Quantum Systems with Special Potentials  

Some Common Types > s.a. quantum systems [with symmetries] / anomalies; hilbert space; quantum oscillator; wigner function.
@ Inverse square: Gozzi & Mauro PLA(05) [scale symmetry, anomaly]; Ávila-Aoki et al PLA(09) [classical and quantum motion]; > s.a. representations [in the polymer representation].
@ δ-function potential: Gosdzinsky & Tarrach AJP(91)jan [and quantum field theory model]; Demiralp & Beker JPA(03) [bound states]; > s.a. relativistic quantum theory.
@ Other central potentials: Ciftci et al JPA(03) [Coulomb + power law]; Martin qp/04 [near r = 0]; Alberg et al PRA(05)qp/04 [1/r4, renormalization]; Hall et al PRA(09)-a0908 [soft Coulomb potential]; Roy a1904-in [general method]; > s.a. relativistic quantum mechanics; quantum states [bound].
@ Other singular potentials: Esposito JPA(98)ht [scattering], FPL(00)qp/99; Landsman gq/98; Schulze-Halberg IJTP(00) [irregular singularity]; Tsutsui & Fülöp qp/03-proc [defects etc]; Fülöp Sigma(07)-a0708-proc [ambiguity in self-adjoint Hamiltonian]; Eckhardt et al JDE(14)-a1401 [δ'-interactions on complicated sets]; Lange a1401 [boundary conditions, distributional theory]; Costa Dias et al JDE-a1601 [Schwartz-distributional formulation]; Samar & Tkachuk a1812 [1/x2, regularization with deformed-space minimal length]; > s.a. perturbation methods.
@ Periodic potentials: Holstein AJP(88)oct [semiclassical]; Khare & Sukhatme JPA(04)qp [rev, solvable]; Pereyra AP(05) [finite-size]; Pavelich & Marsiglio AJP(15)sep-a1411 [general repeated confining potential]; > s.a. coherent states.
@ Periodic in time: Costin et al JPA(00)mp/06 [bound state survival probability], JPA(02)mp/06, JSP(04)mp/06, mp/06; López qp/06; Duclos et al RVMP(08)-a0710 [stability]; Martin & Poertner a2104 [Floquet's theorem].

Potential Steps, Wells and Similar Systems > s.a. pilot-wave phenomenology; quantum systems [time-dependent boundaries].
@ Potential steps: Ahmed PLA(96); Boonserm & Visser JPA(09)-a0808 [transmission probabilities]; Yearsley JPCS(09)-a0901 [propagator, path integral]; Jaffe AJP(10)jun [reflection above the barrier as tunneling in momentum space]; Garrido et al AJP(11)dec [reflection at a downward potential step].
@ Infinite well / box: Leyvraz et al AJP(97)nov [accidental degeneracy]; Ni qp/98 [Einstein, Pauli, Yukawa]; Colanero & Chu PRA(99)qp [oscillating]; Sankaranarayanan et al PRE(01)nl [periodic pulsing and chaos]; Waldenstrøm et al PS(03) [revivals]; Garbaczewski & Karwowski AJP(04)jul-mp/03; García de León et al PLA(08) [coherent state approach]; Pedram & Vahabi AJP(10)aug [with a δ-function potential]; Ogren & Carlsson EJP(11)-a1103 [lower energy bound]; Gaddah EJP(13) [2D, equilateral triangle]; Belloni & Robinett PRP(14) [and Dirac delta, as pedagogical models]; Alberto et al EJP(18)-a1711 [relativistic, Klein-Gordon vs Dirac equations]; > s.a. path integrals.
@ Finite square well: Bender et al JPA(99) [complex]; Blümel JPA(05) [analytical solution]; Roberts & Valluri CJP(17)-a1403 [using the Lambert W function]; Naqvi & Waldenstrøm a1505 [tutorial review]; > s.a. modified quantum theory [PT-symmetric].
@ 2D billiard: Cohen & Wisniacki PRE(03)nl/02 [moving walls]; Gutkin JPA(03) [plane waves and solutions].
@ Double well: Holstein AJP(88)apr [semiclassical]; Razavy NCB(01) [Heisenberg equation of motion]; Friedberg et al AP(01)qp; Roy & Bhattacharjee PLA(01)qp [chaos]; Tian & Zhong ChPL(10)-a1003 [new model]; > s.a. coherent states.

