Quantum Oscillators  

Quantum Harmonic Oscillator > s.a. feynman propagator; formulations [combinatorial]; oscillators; Polymer Representation; time in quantum theory.
* Propagator: The Feynman propagator for the quantum harmonic oscillator is (Δt:= t2t1)

DF(x2, t2; x1, t1) = [mω / (2πi sin(ωΔt))1/2] exp{i mω/2 [(x22+x12) · (ωΔt) – 2 x2x1 csc(ωΔt)] } .

@ General references: Jauch & Hill PR(40); Leubner et al AJP(88)dec [vs classical]; Boya et al IJMPA(98) [inequivalent]; Muñoz AJP(98)mar [and integral equations]; Jordan AJP(01)oct [simple solution]; Baker et al AJP(02)may [applications, numerical]; Kastrup AdP(07)qp/06 [new look]; Plimak & Stenholm AP(08) [response properties]; Marsiglio AJP(09)mar [and bound states of short-range potentials]; Andrews AJP(16)apr-a1509 [evolution of wave functions]; Zubizarreta et al a1607 [structure of the Hilbert space].
@ 2D: Chen & Huang JPA(03) [coherent states, vortex structure]; Montesinos & Torres del Castillo PRA(04), comment Latimer PRA(07)qp/06 [different symplectic structures]; Doll & Ingold AJP(07)mar [semiclassical, Lissajous curves]; Borondo et al JPA(09)-a0907 [and Bohmian trajectories].
@ Entanglement: Kim & Iafrate FPL(04) [coupled]; Rios JPA(07) [non-interacting]; Jost et al Nat(09)jun-a0901 [non-interacting, demonstration].
@ Coupled oscillators: Maxson ht/03, ht/03, ht/03, ht/03 [correlated]; Colosi a0706 [general relativistic, two-point function]; Bhattacharya et al AJP(13)apr.
@ Coupled to a heat bath: Ford & O'Connell PhyE(05)qp/06; Isar & Scheid PhyA(07)-a0705 [decoherence and classical correlations]; Ford & O'Connell PRA(14)-a1408 [two oscillators].
@ Different approaches: Donoghue & Holstein AJP(88)mar [path integral]; Koikawa PTP(01)ht, PTP(02)ht/01 [Moyal]; Shiri-Garakani & Finkelstein JMP(06) [general quantization]; Finkelstein & Shiri-Garakani IJTP(11)-qp/06 [as model for spacetime decondensation]; Nettel & Quevedo mp/06-proc [topological quantization]; Vicary IJTP(08) [categorical framework]; Pimentel & de Castro EJP(13)-a1211 [Laplace-transform approach]; Ahmadzadegan et al PRA(16)-a1510 [Koopman and Moyal formalisms, classical and quantum aspects]; > s.a. path integrals [non-standard].
@ Green function: Khrebtukov & Macek JPA(98); Shao AJP(16)oct [elementary derivation]; > s.a. feynman propagator.
@ Related topics: de la Peña & Cetto JMP(79) [and stochastic electrodynamics]; Dattoli & Torre NCB(95) [phase space, coherent states]; Lorente PLA(01)qp/04 [discrete model]; Morigi et al PRA(02) [irreversibility]; Leshem & Gat PRL(09) [violation of macroscopic realism]; Betz & Castrigiano CMP(11) [coupled to a photon field, density of states]; Wang a1303 [new approach]; Suslov PS(13) [variant of Berry's phase].
> States: see fock space; coherent states; hilbert space [inverted]; representations of quantum mechanics [Bargmann].
> Other topics: see approaches to quantum mechanics; Born-Jordan Quantization; Coarse-Graining; deformation quantization; geometric quantization; quantum-classical coupling; resonance; statistical-mechanical systems.

Anharmonic Oscillators
@ Cubic: Ferreira & Sesma JPA(14) [eigenstates].
@ Quartic: Pesquera & Claverie JMP(82) [in stochastic electrodynamics]; Matamala & Maldonado PLA(03), Liverts et al JMP(06) [spectrum and eigenfunctions, analytic].
@ Other / non-linear / perturbed: Pathak JPA(00)mp/02; Speliotopoulos JPA(00) [spectrum]; Calogero & Graffi PLA(03); Calogero PLA(03); Bhattacharyya & Bhattacharjee PLA(04) [subharmonic, V ∝ |x|r]; Gómez & Sesma in(04)qp [bound states], JPA(05)qp [new approach]; Cariñena et al AP(07) [solvable]; Liverts & Mandelzweig PS(08) [approximate solution]; Fernández a0804 [eigenvalues, upper and lower bounds]; Midya & Roy JPA(09) [exactly solvable, quasi-exactly solvable et al]; Wang & Liu IJTP(09); Nachtergaele et al RVMP(10) [anharmonic oscillator lattice systems]; Tosto a1105 [simple quantum model]; Wójcik APPB-a1210 [numerical renormalization group procedure]; Fernández & García a1310 [V(x, y) = x2y2; eigenvalues and eigenfunctions; the spectrum seems to be discrete]; > s.a. coherent states.
@ Semiclassical states: Matzkin & Lombardi JPA(05)-a0706 [quantum and semiclassical phase functions]; Moncrief et al JMP(12)-a1201 [semiclassical approach]; Brizuela PRD(14)-a1411 [dynamical evolution of classical and quantum probability distributions in terms of moments].
@ Perturbation methods: Cicuta ht/97; Amore mp/04-proc [classical and quantum]; Voros mp/06-proc; > s.a. Perturbation Methods.

