Wave Propagation Speed And Schr(?)dinger Equation On Fractals

The main work of this thesis consists of two parts.In the first part,we con-sider the propagation speed problem for wave equations defined by non-negative self-adjoint operators,including both finite and infinite propagation speeds.In the second part,we study Bohr's formula for Schr(?)dinger operators on fractals with overlaps.This thesis consists of five chapters.Chapter 1 is an introduction.We first introduce the background of fractal ge-ometry and fractal analysis.Then we summarize some notation,definitions,and preliminary results that will be used throughout this thesis.Finally we state the main results of this thesis.In Chapter 2,we study the infinite wave propagation speed problem on fractals,emphasizing on self-similar sets that are not postcritically finite.Strichartz[73]con-jectured that on certain fractals,waves may propagate with infinite speed.Y.-T.Lee[52]recently proved that on a class of self-similar sets satisfying the post-critically finite(p.c.f.)condition,the conjecture is true.We prove that a sub-Gaussian lower heat kernel estimate leads to infinite propagation speed,extending the result of Y.-T.Lee to include bounded and unbounded generalized Sierpinski carpets as well as some fractal blowups,which are not postcritically finite.In Chapter 3,we investigate the finite wave propagation speed problem on fractals,focusing on Laplacians defined by a measure ?,denoted by-??.We prove that if ? is equivalent to Lebesgue measure,and its density d?/dx satisfies suitable conditions,then the heat kernel of-?? has a Gaussian upper estimate,which leads to finite propagation speed.Our main results can be applied to two classes of self-similar measures defined by iterated function systems with overlaps,including the classical infinite Bernoulli convolutions.As a by-product,we show that a class of infinite Bernoulli convolutions have continuous and positive density on the interval(0,1).In Chapter 4,we consider Bohr's formula for Schr(?)dinger operators-??? +V,where-??? is a direct sum of a countably infinite of fractal Laplace operator,and V is some potential function.In[13],sufficient conditions for Bohr's formula to hold on fractals are obtained,and these conditions are verified for nested fractals.Unfortunately,fractals with overlaps usually do not,or not known to,satisfy all these conditions.In this thesis,we derive an analog of Bohr's formula for fractals with overlaps by modifying their conditions.Finally,we show that our assumptions hold for three classes of fractals with overlaps.In Chapter 5,we state some comments and questions related to this thesis for further studies.