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Renormalized oscillator Hamiltonians

✍ Scribed by John C Baez; Zhengfang Zhou

Publisher: Elsevier Science
Year: 1992
Tongue: English
Weight: 942 KB
Volume: 92
Category: Article
ISSN: 0001-8708
DOI: 10.1016/0001-8708(92)90062-p

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✦ Synopsis

Let pj, q,, denote the self-adjoint operators -iajax, and multiplication by x, on L'(W), respectively. Let H,, = (l/2) xyr= r o,(pf +qj-l)whereo,>O,andletPbe a polynomial in the q's that is bounded below. The existence and uniqueness problems are considered for H, + IF'(q) renormalized relative to its own ground state. For example, suppose that P has degree 2d and P(q) >~(qf + ... +qz)"-k for some E, k z 0. Then for some lo > 0 there exists a unique continuous function u: [0, A,] + Z,*(W) such that u(n) is the nonnegative normalized ground state of Ho + 2 :P(q):,,,, and (u(A), P(q) u(1)) is bounded on [0,&l. If u is continuous only on (0, A,] uniqueness may fail.

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