Comparison of Martin boundaries for Schrödinger operators
Mitsuru Nakai · Hokkaido Mathematical Journal · 1989
We consider the Martin compactification R_{P}^{*} for an admissible Schr\"odinger operator -\Delta+P on a Riemann surface R with singular but nonnegative potentials P on R and study how R_{P}^{*} varies according to a small perturbation of the potential P. One reason of the importance of the study of this kind lies in the following instance.For a detailed study of R_{P}^{*} it often occurs the need to construct a potential P such that R_{P}^{*} possesses a property given in advance (cf.e. g. [10], [9], [11] among many others).In this construction it is easier to seek an appropriate P among potentials which are allowed to be discontinuous than to do among only those that are restricted to be smooth.However our primary concern is about R_{P}^{*} with smooth P. Thus one natural procedure may be as follows.First find a P among discontinuous potentials such that R_{P}^{*} has a desired property.Then approximate P by smooth potentials, e.g. by \rho_{\epsilon}*P(\epsilon\downarrow 0) with \rho_{\epsilon}*the Frie- drichs mollifier in a suitable sense (cf.no. 15 below), and then we expect R_{\rho\epsilon*P}^{*} to be identical with R_{P}^{*} from the view point of the Martin theory if the approximation is made close enough.This is the motivation of our present study.In this paper we will give a theorem asserting R_{P}^{*}=R_{Q}^{*} under a certain closeness condition on potentials P and Q on R .Any theorem of this kind (cf.e. g. [16], [9], etc.) would not be considered as natural if it did not imply the following two facts: a. R_{P}^{*}=R_{Q}^{*} if P=Q on R outside a compact subset: b .R_{P}^{*}=R_{\rho\epsilon*P}^{*} if \epsilon is small enough in a suitable sense.Our theorem certainly contains these two facts and especially the validity of the latter of the above must be useful in actual constructions as mentioned above.After preliminary discussions in nos. 1-4,the main comparison theorem is stated in no. 5 and proven in nos. 6-11.The fact a above is deduced in no.12, and the fact b above is stated in no. 13 and proven in nos. 14-15.Although we state and prove our results for Schr\"odinger operators