277,530 research outputs found
Formulation of Complex Action Theory
We formulate a complex action theory which includes operators of coordinate
and momentum and being replaced with non-hermitian
operators and , and their eigenstates and with complex eigenvalues and .
Introducing a philosophy of keeping the analyticity in path integration
variables, we define a modified set of complex conjugate, real and imaginary
parts, hermitian conjugates and bras, and explicitly construct ,
, and by formally
squeezing coherent states. We also pose a theorem on the relation between
functions on the phase space and the corresponding operators. Only in our
formalism can we describe a complex action theory or a real action theory with
complex saddle points in the tunneling effect etc. in terms of bras and kets in
the functional integral. Furthermore, in a system with a non-hermitian
diagonalizable bounded Hamiltonian, we show that the mechanism to obtain a
hermitian Hamiltonian after a long time development proposed in our letter
works also in the complex coordinate formalism. If the hermitian Hamiltonian is
given in a local form, a conserved probability current density can be
constructed with two kinds of wave functions.Comment: 29 pages, 2 figures, references added, presentation improved, typos
corrected. (v5)The definition of and are
corrected by replacing them with their hermitian conjugates. The errors and
typos mentioned in the errata of PTP are corrected. arXiv admin note:
substantial text overlap with arXiv:1009.044
Baxter operator formalism for Macdonald polynomials
We develop basic constructions of the Baxter operator formalism for the
Macdonald polynomials associated with root systems of type A. Precisely we
construct a dual pair of mutually commuting Baxter operators such that the
Macdonald polynomials are their common eigenfunctions. The dual pair of Baxter
operators is closely related to the dual pair of recursive operators for
Macdonald polynomials leading to various families of their integral
representations. We also construct the Baxter operator formalism for the
q-deformed gl(l+1)-Whittaker functions and the Jack polynomials obtained by
degenerations of the Macdonald polynomials associated with the type A_l root
system. This note provides a generalization of our previous results on the
Baxter operator formalism for the Whittaker functions. It was demonstrated
previously that Baxter operator formalism for the Whittaker functions has deep
connections with representation theory. In particular the Baxter operators
should be considered as elements of appropriate spherical Hecke algebras and
their eigenvalues are identified with local Archimedean L-factors associated
with admissible representations of reductive groups over R. We expect that the
Baxter operator formalism for the Macdonald polynomials has an interpretation
in representation theory of higher-dimensional arithmetic fields.Comment: 22 pages, typos are fixe
Operators on the Fréchet sequence space ces(p+),
[EN] The Fréchet sequence spaces ces(p+) are very different to the Fréchet sequence spaces ¿p+,1¿pp}\ell ^q ℓ p + = ∩ q > p ℓ q . Math. Nachr. 147, 7–12 (1990)Pérez Carreras, P., Bonet, J.: Barrelled Locally Convex Spaces. North Holland, Amsterdam (1987)Pitt, H.R.: A note on bilinear forms. J. Lond. Math. Soc. 11, 171–174 (1936)Ricker, W.J.: A spectral mapping theorem for scalar-type spectral operators in locally convex spaces. Integral Equ. Oper. Theory 8, 276–288 (1985)Robertson, A.P., Robertson, W.: Topological Vector Spaces. Cambridge University Press, Cambridge (1973)Waelbroeck, L.: Topological vector spaces and algebras. Lecture Notes in Mathematics, vol. 230. Springer, Berlin (1971
Regularity of oscillatory integral operators
In this paper, we establish the global boundedness of oscillatory integral
operators on Besov-Lipschitz and Triebel-Lizorkin spaces, with amplitudes in
general -classes and non-degenerate phase
functions in the class \textart F^k. Our results hold for a wide range of
parameters , , ,
and . We also provide a sufficient condition for the boundedness of
operators with amplitudes in the forbidden class in
