Scattering and Collisions  

Scattering in General > s.a. diffraction [in field theory]; huygens' principle; resonance.
* Geometric collisions: For particles of diameter d, mean speed v and number density n, the mean free path is l = (√2π d2n)–1, and the collision frequency f = v/l = √2πd2vn; The mean free time is of course t = 1/; > s.a. brownian motion.
* Optical theorem: (a.k.a. optical cross-section theorem) The total cross section for an elastic scattering and absorption process (the potential can possibly be complex) is

σtot = (4π/k) Im f(k, k) ;

The result is often credited to N Bohr, R Peierls and G Placzek, but it is actually due to E Feenberg and Lord Rayleigh.
@ Geometric collisions: Jakoby EJP(09) [relaxation time, mean free path, and electronic conductivity]; Palk AJP(14)jun [mean free path as statistical mean of the distribution of free path lengths].
@ Optical theorem: in Schiff 68; Newton AJP(76)jul; Bussey PLA(86) [and wave-function collapse]; Mansuripur AJP(12)apr [new perspective].
@ Techniques: Barlette et al AJP(01)sep [integral equations, partial waves]; Liu et al JHEP(14)-a1403 [without large-distance asymptotics].

Special Cases and Applications
* Important experiments: Thomson scattering; Rutherford scattering; Deep inelastic scattering, that showed the composite nature of baryons.
* Deep inelastic scattering: High-energy scattering of electrons off nucleons; Provides direct evidence for the existence of quarks inside the proton; Parameters are x = fraction of nuclear momentum carried by q, Q2 = square of momentum transfer between nucleon and beam particle; > s.a. critical phenomena.
@ Coulomb potential: Yafaev JPA(97) [n-dimensional, quantum]; Ahmed qp/03 [quantum]; Mineev TMP(04) [1D, self-adjoint extension]; Glöckle et al PRC(09) [screening and renormalization factor]; Neilson & Senatore ed JPA(09)#21; Abramovici & Avishai JPA(09) [1D].
@ Gravitational, relativistic: Barrabès & Hogan CQG(04); Barbieri & Guadagnini NPB(05)gq [massless particles off rotating bodies]; Nikishov a0807 [classical and quantum]; Betti a1411-th [transplackian scattering]; > s.a. motion of gravitating bodies.
@ Bounded / point obstacle: Athanasiadis et al JMP(02) [acoustic and electromagnetic waves].
@ Off defects: Katanaev & Volovich AP(99); Spinelly et al CQG(01) [conical, cosmic string].
> Specific processes: see Bhabha Scattering; Drell-Yan Process; neutron; Preons; Rutherford, Superradiant, Thomas, Thomson Scattering.
> Specific theories: see atomic physics; dirac fields; graviton; molecular physics; photon phenomenology; pilot-wave quantum mechanics; wave phenomena.

Multiple Scattering > s.a. Debye Length; light; Rayleigh.
@ General references: Huang PLA(04) [perturbative]; Ramm mp/06, PLA(07)mp/06, JPA(08) [off many small bodies], PLA(07), PLA(08) [waves off many particles]; Ramm & Rona a0910; Ramm RPMP(13) [transmission boundary conditions].
@ Random scatterers: Mathur & Yeh JMP(64) [finite size, electromagnetic waves]; Külske mp/01, Dean et al JPA(04) [point scatterers]; Field 09; Basile et al a1307 [diffusion limit]; Ramm JMP(14)-a1402 [electromagnetic wave scattering by small perfectly conducting particles].

