Let $H$ be an Schrödinger operator $(\Delta+V(x))$ with a radially symmetric and smooth potential and $H$ is e.s.a. on $C^\infty_0(\mathbb{R}^n)$, furthermore let $\lambda$ be an element of the resolvent set of $H$. Consider some rapidly decaying function $\varphi\in\mathscr{S}$, because $\lambda\in\rho(H)$, there exists a smooth solution $u$ of the inhomogeneous PDE $$ (H\lambda)u=\psi $$Does this already imply, that $u\in\mathscr{S}$? I need some estimate that tells me, that $u(x)\approx C \exp(\alpha r)$ at infinity. I could imagine that one could need some additional properties for $V$. I know there exist some results by Kato for the homogeneous case ("Growth Properties of Solutions of the Reduced Wave Equation With a Variable Coefficient"), but they dont seem to by applicable to my problem (or I am too dumb to see it). Has anyone stumbled across something similar for my problem?
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1$\begingroup$ The resolvent is an integral operator with exponentially decaying kernel (with decay rate depending on $\textrm{dist}(\lambda, \sigma(H)$): this kind of statement is known as CombesThomas estimate. I don't see any reason why we would get exponential decay of $u$ in general (if $\varphi$ is just a Schwartz function), but I think you can get $u\in S$ from this. $\endgroup$ – Christian Remling Sep 20 '14 at 22:13