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
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Date
2014
DOI
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Massey University
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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.
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Keywords
Lp-Hardy inequality, Non-empty boundary, Self-adjointness, Distance function