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Full-Text Articles in Physical Sciences and Mathematics
Survey Of Results On The Schrodinger Operator With Inverse Square Potential, Richardson Saint Bonheur
Survey Of Results On The Schrodinger Operator With Inverse Square Potential, Richardson Saint Bonheur
Electronic Theses and Dissertations
In this paper we present a survey of results on the Schrodinger operator with Inverse ¨ Square potential, La= −∆ + a/|x|^2 , a ≥ −( d−2/2 )^2. We briefly discuss the long-time behavior of solutions to the inter-critical focusing NLS with an inverse square potential(proof not provided). Later we present spectral multiplier theorems for the operator. For the case when a ≥ 0, we present the multiplier theorem from Hebisch [12]. The case when 0 > a ≥ −( d−2/2 )^2 was explored in [1], and their proof will be presented for completeness. No improvements on the sharpness …
Weighted Inequalities For Dyadic Operators Over Spaces Of Homogeneous Type, David Edward Weirich
Weighted Inequalities For Dyadic Operators Over Spaces Of Homogeneous Type, David Edward Weirich
Mathematics & Statistics ETDs
A so-called space of homogeneous type is a set equipped with a quasi-metric and a doubling measure. We give a survey of results spanning the last few decades concerning the geometric properties of such spaces, culminating in the description of a system of dyadic cubes in this setting whose properties mirror the more familiar dyadic lattices in R^n . We then use these cubes to prove a result pertaining to weighted inequality theory over such spaces. We develop a general method for extending Bellman function type arguments from the real line to spaces of homogeneous type. Finally, we uses this …