14:16
• Fractional Laplacian $(-\Delta)^s$ are non local. It is very unlikely that the result you mention should hold. – RaphaelB4 May 18 '18 at 8:07
• @RaphaelB4 Indeed for the pure fractional laplacian $(-\Delta)^s$ is it indeed generically false, but it can be true for different functions of $-\Delta$. It is true, under some additional assumptions, for the "Bessel" operator $(I-\Delta)^s$ and for the relativistic fractional operator $\sqrt{m^2-\Delta}-m$. – Siminore May 18 '18 at 11:29
• If your operators are local then you have the usefull "Combes Thomas estimate" (see the proof for example in the book "caught by disorder") applied with the Cauchy formula – RaphaelB4 May 18 '18 at 11:43

I believe only partial results are available, and the decay is typically slower than exponential. Here is what I am aware of when $V(x) \to \infty$ as $|x| \to \infty$ (in this case the spectrum is purely discrete).
For $(-\Delta)^s + V(x)$ the eigenfunctions typically decay as $(1+|x|)^{-n - 2s} (V(x))^{-1}$. This is proved in the paper Intrinsic ultracontractivity for Schrödinger operators based on fractional Laplacians by my colleagues, Kaleta and Kulczycki. Theorems 1 and 4 therein deal with the first eigenfunction only, but the upper bounds are quite general.
This was generalised in a follow-up Pointwise eigenfunction estimates and intrinsic ultracontractivity-type properties of Feynman-Kac semigroups for a class of Levy processes by Kaleta and Lőrinczi. The class of operators considered in this paper includes $(-\Delta + m^2)^s + V(x)$, if I remember correctly.
• I suspected that a strong result is still missing. For the relativistic operator $(-\Delta+m^2)^s-m^{2s}$ the decay is expected to be exponential, but indeed I can't find a general statement as Simon's. – Siminore May 17 '18 at 9:50