A quasi-Hermitian operator is an operator that is similar to its adjoint in some sense, via a metric operator, i.e., a strictly positive self-adjoint operator. Whereas those metric operators are in general assumed to be bounded, we analyze the structure generated by unbounded metric operators in a Hilbert space. Following our previous work, we introduce several generalizations of the notion of similarity between operators. Then we explore systematically the various types of quasi-Hermitian operators, bounded or not. Finally, we discuss their application in the so-called pseudo-Hermitian quantum mechanics.
REFERENCES
1.
J.
Dieudonné
, “Quasi-Hermitian operators
,” in Proceedings of the International Symposium on Linear Spaces, Jerusalem, 1960
(Pergamon Press
, Oxford
, 1961
), pp. 115
–122
.2.
P.
Siegl
and D.
Krejčiřík
, “On the metric operator for the imaginary cubic oscillator
,” Phys. Rev. D
86
, 121702
(R) (2012
).3.
F.
Bagarello
, “From self-adjoint to non-self-adjoint harmonic oscillators: Physical consequences and mathematical pitfalls
,” Phys. Rev. A
88
, 032120
(2013
).4.
F.
Bagarello
and A.
Fring
, “Non-self-adjoint model of a two-dimensional noncommutative space with an unbounded metric
,” Phys. Rev. A
88
, 042119
(2013
).5.
F.
Bagarello
and M.
Znojil
, “Nonlinear pseudo-bosons versus hidden Hermiticity. II. The case of unbounded operators
,” J. Phys. A: Math. Theor.
45
, 115311
(2012
).6.
A.
Mostafazadeh
, “Pseudo–Hermitian quantum mechanics with unbounded metric operators
,” Philos. Trans. R. Soc. London
371
, 20120050
(2013
).7.
J.-P.
Antoine
, and C.
Trapani
, “Partial inner product spaces, metric operators and generalized Hermiticity
,” J. Phys. A: Math. Theor.
46
, 025204
(2013
);J.-P.
Antoine
, and C.
Trapani
, Corrigendum, J. Phys. A: Math. Theor.
46
, 329501
(2013
).8.
J.-P.
Antoine
and C.
Trapani
, Partial Inner Product Spaces: Theory and Applications
, Lecture Notes in Mathematics
Vol. 1986
(Springer-Verlag
, Berlin
, 2009
).9.
N.
Dunford
and J. T.
Schwartz
, 1957, 1963, 1971, Linear Operators. Part I: General Theory; Part II: Spectral Theory; Part III: Spectral Operators
(Interscience
, New York
, 1957
, 1963, 1971).10.
C. M.
Bender
, “Making sense of non-Hermitian Hamiltonians
,” Rep. Prog. Phys.
70
, 947
–1018
(2007
).11.
C. M.
Bender
, A.
Fring
, U.
Günther
, and H.
Jones
, “Quantum physics with non-Hermitian operators (Preface to a special issue)
,” J. Phys. A: Math. Theor.
45
, 440301
(2012
).12.
J.
Bergh
and J.
Löfström
, Interpolation Spaces
(Springer-Verlag
, Berlin
, 1976
).13.
The space
${\mathcal {H}}(R_G^{-1})$
is (three times) erroneously denoted ${\mathcal {H}}(R_{G^{-1}})$
in Ref. 7, p. 4
(see Corrigendum).14.
15.
S.
Kantorovitz
, “On the characterization of spectral operators
,” Trans. Am. Math. Soc.
111
, 152
–181
(1964
).16.
F. G.
Scholtz
, H. B.
Geyer
, and F. J. W.
Hahne
, “Quasi-Hermitian operators in Quantum Mechanics and the variational principle
,” Ann. Phys. (N.Y.)
213
, 74
–101
(1992
).17.
H. B.
Geyer
, W. D.
Heiss
, and F. G.
Scholtz
, “Non-Hermitian Hamiltonians, metric, other observables and physical implications
,” preprint arXiv:0710.5593v1 (2007
).18.
R.
Kretschmer
and L.
Szymanowski
, “Quasi-Hermiticity in infinite-dimensional Hilbert spaces
,” Phys. Lett. A
325
, 112
–117
(2004
).19.
A.
