Selfadjointness of the momentum operator with a singular term

Watanabe, Michiaki; Watanabe, Shuji

doi:10.1090/S0002-9939-1989-0984821-8

Selfadjointness of the momentum operator with a singular term
HTML articles powered by AMS MathViewer

by Michiaki Watanabe and Shuji Watanabe PDF

Proc. Amer. Math. Soc. 107 (1989), 999-1004 Request permission

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 $H = ({p^2} + {x^2})/2$.

References

Jerome A. Goldstein, Semigroups of linear operators and applications, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1985. MR 790497
Tosio Kato, Perturbation theory for linear operators, Die Grundlehren der mathematischen Wissenschaften, Band 132, Springer-Verlag New York, Inc., New York, 1966. MR 0203473
Yoshio Ohnuki and Susumu Kamefuchi, Quantum field theory and parastatistics, University of Tokyo Press, Tokyo; Springer-Verlag, Berlin, 1982. MR 687830, DOI 10.1007/978-3-642-68622-1
Noboru Okazawa, On the perturbation of linear operators in Banach and Hilbert spaces, J. Math. Soc. Japan 34 (1982), no. 4, 677–701. MR 669276, DOI 10.2969/jmsj/03440677
Michael Reed and Barry Simon, Methods of modern mathematical physics. I, 2nd ed., Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1980. Functional analysis. MR 751959

84

Kôsaku Yosida, Functional analysis, 6th ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 123, Springer-Verlag, Berlin-New York, 1980. MR 617913
Shuji Watanabe, A harmonic oscillator system with the generalized commutation relations, J. Math. Phys. 30 (1989), no. 2, 376–379. MR 978552, DOI 10.1063/1.528455