Exact solutions for the general nonstationary oscillator with a singular perturbation

Abstract

Three linearly independent Hermitian invariants for the nonstationary generalized singular oscillator (SO) are constructed and their complex linear combination is diagonalized. The constructed family of eigenstates contains as subsets all previously obtained solutions for the SO and includes all Robertson and Schrödinger intelligent states for the three invariants. It is shown that the constructed analogues of the SU (1, 1) group-related coherent states for the SO minimize the Robertson and Schrödinger uncertainty relations for the three invariants and for every pair of them simultaneously. The squeezing properties of the new states are briefly discussed.

Photon-added coherent states for exactly solvable Hamiltonians

“…The main task of this work is to give a general method to construct Photon-added coherent states by adding photons to Gazeau-Klauder and Klauder-Perelomov coherent states. The term Klauder-Perelomov do not should be confused with one previously used in [22] in another context. Note also that the term "Photonadded" is used in the sense discussed in [21].…”

Quantum features of a charged particle in ionized plasma controlled by a time-dependent magnetic field

“…Hence the system is a kind of TDHSs that have attracted wide interest in the physical society [5,6,[17][18][19][20][21][22][23][24][25][26]. To derive quantum solutions of a TDHS, it is convenient to introduce an invariant operator [5,6] because the quantum properties of such system can be investigated via the eigenstates of the invariant operator.…”

Diagonalization of Hamiltonians, Uncertainty Matrices and Robertson Inequality

“…For example constant Ω corresponds to the stationary oscillator and to the oscillators with varying mass (damped oscillators) m(t) = m 0 exp(−2bt) and m(t) = m 0 cos 2 bt, considered later by many authors (see refs. in [3,6,9]). Analytical solutions to the equation of z(t) are known for a variety of "frequencies" Ω(t).…”

“…Quadratic Hamiltonians model many quantum (and classical) systems: from free particle and free electromagnetic field to the waves in nonlinear media, molecular dynamics and gravitational waveguide [3,4,5,6]. A considerable attention to quadratic classical and/or quantum systems is paid in the literature for a long period of time (see, for example, [7,3,6,8,9] and references therein).…”