# One question about Schrodinger Semigroups-(B. Simon)

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 $$u$$.

• Assuming $u\in \mathbb{R}^n$, this comes from an argument using the Agmon metric: since this bound is easier to show in the 1-dimensional case, you extend the result to higher dimensions by approximating your eigenfunction using the WKB method. A good place to read about this is Hislops "Introduction to Spectral Theory." Theorem 3.10 is the result you want, and the geometric hypothesis (the condition on $\text{supp}(E-V(x))_+$) of this theorem requires is implied by your potential $V$ being bounded. The constant $C$ does depend on $\delta$. – Hadrian Quan Mar 4 at 15:35
• @HadrianQuan Thank you for the comment and reference. But I still have one question, if we multiply any positive number $k$ with the function $u$, it is still a solution of that equation.So, how to get the constant ($\delta$ comes from the spectrum of $H$). – DLIN Mar 5 at 2:28
• @DLIN $u$ is assumed to be normalized. This also gives you a hint on the dependence on $\delta$ – lcv Mar 5 at 14:47
• @lcv Well, from that paper, I have not seen any hypothesis on the norm of $u$. By the way, do you mean the theorem needs the assumption that $\|u\|_{L^2}=1$. – DLIN Mar 6 at 0:44
• The theorem as stated does not need it. But if you want to see how $C$ depends on $\delta$ you can simply normalize both sides. – lcv Mar 6 at 4:41