Isochronous n-dimensional nonlinear PDM-oscillators: linearizability, invariance and exact solvability

Confined Klein–Gordon oscillator from a (2+1)-dimensional Gürses to a Gürses or a pseudo-Gürses space-time backgrounds: Invariance and isospectrality

“…On the other hand, the introduction of Mathews-Lakshmanan oscillator [25] has activated intensive research studies on "effective" position-dependent mass (PDM in short), both in classical and quantum mechanics [25][26][27][28][29][30][31][32][33][34][35][36][37][38][39][40][41][42]. PDM is a metaphoric manifestation of the coordinate deformation/transformation [29][30][31]. Nevertheless, Khlevniuk [34] has argued that a point mass in the curved space may effectively be transformed into a PDM in Euclidean space.…”

“…Nevertheless, Khlevniuk [34] has argued that a point mass in the curved space may effectively be transformed into a PDM in Euclidean space. Such coordinate transformation/deformation affects, in turn, the form of the canonical momentum in classical and the momentum operator in quantum mechanics (e.g., [29,30,33,37] and related references therein). In classical mechanics, it has been shown that negative the gradient of the potential force field is no longer the time derivative of the canonical momentum p = m (x) ẋ, but it is rather related to the time derivative of the pseudo-momentum (also called Noether momentum) π (x) = √ m (x) ẋ [30].…”

“…Such coordinate transformation/deformation affects, in turn, the form of the canonical momentum in classical and the momentum operator in quantum mechanics (e.g., [29,30,33,37] and related references therein). In classical mechanics, it has been shown that negative the gradient of the potential force field is no longer the time derivative of the canonical momentum p = m (x) ẋ, but it is rather related to the time derivative of the pseudo-momentum (also called Noether momentum) π (x) = √ m (x) ẋ [30]. In quantum mechanics, however, the PDM momentum operator is constructed [33] and used to find the PDM creation and annihilation operators for the Schrödinger oscillator [29].…”

PDM Damped-Driven Oscillators: Exact Solvability, Classical States Crossings, and Self-Crossings

“…Early on, we have reported the phase-space trajectories crossing (i.e., classical states {x(t), p(t)} crossings) as a phenomenon associated with dissipative forces (c.f., e.g., [8,9,39]) which may as well emerge as a result of the point canonical transformation (12) into PDM-settings. In the current study, we witness yet another type of classical states crossing.…”

“…This would keep the longstanding gain-loss balance correlation between the kinetic and potential energies of the system and, consequently, the total energy remains an integral of motion (i.e., the standard structure the textbook Lagrangians/Hamiltonians for conservative systems of course). Such a transformation/deformation recipe would in effect introduce the so called position-dependent effective mass concept (or PDM in short) into classical and quantum mechanics (c.f., e.g., sample of references [3][4][5][6][7][8][9][10][11][12][13][14][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39] ). It is, therefore, interesting to study and investigate PDM settings on such DDOs.…”