Norm-Resolvent Estimates and Perforated Domains

Abstract

In 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 Schrödinger operator H = −∆+V on L²(Rᵈ), where Re(V(x))≥c|x|²−b for some c,b>0.
The spectrum of this operator is discrete and its real part is bounded below by −b. In general, the ε-pseudospectrum of H will have an unbounded component for any ε > 0 and thus will not approximate the spectrum in a global sense.
By exploiting the fact that the semigroup exp(−tH) is immediately compact, we show a complementary result, namely that for every δ > 0, R > 0 there exists an ε > 0 such that the ε-pseudospectrum is contained in the union of the half plane and disks of radius δ around the eigenvalues.
In particular, the unbounded component of the pseudospectrum escapes towards +∞ as ε decreases. Additionally, we give two examples of non-selfadjoint Schrödinger operators outside of our class and study their pseudospectra in more detail.
In Part IV, we prove norm-resolvent convergence, as ε→0, for the operator −∆ in domain perforated ε-periodically, to the limit operator −∆+μ on L²(Ω), where μ∈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.