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.

functionsof $-\Delta$. It is true, under some additional assumptions, for the "Bessel" operator $(I-\Delta)^s$ and for the relativistic fractional operator $\sqrt{m^2-\Delta}-m$. $\endgroup$ – Siminore May 18 '18 at 11:29