Harmonic oscillator with time-dependent mass and frequency and a perturbative potential

Inter-Universal Entanglement

“…We can then follow the theory of invariants developed by Lewis [43] and others [11,41,55,57,67,69,76], and find a Hermitian invariant operator,…”

Effects of a scalar field on the thermodynamics of interuniversal entanglement

“…(22) can be expressed in terms of the eigenfunctions ∆ n (t, ϕ) of the harmonic oscillator with time dependent mass [46][47][48][49][50][51], i.e., ∆(t, ϕ) = n B n ∆ n (t, ϕ), with B n constant coefficients and ∆ n (t, ϕ) the normalized eigenfunctions that can be expressed as (see, for instance, Sec. 4.2 of Ref.…”

“…This can be given by the so called Lewis representation [46] (see also, Refs. [49,50,52,53]), for which the creation and annihilation operators are defined for each single mode n as [25] …”

Time-dependent formulation of the linear unitary transformation and the time evolution of general time-dependent quadratic Hamiltonian systems

“…Many problems of fundamental importance in various fields of physics, such as time-dependent harmonic oscillators with damping and driving [5][6][7][8][9][10][11][12][13], Paul trap [14][15][16], and motion of a particle in a time-dependent linear potential [17][18][19][20][21][22] turn out to be typical examples modeled by such a type of timedependent quadratic Hamiltonians. It has already been shown that the Lie algebraic approach is a powerful tool to study the time evolution of systems with an explicitly time-dependent Hamiltonian that can be written as a linear combination of time-independent operators which span a finite-dimensional Lie algebra.…”