Atoms and Systems in External Fields
@ Hydrogen atom: Hofer qp/98 [different]; Parfitt & Portnoi JMP(02)mp [2D]; Alves et al PRA(03)ht/05 [between parallel plates]; Palma & Raff CJP(06)qp [1D]; Zhao et al PRD(07)-a0705 [in Schwarzschild metric]; Martínez-y-Romero et al AJP(07)jul [with group theory methods]; Jaramillo et al PLA(09) [1D]; Machet & Vysotsky PRD(11)-a1011 [in a superstrong magnetic field]; Sharipov a1308 [proton and electron coupled to the electromagnetic field]; Ferreyra & Proetto AJP(13)nov [compressed in a spherical well]; Ogawa a1607 [algebraic method]; Chua a1805 [using the Runge-Lenz vector]; > s.a. born-infeld theory; momentum representation; topological defects.
@ Helium atom: Schwartz mp/06 [ground state]; Withers & Nadarajah RPMP(11) [solutions of Schrödinger's equation]; Esposito & Naddeo FP(12)-a1207 [Majorana and the two-electron problem].
@ In an electric field: Karasev & Osborn JMP(02)qp/00 [electromagnetic fields]; Matteucci EJP(07) [intro].
@ In a gravitational field: Chacón-Acosta et al a1904 [particle falling].
@ In a magnetic field: Krause PRA(96) [constant]; Schmiedmayer & Scrinzi PRA(96) [linear current]; Thienel AP(00)qp/98; Nambu NPB(00) [2D, vortices and field]; Schuch & Moshinsky JPA(03) [coherent states]; Chiou et al mp/04 [self-linking B field]; > s.a. aharonov-bohm effect.

Other Types of Potentials > s.a. integrable quantum systems; relativistic quantum theory [non-local]; schrödinger equation.
* 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.
@ Exactly solvable: Fernández IJMPA(97)qp/96 [supersymmetric]; Rosas-Ortiz JPA(98)qp, qp/98-proc; Alhaidari mp/03 [larger class]; de Prunelé JPA(06) [2D]; Tremblay et al JPA(09) [and integrable, 2D, infinite family]; Odake & Sasaki PLB(09)-a0906; Alhaidari PS(10)-a1004, Bahlouli & Alhaidari PS(10)-a1004 [larger class]; Quesne JPCS(12)-a1111 [and exceptional orthogonal polynomials]; > s.a. coherent states.
@ Conditionally exactly solvable: Roychoudhury et al JMP(01).
@ Quasi-exactly solvable / integrable: Turbiner CMP(88), JPA(89); Ushveridze SJPP(89); 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 AP(02)qp, IJMPA(02)qp [Darboux transformations], IJMPA(03)qp/02; Geojo et al JPA(03)qp/02 [Hamilton-Jacobi method]; Atre & Panigrahi PLA(03) [approach]; Bender & Monou JPA(05)qp [sextic]; Koc & Koca mp/05 [Pöschl-Teller et alia], mp/05 [Eckart-type potentials]; Klishevich mp/06-conf [conditions].
@ Random potentials: Yannacopoulos et al PS(02) [2D]; Germinet & Klein mp/05, mp/06 [localization]; Baker et al CMP(08) [deformed lattice]; > s.a. Anderson Localization.
@ PT-invariant: Weigert CzJP(04)qp; Ahmed JPA(05) [classical orbits and quantization].
@ Other complex potentials: Muga et al PRP(04) [scattering, absorption].
@ Other types: Damanik et al CMP(05)mp/04 [with finitely-many bound states]; Smilga JPA(09)-a0808 [exceptional points]; Fabre & Guéry-Odelin AJP(11)jul-a1012 [exactly solvable, supersymmetry and approximation schemes]; Di Martino et al JPA(13)-a1306, Cooney a1703 [box with moving walls]; Teichert et al a1910 [equidistant energy levels]; > s.a. Bloch Theory; Pöschl-Teller Potential; quantum oscillators; Rotor; Yukawa Potential.

main pageabbreviationsjournalscommentsother sitesacknowledgements
send feedback and suggestions to bombelli at olemiss.edu – modified 15 apr 2021