In Quantum Theory > s.a. quantum mechanical tunneling [delay time]; S-Matrix.
* Levinson's theorem: A fundamental theorem in quantum scattering theory, which shows the relation between the number of bound states and the phase shift at zero momentum for the Schrödinger equation.
* Scattering amplitudes in quantum field theory: They can be expressed using a path integral over all possible classical field configurations, or starting from first principles and using recursion relations.
@ General references: Amrein qp/01 [large-time behavior]; Ignatovich qp/04 [problems?]; Cannata et al AP(07) [PT-symmetric quantum mechanics, 1D]; Hussein et al JPA(08)-a0807 [new formulation]; Norsen a0910; Carron & Rosenfelder NJP(10)-a0912 [path-integral description]; > s.a. quantum systems.
@ Levinson's theorem: Wellner AJP(64)oct; Lin PRA(97)qp/98, PRA(98)qp/98 [2D]; Rosu NCB(99)gq/97 [in quantum cosmology]; Dong & Ma IJTP(00) [1D Schrödinger equation]; Sheka et al PRA(03)qp/02; Boya & Casahorrán IJTP(07)qp/05-conf [from spectral density]; Kellendonk & Richard qp/05, JPA(08)-a0712 [topological version]; Ma JPA(06) [rev]; Jia et al a1007 [for potentials with critical decay 1/r 2]; Kellendonk & Richard a1009; Childs & Strouse JMP(11)-a1103, Childs & Gosset JMP(12) [for scattering on a graph]; Nicoleau et al a1611 [extended version for systems with complex eigenvalues]; > s.a. topology in physics.
@ Semiclassical: Ford & Wheeler AP(00); Rothstein JMP(04) [1D and 2D]; Berera NPA(07)ht [scattering of large objects in quantum field theory and classical description]; Adhikari & Hussain AJP(08)dec [2D].
@ In quantum field theory: Buchholz & Summers mp/05-en; Biswas a0807 [alternative approach]; Toth CEJP(12)-a0904 [definition of scattering states]; Rubtsov et al PRD(12)-a1204 [Lorentz-violating theories]; Arkani-Hamed et al 16 [Grassmannian geometry]; Taylor a1703 [gauge theories]; > s.a. bogoliubov transformations.
@ Recursion relations for scattering amplitudes: Cheung et al PRL(16) + Kosower Phy(16) [effective field theories].
@ Electron scattering: Tyutin pr(74)-a0801 [by solenoid]; Dybalski a1706 [non-perturbative description of colliding electrons].
@ Yang-Mills theory scattering amplitudes: Britto et al PRL(05); Bjerrum-Bohr a1605 [analytic expressions].
@ Other types of situations: Mostafazadeh PRA(96) [on curved surfaces]; de la Torre AJP(97)feb [wave packet in a central potential, distorsion]; Esposito JPA(98)ht [singular potentials]; Pérez Prieto et al JPA(03) [Gaussian wave function and square barrier]; Roux & Yafaev JMP(03) [long-range Vs].

References > s.a. Inverse Scattering; Perturbation Methods; special potentials.
@ General: Reed & Simon 79; Ramm in(80)mp/00 [scalar + vector waves, arbitrary shapes].
@ Relativistic: Aichelburg et al CQG(04)gq/03 [ultrarelativistic charges].
@ In curved spaces: Beig APA(82) [scalar fields, Schwarzschild spacetime]; Ito & Skibsted a1109 [on non-compact, connected, complete Riemannian manifolds].
@ In non-commutative theories: Alavi MPLA(05)ht/04, Bellucci & Yeranyan PLB(05)ht/04 [quantum]; Kumar & Rajaraman PRD(06)ht/05.
@ Bohm-Gadella theory controversy: de la Madrid JPA(06)qp; Gadella & Wickramasekara JPA(07); de la Madrid JPA(07)-a0704; Baumgärtel a0704; de la Madrid a0705.
@ Related topics: Fabbrichesi et al NPB(94) [Planck energies]; Visser & Wolf PLA(97) [with field discontinuities]; Laura IJTP(97)qp/99; de Vries et al RMP(98) [waves, point scatterer]; Horan et al JMP(00) [weak convergence]; Albeverio & Gottschalk CMP(01)mp/05, mp/05 [with indefinite metric]; Stetsko a0912 [in spaces with minimal length]; Sassoli de Bianchi CEJP(12)-a1010 [time delay, introduction]; > s.a. Lennard-Jones Potential [scattering length].

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