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Ruffing, A.* ; Simon, M.

Analytic aspects of q-delayed exponentials: minimal growth, negative zeros and basic ghost states.

J. Difference Equ. Appl. 14, 347-366 (2008)
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Throughout the paper we discuss results related to the class of entire functions phi(N)(z;lambda,q) for N is an element of N defined by phi(N)(0) = 1 and phi '(N)(z) = lambda Nz(N-1) phi(N)(qz) for all z is an element of C where lambda not equal 0 and while the basic parameter q is an element of (0,1) assigns a fixed delay. In that sense the functions phi(N) may be regarded as q-delayed analogs of the standard exponentials exp(lambda z(N)). We shall see that all functions phi(N) are of order zero, whereas their continuum counterparts show growth (N,vertical bar lambda vertical bar). Some more examples for the order and type of discrete exponentials are given for comparison. It is shown that the q-exponential function E(q()z) phi(1()z;1,q) possesses an infinite sequence of zeros along the negative real axis. Finally, for alpha > 0 we construct q-ladder operators A and A(dagger) based on the ground state psi(0)(x;alpha) root x phi(2()x;-(alpha/2),q). The corresponding q-analog of the harmonic oscillator from Schrodinger theory has "excited states" psi(n)(x;alpha) for n is an element of N, which correspond to negative eigenvalues lambda(n) = alpha(2)q(-2)(1-q(-4n)) of A(dagger)A. This in turn seems to imply that none of the psi(n) lies in L-2(R+). We even prove that the q-delayed Gaussian bell E-q(x;alpha) phi(2)(x;-alpha/2),q) is not square integrable. Towards the end, we discuss some actual physical perspectives in quantum optics and coherent state theory.
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Publikationstyp Artikel: Journalartikel
Dokumenttyp Wissenschaftlicher Artikel
Schlagwörter q-delayed exponentials; minimal growth; negative zeros; harmonic oscillator; ghost states
ISSN (print) / ISBN 1023-6198
e-ISSN 1563-5120
Quellenangaben Band: 14, Heft: 4, Seiten: 347-366 Artikelnummer: , Supplement: ,
Verlag Taylor & Francis
Begutachtungsstatus Peer reviewed