Friedrichs extensions of Schrödinger operators with singular potentials |
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Authors: | Attila B von Keviczky Richard L Hall |
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Institution: | a Department of Mathematics and Statistics, Concordia University, 1455 de Maisonneuve Boulevard West, Montréal, PQ, Canada H3G 1M8 b Department of Mathematics and Statistics, University of Prince Edward Island, 550 University Avenue, Charlottetown, PE, Canada C1A 4P3 |
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Abstract: | The Friedrichs extension for the generalized spiked harmonic oscillator given by the singular differential operator −d2/dx2+Bx2+Ax−2+λx−α (B>0, A?0) in L2(0,∞) is studied. We look at two different domains of definition for each of these differential operators in L2(0,∞), namely C0∞(0,∞) and D(T2,F)∩D(Mλ,α), where the latter is a subspace of the Sobolev space W2,2(0,∞). Adjoints of these differential operators on C0∞(0,∞) exist as result of the null-space properties of functionals. For the other domain, convolutions and Jensen and Minkowski integral inequalities, density of C0∞(0,∞) in D(T2,F)∩D(Mλ,α) in L2(0,∞) lead to the other adjoints. Further density properties C0∞(0,∞) in D(T2,F)∩D(Mλ,α) yield the Friedrichs extension of these differential operators with domains of definition D(T2,F)∩D(Mλ,α). |
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Keywords: | Generalized spiked harmonic oscillators Singular potentials Friedrichs extension Self-adjoint extension Jensen inequality Minkowski inequality |
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