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Selfadjointness of the momentum operator with a singular term

Authors: Michiaki Watanabe and Shuji Watanabe
Journal: Proc. Amer. Math. Soc. 107 (1989), 999-1004
MSC: Primary 81C10; Secondary 47B15
MathSciNet review: 984821
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Abstract: Self-adjointness is shown of the momentum operator $ \{ u \in {H^1}({{\mathbf{R}}^1}):u/x \in {L^2}({{\mathbf{R}}^1})\} $, with domain $ \left\{ {u \in {H^1}({{\mathbf{R}}^1}):u/x \in {L^2}({{\mathbf{R}}^1})} \right\}$ when $ c > 1$ or $ c < - 1$. This operator appears in a harmonic oscillator system with the generalized commutation relations by Wigner: $ ip = [x,H]$ and $ - ix = [p,H]$ for the Hamiltonian $ H$ and the multiplication operator $ x$.

The proof is carried out by generation of a unitary group in terms of ip, based on the Hille-Yosida theorem and Stone's theorem. The result is applied to the self-adjoitness of $ H = ({p^2} + {x^2})/2$.

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Keywords: Self-adjointness, generation of unitary group, perturbation, momentum operator, Hamiltonain
Article copyright: © Copyright 1989 American Mathematical Society

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