Potential

In General
* Idea: Originally, a function whose gradient gives a physical field; Then extended to the mathematical notion of a function (or sometimes a higher-rank tensor field) which gives, by differentiation, a field of interest, possibly a dynamical tensor field.
> Vector potential: see aharonov-bohm, connection, electromagnetism.

In Classical Physics > s.a. scattering.
* Retarded potential: It has to be used for systems with large velocities (corrections are of order v²/c²), or pairs of systems with large separations compared to the internal motions (even if slow).
@ References: Kellogg 29; Grant & Rosner AJP(94)apr [orbits in power law V].
@ Retarded potential: Spruch & Kelsey PRA(78) [elementary derivation]; > s.a. arrow of time.
> Specific systems: see classical systems; Coulomb Potential; electromagnetism; newtonian gravitation.
> Results: see Bertrand's Theorem.

In Quantum Theory > s.a. effective field theories [effective potential]; quantum field theory; quantum systems [special types].
* Quasi-integrable: A spectral problem depending on a parameter, such that a finite set of eigenvalues can be obtained algebraically for special values of the parameter.
@ Quantum potential: Dewdney & Hiley FP(82) [and 1D scattering]; Garbaczewski PLA(92) [from realistic Brownian particle motions]; Carroll qp/04, qp/04 [survey], gq/05 [and quantum fluctuations, Weyl tensor]; Delle Site PhyB(04)qp [many-particle Bohm quantum potential]; Grössing PhyA(09)-a0808 [thermodynamic origin]; > s.a. measurement, pilot-wave interpretation.
@ PT-invariant: Weigert CzJP(04)qp; Ahmed JPA(05) [classical orbits and quantization].
@ Other complex: Muga et al PRP(04) [scattering, absorption].

Specific Types > s.a. oscillator; Pöschl-Teller; relativistic quantum mechanics [non-local]; schrödinger equation; Yukawa.
@ Inverse square: Gozzi & Mauro PLA(05) [scale symmetry, anomaly]; Ávila-Aoki et al PLA(09) [classical and quantum motion].
@ Exactly solvable: Fernández IJMPA(97)qp/96 [supersymmetric]; Rosas-Ortiz JPA(98)qp, qp/98-in; de Prunelé JPA(06) [2D]; Tremblay et al JPA(09) [and integrable, 2D, infinite family]; Odake & Sasaki a0906; > s.a. coherent states.
@ Conditionally exactly solvable: Roychoudhury et al JMP(01).
@ Quasi-exactly solvable / integrable: Turbiner CMP(88), JPA(89); Ushveridze SJPP(89), 94; Lazutkin 93 [nearly integrable, IV]; Braibant & Brihaye JMP(93) [applications]; Ushveridze 94; Bender & Dunne JMP(96)ht/95; Bender & Boettcher JPA(98)phy [quartic]; Debergh et al qp/02, qp/02 [Darboux transformations], qp/02; Geojo et al JPA(03)qp/02 [Hamilton-Jacobi method]; Atre & Panigrahi PLA(03) [approach]; Bender & Monou qp/05 [sextic]; Koc & Koca mp/05 [Pöschl-Teller et alia], mp/05 [Eckart-type potentials]; Klishevich mp/06-in [conditions].

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