Mostafazadeh
, “Pseudo-Hermitian representation of quantum mechanics
,” Int. J. Geom. Methods Mod. Phys.
07
, 1191
–1306
(2010
).20.
S.
Albeverio
, U.
Günther
, and S.
Kuzhzel
, “J-self-adjoint operators with
,” $\mathcal {C}$
-symmetries: An extension theory approachJ. Phys. A: Math. Theor.
42
, 105205
(2009
).21.
J.
Bognar
, Indefinite Inner Product Spaces
(Springer-Verlag
, Berlin/New York
, 1974
).22.
C. M.
Bender
and S.
Kuzhel
, “Unbounded
,” ${\mathcal {C}}$
-symmetries and their nonuniquenessJ. Phys. A: Math. Theor.
45
, 444005
(2012
).23.
E. B.
Davies
, Linear Operators and Their Spectra
(Cambridge University Press
, Cambridge, UK
, 2007
).24.
B.
Sz.-Nagy
and C.
Foiaş
, Harmonic Analysis of Operators in Hilbert Space
(North-Holland
, Amsterdam, and Akadémiai Kiadó, Budapest
, 1970
).25.
L.
Tzafriri
, “Quasi-similarity for spectral operators on Banach spaces
,” Pac. J. Math.
25
, 197
–217
(1968
).26.
N.
Dunford
, “A survey of the theory of spectral operators
,” Bull. Am. Math. Soc.
64
, 217
–274
(1958
).27.
A. N.
Feldzamen
, “Semi-similarity invariants for spectral operators on Hilbert space
,” Trans. Am. Math. Soc.
100
, 277
–323
(1961
).28.
S.
Ôta
and K.
Schmüdgen
, “On some classes of unbounded operators
,” Integral Equ. Oper. Theory
12
, 211
–226
(1989
).29.
There is a misprint in that paper, on page 2, l.-2. The correct statement is Θ[Dom(H)] ⊂ Dom(H†), which indeed satisfies the relation H ⊣ H†.
30.
E. B.
Davies
, “Semi-classical states for non-self-adjoint Schrödinger operators
,” Commun. Math. Phys.
200
, 35
–41
(1999
).31.
B. F.
Samsonov
, “Hermitian Hamiltonian equivalent to a given non-Hermitian one: Manifestation of spectral singularity
,” Philos. Trans. R. Soc. London
371
, 20120044
(2013
).32.
M.
Reed
and B.
Simon
, Methods of Modern Mathematical Physics. I. Functional Analysis
(Academic Press
, New York/London
, 1972
, 1980).33.
K.
Schmüdgen
, Unbounded Self-Adjoint Operators on Hilbert Space
(Springer-Verlag
, Dordrecht/Heidelberg
, 2012
).34.
M.
Reed
and B.
Simon
, Methods of Modern Mathematical Physics. IV. Analysis of Operators
(Academic Press
, New York and London
, 1978
).35.
J.
Weidmann
, Linear Operators in Hilbert Spaces
(Springer
, New York
, 1980
).36.
A.
Inoue
and C.
Trapani
, “Non-self-adjoint resolutions of the identity and associated operators
,” preprint arXiv:1312.7090v1[math FA].37.
G. W.
Mackey
, Commutative Banach Algebras
, Notas de Matematica
Vol. 17
(Instituto de Matemática Pura e Aplicada
, Rio de Janeiro, RJ, Brasil
, 1959
).38.
Using Dunford's result (Ref. 9, Sec. XV.6), we may conclude directly that A = T−1ST, with S self-adjoint. The rest follows by putting again T = UG1/2.
39.
C.
Burnap
and P. F.
Zweifel
, “A note on the spectral theorem
,” Integral Equ. Oper. Theory
9
, 305
–324
(1986
).41.
A. O.
Barut
and R.
Rączka
, Theory of Group Representations and Applications
(PWN—Polish Scientific Publishers
, Warszawa
, 1977
).42.
E.
Nelson
, “Analytic vectors
,” Ann. Math.
70
, 572
–615
(1959
).43.
J.
Schwartz
, “Some non-selfadjoint operators
,” Commun. Pure Appl. Math.
13
, 609
–639
(1960
).© 2014 AIP Publishing LLC.
2014
AIP Publishing LLC
You do not currently have access to this content.
