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Information Center for Mathematical Science

논문검색

Information Center for Mathematical Science

논문검색

Manuscripta Mathematica.
( Vol. 76 NO.3 / (1992))
Schrodinger Operators with Highly Singular, Oscillating Potentials
Karl-Theodor Sturm,
Pages. 367-395
Abstract We investigate the Feynman-Kac semigroup $P_t^V$ and its density $p^V(t,..),t>0$, associated with the Schrodinger operator $-frac{1}{2}Delta + V$ on $mathbb{R}^d setminus {0}$.\$V$ will be a highly singular, oscillating potential like\$$V(x)=k cdot ||x||^{-l}cdot sin(||x||^{-m})$$\with arbitrary $k,l,m>0$. We derive conditions (on $k,l,m)$ which are sufficient $and$ necessary for the existence of constants $alpha, eta,gamma in mathbb{R}$ such that for all $t,s,y$\$$p^V(t,x,y) leq gamma cdot p(eta t,x,y) cdot e^{alpha t}.$$\On the other hand, also conditions are derived which imply that $p^V(t,x,y) equiv infty$ for all $t,s,y$. The aim is to see to which extent quick oscillations can lead to annihilations of the singularities of $V$. For this purpose, we analyse the above example in great detail. Note that for $l geq 2$ the potential is so singular that none of the usual perturbation techniques applies.
Contents 1. Introduction 2. Two results on uniform boundedness 3. Three results on unboundedness 4. The compatibility condition 5. Further examples and applications
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