Scattering and Collisions  

Scattering in General > s.a. diffraction [in field theory]; huygens; resonance.
* Important experiments: Thomson scattering; Rutherford scattering; Deep inelastic scattering, that showed the composite nature of baryons; > s.a. Preons.
* Geometric collisions: For particles of diameter d, mean speed v and number density n, the mean free path is = (d2n)–1, and the collision frequency f = v/ = d2vn; the mean free time is of course t = 1/f; > s.a. brownian motion.
* Optical theorem: The total cross section for an elastic scattering and absorption process (i.e., the potential can possibly be complex) is

tot = (4/k) Im f(k,k) ;

Often credited to N Bohr, R Peierls and G Placzek, but actually due to E Feenberg and Lord Rayleigh.
@ Optical theorem: in Schiff 68; Newton AJP(76); Bussey PLA(86) [and wave function collapse].
@ Techniques: Barlette et al AJP(01) [integral equations, partial waves].

Special Cases > s.a. Bhabha; neutron; Rutherford; Superradiant; Thomas; Thomson.
* Deep inelastic scattering: Parameters are x = fraction of nuclear momentum carried by q, Q2 = square of mom transfer between nucleon and beam particle.
@ Coulomb potential: Yafaev JPA(97) [n-dimensional, quantum]; Ahmed qp/03 [quantum]; Mineev TMP(04) [1D, self-adjoint extension].
@ Gravitational, relativistic: Barrabès & Hogan CQG(04); Barbieri & Guadagnini NPB(05)gq [massless particles off rotating bodies]; > s.a. orbits.
@ 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 theories: see atomic physics; dirac fields; graviton; photon; pilot wave quantum mechanics.

Multiple Scattering > s.a. Debye Length; light; Rayleigh.
@ General references: Huang PLA(04) [perturbative]; Ramm mp/06, mp/06, JPA(08) [off many small bodies], PLA(07), PLA(08) [waves off many particles].
@ Random scatterers: Mathur & Yeh JMP(64) [finite size, electromagnetic waves]; Kuelske mp/01, Dean et al JPA(04) [point scatterers].

In Quantum Theory
* 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.
@ General references: Tyutin pr(74)-a0801 [electron scattering by solenoid]; Mostafazadeh PRA(96) [on curved surfaces]; de la Torre AJP(97) [wave packet in central potential, distorsion]; Esposito JPA(98)ht [singular V's]; Amrein qp/01 [large-t behavior]; Pérez Prieto et al JPA(03) [Gaussian wave function and square barrier]; Roux & Yafaev JMP(03) [long-range V's]; Ignatovich qp/04 [problems?]; Cannata et al AP(07) [PT-symmetric quantum mechanics, 1D]; Hussein et al a0807 [new formulation]; > s.a. quantum systems.
@ Levinson's theorem: Wellner AJP(64); 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 qp/02; Boya & Casahorrán IJTP(07)qp/05-in [from spectral density]; Kellendonk & Richard qp/05, JPA(08)-a0712 [topological version]; Ma JPA(06) [rev]; > s.a. topology in physics.
@ Semiclassical: Ford & Wheeler AP(00); Rothstein JMP(04) [1D and 2D]; Berera NPA-ht/07 [scattering of large objects in quantum field theory and classical description].
@ In quantum field theory: Buchholz & Summers mp/05-in.

References > s.a. Perturbation Methods; potential.
@ 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 spacetime: Beig APP(88) [scalar fields].
@ 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].

Inverse Scattering
* Idea: Obtaining the scattering potential from the scattered wave.
* And solution of non-linear pde's: An approach in which the equation appears as an integrability condition for a pair of linear de's with a spectral parameter, a stationary and an evolution equation.
@ General theory: Schroer AP(03)ht/01 [uniqueness in local quantum theory].
@ For Einstein's equation: Belinsky & Zakharov JETP(78); Belinsky JETP(79); Zakharov & Shabat FAA(79); Flaschka & Newell CMP(80); > s.a. solutions of general relativity with symmetries [stationary].

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