Damped Oscillator > s.a. Lindblad Equation; quantum probability.
* Idea: A dissipative system; > s.a. dissipation.
@ General references: Pedrosa & Baseia PRD(84) [+ oscillator bath]; Milburn & Holmes PRL(96); Isar & Sandulescu RJP(92)qp/06 [rev]; de Brito et al NCB(98); Um et al PRP(02); Banerjee & Mukherjee JPA(02)qp/01 [canonical approach]; Blasone et al qp/03-conf; Montesinos PRA(03) [Heisenberg picture]; Latimer JPA(05)qp/04; López & López IJTP(06)qp/05; Endo et al IJGMP(08)-a0710, Fujii & Suzuki IJGMP(09)-a0806 [general solution]; Cordero-Soto et al a0905; Baldiotti et al PLA(11)-a1005; Philbin NJP(12) [with a continuum of oscillators as reservoir]; Barnett et al a1508 [strongly damped].
@ Bateman's dual system: (A damped simple harmonic oscillator coupled to its time-reversed image) Blasone & Jizba AP(04)qp/01.
@ Related topics: Kheirandish & Amooshahi MPLA(05)qp [radiation reaction]; Chruściński & Jurkowski AP(06) [resonances]; Isar O&S(07)qp/06-conf [decoherence, Lindblad theory].
@ In alternative approaches: Vandyck JPA(94) [pilot-wave theory]; Dito & Turrubiates PLA(06)qp/05 [deformation quantization]; Fujii qp/07-conf [complex time]; > s.a. wigner functions.

Other Types and Effects > s.a. decoherence; Dirac Oscillator; histories formulation; non-commutative systems; Perturbation Methods.
@ Relativistic: Guerrero & Aldaya MPLA(99) [perturbative]; Toyama & Nogami PRA(99); Bars PRD(09)-a0810; Arbab a1709 [quaternionic].
@ Inverted: Blume-Kohout & Zurek PRA(03)qp/02 [upside down, decoherence]; Chruściński JMP(04)mp/03; Yuce et al PS(06).
@ On the sphere and hyperbolic space: Cariñena et al AP(07)-a0709 [2D]; Mardoyan a0708-proc [in d dimensions]; Quesne PLA(15)-a1411.
@ Forced, time-dependent: Dodonov PLA(96) [kicked]; Graffi & Yajima CMP(00)mp [forced]; Kim & Yee PRA(02)ht; Moya-Cessa & Fernández-Guasti PLA(03)qp [sudden change, coherent states]; Adler JPA(05)qp/04 [stochastic collapse and decoherence]; Gómez & Villaseñor AP(09) [and quantum field theory]; Velasco-Martínez et al a1409 [unitary approach]; > s.a. stochastic quantization.
@ With gup, minimal length: Chang et al PRD(02)ht/01; Nouicer PLA(06); Gemba et al a0712 [algebraic solution, deformed su(1,1) algebra]; Fakel & Merad JMP(09); Lewis & Takeuchi PRD(11); Valtancoli MPLA(12)-a1205 [with a minimal uncertainty in position]; Valtancoli a1306 [with a minimal length]; Das et al CJP(16)-a1412 [in phase space]; Quintela et al BJP(16)-a1510 [classical limit].
@ Other deformed oscillators: Man'ko et al qp/97-proc; De Freitas & Salamó ht/99; Gruver PLA(99); Sogami & Koizumi PTP(02)mp/01; Isar & Scheid PhyA(02)qp/07, PhyA(04)qp/07 [in dissipative environment]; Albanese & Lawi JPA(04)ht/03; Narayana Swami qp/04 [and intermediate statistics]; Jafarov et al JPA(07)mp [Wigner function]; Batouli & El Baz FP(14) [classical interpretation]; Sadurní & Rivera-Mociños JPA(15)-a1504 [with fractal position spectrum]; > s.a. modified coherent states [including Grassmann].
@ Pais-Uhlenbeck oscillator: (An example of higher-derivative theory) Mannheim & Davidson PRA(05)ht/04 [Dirac quantization]; Andrzejewski et al PTP(11)-a0904 [Euclidean path-integral approach].
@ Supersymmetric: Thienel JPA(96) [Bargmann representation]; > s.a. coherent states.
@ Other types: Dragovich TMP(94)ht/04, IJMPA(95)ht/04 [adelic]; Banerjee & Ghosh JPA(98) [chiral]; Badescu & Landsberg JPA(02) [τ-oscillator]; Kim & Page qp/02 [generalized]; Blasone et al PLA(03)qp/02 [group contraction]; Guido a1208 [Intrinsic Quantum Oscillator (IQuO)]; Valtancoli PTEP(13)-a1306 [in a Snyder geometry]; Bender et al PRA(14)-a1406 [coupled PT-symmetric oscillators]; > s.a. stochastic quantization [Fermi oscillator].

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