The absolute continuity of phase operators

Authors:
J. Dombrowski and G. H. Fricke

Journal:
Trans. Amer. Math. Soc. **213** (1975), 363-372

MSC:
Primary 47B15

MathSciNet review:
0377573

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Abstract: This paper studies the spectral properties of a class of operators known as phase operators which originated in the study of harmonic oscillator phase. Ifantis conjectured that such operators had no point spectrum. It was later shown that certain phase operators were, in fact, absolutely continuous and that all phase operators at least had an absolutely continuous part. The present work completes the discussion by showing that all phase operators are absolutely continuous.

**[1]**J. Dombrowski,*Spectral properties of phase operators*, J. Mathematical Phys.**15**(1974), 576–577. MR**0334757****[2]**Paul R. Halmos,*Introduction to Hilbert space and the theory of spectral multiplicity*, AMS Chelsea Publishing, Providence, RI, 1998. Reprint of the second (1957) edition. MR**1653399****[3]**Evangelos K. Ifantis,*Abstract formulation of the quantum mechanical oscillator phase problem*, J. Mathematical Phys.**12**(1971), 1021–1026. MR**0280113****[4]**E. C. Lerner,*Harmonic-oscillator phase operators*, Nuovo Cimento**56B**(1968), 183-186.**[5]**E. C. Lerner, H. W. Huang, and G. E. Walters,*Some mathematical properties of oscillator phase operators*, J. Mathematical Phys.**11**(1970), 1679–1684. MR**0260346****[6]**C. R. Putnam,*Commutation properties of Hilbert space operators and related topics*, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 36, Springer-Verlag New York, Inc., New York, 1967. MR**0217618**

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DOI:
http://dx.doi.org/10.1090/S0002-9947-1975-0377573-0

Keywords:
Phase operators,
absolute continuity of phase operators,
real parts of weighted shifts

Article copyright:
© Copyright 1975
American Mathematical Society