Zeros of Bargmann Analytic Representation in the Complex Plane
The paper contains an investigation of zeros Of Bargmann analytic representation. A brief introduction to Harmonic oscillator formalism is given. The Bargmann analytic representation has been studied. The zeros of Bargmann analytic function are considered. The Q or Husimi functions are introduced. The The Bargmann functions and the Husimi functions have the same zeros. The Bargmann functions f(z) have exactly q zeros. The evolution time of the zeros μn are discussed. Various examples have been given.
[1] A. M. Perelomov. Generalized Coherent States and Their Applications. Springer-Verlag, Berlin, 1986. [2] A. Vourdas. Analytic representations in quantum mechanics. J. Phys. A: Math. Gen., 39:R65, 2006. [3] A. Vourdas and R. F. Bishop. Thermal coherent states in the Bargmann representation. Phys. Rev. A, 50:3331, 1994. [4] V. Bargmann. On a Hilbert space of analytic functions and an associated integral transform Part I. Commun. Pure Appl. Math., 14:187, 1961. [5] V. Bargmann. On a Hilbert space of analytic functions and an associated integral transform Part II. Commun. Pure Appl. Math., 120:1, 1967. [6] S. Schweber. On the application of Bargmann Hilbert spaces to dynamical problems. Ann. Phys., 41:205, 1967. [7] P. Leboeuf and A. Voros. Chaos-revealing multiplicative representation of quantum eigenstates. J. Phys. A, 23:1765, 1990. [8] P. Leboeuf. Phase space approach to quantum dynamics. J. Phys. A: Math. Gen., 24:4575, 1991. [9] M. B. Cibils, Y. Cuche, P. Leboeuf, and W. F. Wreszinski. Zeros of the Husimi functions of the spin-boson model. Phys. Rev. A, 46:4560, 1992. [10] J. M. Tualle and A. Voros, Chaos. Normal modes of billiards portrayed in the stellar (or nodal) representation. Solitons and Fractals. 5:1085, 1995 [11] S. Nonnenmacher and A. Voros. Chaotic eigenfunctions in phase space. J. Phys. A, 30:L677, 1997. [12] H. J. Korsch, C. M´uller, and H. Wiescher. On the zeros of the Husimi distribution. J. Phys. A:Math. Gen., 30:L677, 1997. [13] F. Toscano and A. M. O. de Almeida. Geometrical approach to the distribution of the zeros for the Husimi function. J. Phys. A:Math. Gen., 32:6321, 1999. [14] R. P. Boas. Entire Functions. Academic Press, New York, 1954. [15] B. Ja Levin. Distribution of Zeros of Entire Functions. American Mathematical Society, Providence, Rhode Island 1964. [16] B. J. Levin, Lecture on Entire Functions. Rhode Island: American Mathematical Society, 1996. [17] A. Vourdas. The growth Bargmann functions and the completeness of sequences of coherent states. J. Phys. A: Math. Gen., 30:4867, 1997. [18] A. Vourdas. The growth and zeros of Bargmann functions. J.Phys. conf.ser.213(2010)012001
[1] A. M. Perelomov. Generalized Coherent States and Their Applications. Springer-Verlag, Berlin, 1986. [2] A. Vourdas. Analytic representations in quantum mechanics. J. Phys. A: Math. Gen., 39:R65, 2006. [3] A. Vourdas and R. F. Bishop. Thermal coherent states in the Bargmann representation. Phys. Rev. A, 50:3331, 1994. [4] V. Bargmann. On a Hilbert space of analytic functions and an associated integral transform Part I. Commun. Pure Appl. Math., 14:187, 1961. [5] V. Bargmann. On a Hilbert space of analytic functions and an associated integral transform Part II. Commun. Pure Appl. Math., 120:1, 1967. [6] S. Schweber. On the application of Bargmann Hilbert spaces to dynamical problems. Ann. Phys., 41:205, 1967. [7] P. Leboeuf and A. Voros. Chaos-revealing multiplicative representation of quantum eigenstates. J. Phys. A, 23:1765, 1990. [8] P. Leboeuf. Phase space approach to quantum dynamics. J. Phys. A: Math. Gen., 24:4575, 1991. [9] M. B. Cibils, Y. Cuche, P. Leboeuf, and W. F. Wreszinski. Zeros of the Husimi functions of the spin-boson model. Phys. Rev. A, 46:4560, 1992. [10] J. M. Tualle and A. Voros, Chaos. Normal modes of billiards portrayed in the stellar (or nodal) representation. Solitons and Fractals. 5:1085, 1995 [11] S. Nonnenmacher and A. Voros. Chaotic eigenfunctions in phase space. J. Phys. A, 30:L677, 1997. [12] H. J. Korsch, C. M´uller, and H. Wiescher. On the zeros of the Husimi distribution. J. Phys. A:Math. Gen., 30:L677, 1997. [13] F. Toscano and A. M. O. de Almeida. Geometrical approach to the distribution of the zeros for the Husimi function. J. Phys. A:Math. Gen., 32:6321, 1999. [14] R. P. Boas. Entire Functions. Academic Press, New York, 1954. [15] B. Ja Levin. Distribution of Zeros of Entire Functions. American Mathematical Society, Providence, Rhode Island 1964. [16] B. J. Levin, Lecture on Entire Functions. Rhode Island: American Mathematical Society, 1996. [17] A. Vourdas. The growth Bargmann functions and the completeness of sequences of coherent states. J. Phys. A: Math. Gen., 30:4867, 1997. [18] A. Vourdas. The growth and zeros of Bargmann functions. J.Phys. conf.ser.213(2010)012001
@article{"International Journal of Engineering, Mathematical and Physical Sciences:57216", author = "Muna Tabuni", title = "Zeros of Bargmann Analytic Representation in the Complex Plane", abstract = "The paper contains an investigation of zeros Of Bargmann analytic representation. A brief introduction to Harmonic oscillator formalism is given. The Bargmann analytic representation has been studied. The zeros of Bargmann analytic function are considered. The Q or Husimi functions are introduced. The The Bargmann functions and the Husimi functions have the same zeros. The Bargmann functions f(z) have exactly q zeros. The evolution time of the zeros μn are discussed. Various examples have been given.
", keywords = "Bargmann functions, Husimi functions, zeros.", volume = "7", number = "8", pages = "1294-5", }
