Coherent States for an Abstract Hamiltonian with a General Spectrum

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