Resolvent Estimates and Perforated Domains
Microsoft Word (PhD Thesis, Frank Rsler) Accepted Version
Available under License Creative Commons Public Domain Dedication CC0 1.0 Universal.
1234KbAbstractIn this thesis we are concerned with norm resolvent estimates for unbounded linear operators. The text is structured into four parts. The first two parts contain mathematical preliminaries, reviews of previous work and an introduction into the two results which constitute parts three and four. In the third part we are concerned with the non normal Schrdinger operator H = +V on L(R), where Re(V(x))cxb for some c,b>0. The spectrum of this operator is discrete and its real part is bounded below by b. In general, the pseudospectrum of H will have an unbounded component for any > 0 and thus will not approximate the spectrum in a global sense. In particular, the unbounded component of the pseudospectrum escapes towards + as decreases. Additionally, we give two examples of non selfadjoint Schrdinger operators outside of our class and study their pseudospectra in more detail. In Part IV, we prove norm resolvent convergence, as 0, for the operator in domain perforated periodically, to the limit operator + on replica hermes rings
L(), where C is a constant depending on the choice of boundary conditions on the holes (we consider Dirichlet, Neumann and Robin boundary conditions). This is an improvement of previous results by [Cioranescu Murat(1997)], [Kaizu(1985)], which show strong resolvent convergence. In particular, our result implies Hausdorff convergence of the spectrum of the resolvent for the perforated domain problem.