On essential self-adjointness, confining potentials & the Lp-Hardy inequality : a thesis presented in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Mathematics at Massey University, Albany, New Zealand

Abstract

Let Ω be a domain in Rm with non-empty boundary and let H = -Δ + V be a Schrödinger operator defined on C[symbol](Ω) where V E L[symbol](Ω). We seek the minimal criteria on the potential V that ensures that H is essentially self-adjoint, i.e. that ensures the closed operator H is self-adjoint. Overcoming various technical problems, we extend the results of Nenciu & Nenciu in [1] to more general types of domain, specifically unbounded domains and domains whose boundaries are fractal. As a special case of an abstract condition we show that H is essentially self-adjoint provided that sufficiently close to the boundary

[equation]

where d(x) = dist(x;δΩ) and the right hand side of the above inequality contains a f nite number of logarithmic terms. The constant μ2(Ω ) appearing in (1) is the variational constant associated with the L2-Hardy inequality and is non-zero if and only if Ω admits the aforementioned inequality. Our results indicate that the existence of an L2-Hardy nequality, and the specific value of μ2(Ω), depend intimately on the (Hausdorff / Aikawa) dimension of the boundary. In certain cases where Ω is geometrically simple, this constant, as well as the constant `1' appearing in front of each logarithmic term, is shown to be optimal with regards to the essential self-adjointness of H.