Triebel-Lizorkin spaces.Comment: arXiv admin note: text overlap with arXiv:2302.0031
Metrical Quantization
Canonical quantization may be approached from several different starting
points. The usual approaches involve promotion of c-numbers to q-numbers, or
path integral constructs, each of which generally succeeds only in Cartesian
coordinates. All quantization schemes that lead to Hilbert space vectors and
Weyl operators---even those that eschew Cartesian coordinates---implicitly
contain a metric on a flat phase space. This feature is demonstrated by
studying the classical and quantum ``aggregations'', namely, the set of all
facts and properties resident in all classical and quantum theories,
respectively. Metrical quantization is an approach that elevates the flat phase
space metric inherent in any canonical quantization to the level of a
postulate. Far from being an unwanted structure, the flat phase space metric
carries essential physical information. It is shown how the metric, when
employed within a continuous-time regularization scheme, gives rise to an
unambiguous quantization procedure that automatically leads to a canonical
coherent state representation. Although attention in this paper is confined to
canonical quantization we note that alternative, nonflat metrics may also be
used, and they generally give rise to qualitatively different, noncanonical
quantization schemes.Comment: 13 pages, LaTeX, no figures, to appear in Born X Proceeding
Dynamics of the Volterra-type integral and differentiation operators on generalized Fock spaces
[EN] Various dynamical properties of the differentiation and Volterra-type integral operators on generalized Fock spaces are studied. We show that the differentiation operator is always supercyclic on these spaces. We further characterize when it is hypercyclic, power bounded and uniformly mean ergodic. We prove that the operator satisfies the Ritt's resolvent condition if and only if it is power bounded and uniformly mean ergodic. Some similar results are obtained for the Volterra-type and Hardy integral operators.J. Bonet was partially supported by the research projects MTM2016-76647-P and GV Prometeo 2017/102 (Spain). M. Worku is supported by ISP project, Addis Ababa University, Ethiopia.Bonet Solves, JA.; Mengestie, T.; Worku, M. (2019). Dynamics of the Volterra-type integral and differentiation operators on generalized Fock spaces. Results in Mathematics. 74(4):1-15. https://doi.org/10.1007/s00025-019-1123-7S115744Abanin, A.V., Tien, P.T.: Differentiation and integration operators on weighted Banach spaces of holomorphic functions. Math. Nachr. 290(8–9), 1144–1162 (2017)Atzmon, A., Brive, B.: Surjectivity and invariant subspaces of differential operators on weighted Bergman spaces of entire functions, Bergman spaces and related topics in complex analysis, Contemp. Math., vol. 404, Amer. Math. Soc., Providence, RI, pp. 27–39 (2006)Bayart, F., Matheron, E.: Dynamics of Linear Operators, Cambridge Tracts in Math, vol. 179. Cambridge Univ. Press, Cambridge (2009)Bermúdez, T., Bonilla, A., Peris, A.: On hypercyclicity and supercyclicity criteria. Bull. Austral. Math. Soc. 70, 45–54 (2004)Beltrán, M.J.: Dynamics of differentiation and integration operators on weighted space of entire functions. Studia Math. 221, 35–60 (2014)Beltrán, M.J., Bonet, J., Fernández, C.: Classical operators on weighted Banach spaces of entire functions. Proc. Am. Math. Soc. 141, 4293–4303 (2013)Bès, J., Peris, A.: Hereditarily hypercyclic operators. J. Funct. Anal. 167, 94–112 (1999)Bonet, J.: Dynamics of the differentiation operator on weighted spaces of entire functions. Math. Z. 26, 649–657 (2009)Bonet, J.: The spectrum of Volterra operators on weighted Banach spaces of entire functions. Q. J. Math. 66, 799–807 (2015)Bonet, J., Bonilla, A.: Chaos of the differentiation operator on weighted Banach spaces of entire functions. Complex Anal. Oper. Theory 7, 33–42 (2013)Bonet, J., Taskinen, J.: A note about Volterra operators on weighted Banach spaces of entire functions. Math. Nachr. 288, 1216–1225 (2015)Constantin, O., Persson, A.