This question comes from the paper: B. Simon, Schrodinger Semigroups, Bull. A.M.S., (1982) Vol. 7 (3).

On the **Theorem C.3.4**(subsolution estimate) of the paper, it says that: Let $Hu=Eu$ and $u\in L^2$, where $H=-\Delta+V$ for some bounded continuous function $V$, if $E$ is in discrete spectrum of $H$. Then for some $C,~\delta>0$
$$|u(x)|\leq Ce^{-\delta|x|}.$$

**Q**

How to get constant $C$, does it depend $\delta$?

Is there any estimate about $C$, especially without the condition on the compactness of $supp(E-V)_+$?

*PS*:
I can follow how to get $\delta$. I can understand that $e^{\delta |x|}u\in L^\infty$. I do not know how to get the constant $C$, as the equation is linear, if we multiply any positive number $k$ with $u$, it is also a solution for the equation, where I can not follow.
And I think $C$ depends on the $L^2$-norm of $u$.