Abstract:
Following the method proposed by Gazeau and Klauder to construct
temporally stable coherent states, CS for short, in recent years, several classes of
CS were constructed for quantum Hamiltonians. The spectrum E(n) of several
solvable quantum Hamiltonians is a polynomial of the label n. In this letter, we
discuss CS with a general spectrum 𝐸 𝑛 = 𝑎𝑘𝑛
𝑘 + 𝑎𝑘−1𝑛
𝑘−1 + ⋯ + 𝑎1𝑛 + 𝑎0
,
of degree k, which is considered as the spectrum of an abtract Hamiltonian. As
special cases of our construction we obtain CS for the quantum Hamiltonians,
namely; Harmonic oscillator, Isotonic oscillator, Pseudoharmonic oscillator,
Infite well potential, Pöschl-Teller potential and Eckart potential. We shall also
exploit the coherent states on a letf quaternionic separable Hilbert space with
the spectrum E(n). Let us introduce the general features of Gazeau-Klauder CS.
Let 𝐻 be a Hamiltonian with a bounded below discrete spectrum 𝑒𝑛
𝑛=0
∞ and it
has been adjusted so that 𝐻 ≥ 0. Further assume that the eigenvalues 𝑒𝑛 are
non-degenerate and arranged in increasing order, 𝑒0 < 𝑒1 < ⋯ . For such a
Hamiltonian, the so-called Gazeau-Klauder coherent states (GKCS for short) are
defined as