-M.: The spectrum of Volterra-type integration operators on generalized Fock spaces. Bull. Lond. Math. Soc. 47, 958–963 (2015)Constantin, O., Peláez, J.-Á.: Integral operators, embedding theorems and a Littlewood–Paley formula on weighted Fock spaces. J. Geom. Anal. 26, 1109–1154 (2016)De La Rosa, M., Read, C.: A hypercyclic operator whose direct sum is not hypercyclic. J. Oper. Theory 61, 369–380 (2009)Dunford, N.: Spectral theory. I. Convergence to projections. Trans. Am. Math. Soc. 54, 185–217 (1943)Grosse-Erdmann, K.G., Peris Manguillot, A.: Linear Chaos. Springer, New York (2011)Harutyunyan, A., Lusky, W.: On the boundedness of the differentiation operator between weighted spaces of holomorphic functions. Studia Math. 184, 233–247 (2008)Krengel, U.: Ergodic Theorems. Walter de Gruyter, Berlin (1985)Lyubich, Yu.: Spectral localization, power boundedness and invariant subspaces under Ritt’s type condition. Studia Mathematica 143(2), 153–167 (1999)Mengestie, T.: A note on the differential operator on generalized Fock spaces. J. Math. Anal. Appl. 458(2), 937–948 (2018)Mengestie, T.: Spectral properties of Volterra-type integral operators on Fock–Sobolev spaces. J. Kor. Math. Soc. 54(6), 1801–1816 (2017)Mengestie, T.: On the spectrum of volterra-type integral operators on Fock–Sobolev spaces. Complex Anal. Oper. Theory 11(6), 1451–1461 (2017)Mengestie, T., Ueki, S.: Integral, differential and multiplication operators on weighted Fock spaces. Complex Anal. Oper. Theory 13, 935–95 (2019)Mengestie, T., Worku, M.: Isolated and essentially isolated Volterra-type integral operators on generalized Fock spaces. Integr. Transf. Spec. Funct. 30, 41–54 (2019)Nagy, B., Zemanek, J.A.: A resolvent condition implying power boundedness. Studia Math. 134, 143–151 (1999)Nevanlinna, O.: Convergence of iterations for linear equations. Lecture Notes in Mathematics. ETH Zürich, Birkhäuser, Basel (1993)Ritt, R.K.: A condition that . Proc. Am. Math. Soc. 4, 898–899 (1953)Ueki, S.: Characterization for Fock-type space via higher order derivatives and its application. Complex Anal. Oper. Theory 8, 1475–1486 (2014)Yosida, K.: Functional Analysis. Springer, Berlin (1978)Yosida, K., Kakutani, S.: Operator-theoretical treatment of Marko’s process and mean ergodic theorem. Ann. Math. 42(1), 188–228 (1941
History operators in quantum mechanics
It is convenient to describe a quantum system at all times by means of a
"history operator" , encoding measurements and unitary time evolution
between measurements. These operators naturally arise when computing the
probability of measurement sequences, and generalize the "sum over position
histories " of the Feynman path-integral. As we argue in the present note, this
description has some computational advantages over the usual state vector
description, and may help to clarify some issues regarding nonlocality of
quantum correlations and collapse. A measurement on a system described by
modifies the history operator, , where is the projector
corresponding to the measurement. We refer to this modification as "history
operator collapse". Thus keeps track of the succession of measurements on a
system, and contains all histories compatible with the results of these
measurements. The collapse modifies the history content of , and therefore
modifies also the past (relative to the measurement), but never in a way to
violate causality. Probabilities of outcomes are obtained as . A similar formula yields probabilities for intermediate
measurements, and reproduces the result of the two-vector formalism in the case
of given initial and final states. We apply the history operator formalism to a
few examples: entangler circuit, Mach-Zehnder interferometer, teleportation
circuit, three-box experiment. Not surprisingly, the propagation of coordinate
eigenstates is described by a history operator containing the
Feynman path-integral.Comment: 18 pages, 3 figures, LaTeX; V2: added section on three-box
experiment, expanded section on history operators, added subsection on
history amplitudes, added ref.s. V3: added ref., corrected typos and improved
notation, matches version to be published in Int. Jou. of Quantum Informatio
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