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

Abstract

Throughout the paper we discuss results related to the class of entire functions for N ∈ ℕ, defined by and for all z ∈ ℂ, where λ ≠ 0 and while the basic parameter q ∈ (0,1) assigns a fixed delay. In that sense the functions ϕ N may be regarded as q-delayed analogs of the standard exponentials exp(λz N ). We shall see that all functions ϕ N are of order zero, whereas their continuum counterparts show growth . Some more examples for the order and type of discrete exponentials are given for comparison. It is shown that the q-exponential function possesses an infinite sequence of zeros along the negative real axis. Finally, for α>0 we construct q-ladder operators 𝒜 and 𝒜† based on the ground state . The corresponding q-analog of the harmonic oscillator from Schrödinger theory has “excited states” for n ∈ ℕ, which correspond to negative eigenvalues of 𝒜†𝒜. This in turn seems to imply that none of the ψ n lies in . We even prove that the q-delayed Gaussian bell is not square integrable. Towards the end, we discuss some actual physical perspectives in quantum optics and coherent state theory.