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arXiv:quant-ph/0309170 · Physical Review A

Self-adjoint extensions of phase and time operators

Gilad Gour, F. C. Khanna, M. Revzen

Published Jan 16, 2004• 10 citations
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Abstract

It is shown that any real and even function of the phase (time) operator has a self-adjoint extension and its relation to the general phase operator problem is analyzed.
DOI
10.1103/physreva.69.014101

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We study a family of Jacobi operators in which the diagonal entries are multiplied by a coupling parameter λ≥0λ\geq0λ≥0. Under suitable conditions, the operator is self-adjoint for every λ>0λ>0λ>0, while the formal limit at λ=0λ=0λ=0 is a symmetric Jacobi operator admitting a one-parameter family of self-adjoint extensions. A central ingredient of our analysis is the derivation of uniform bounds for square-summable generalized eigenvectors in the small-λλλ regime, which combines discrete WKB methods with Airy-function asymptotics. Using these estimates, we analyze the limiting behavior λ→0λ\to0λ→0 in the strong resolvent sense, proving that for every sequence λj→0λ_j\to0λj​→0 one can extract a subsequence along which the corresponding Jacobi operators converge to some self-adjoint extension of the limiting operator; conversely, every such extension can be obtained in this way. We call this behavior an essentially singular limit, by analogy with essential singularities in complex analysis. As an application, we study higher-order squeezing operators arising in quantum optics. Using the connection with Jacobi operators, we show that when the relative strength of the free-field term tends to zero, different self-adjoint extensions of the squeezing operator are selected along different sequences. In particular, this limit does not single out a physically distinguished self-adjoint extension, but instead identifies a distinguished subclass of extensions compatible with the underlying symmetry.