Resonances |

**In General** > s.a. solar
planets; chaos; oscillator.

* __In particle physics__:
A resonance is an unstable particle whose existence is inferred from a peak
in the invariant mass distribution of other sets of particles into which
it decays; The peak width Γ and lifetime *τ* are related
by Γ = \(\hbar\)/*τ*; > s.a. quantum mechanics.

* __Relativistic, theory__:
A pole of the S-matrix at a complex value *s*_{R} of the energy squared *s*.

* __Descriptions__: One can use
Gamow vectors, the S-matrix, or the Green function; They are usually associated
with time asymmetry.

@ __Mechanical and general resonances__: Rosas-Ortiz et al AIP(08)-a0902 [primer];
Bleck-Neuhaus a1811 [mechanical, history].

@ __In particle physics__: Mosini SHPMP(00) [history];
Bohm & Sato PRD(05) [general theory, properties];
de la Madrid AIP(07)qp/06 [rigged-Hilbert-space description];
Hatano et al Pra(09)-a0904-proc [probabilistic interpretation];
Gluskin et al a1003 [graphical convolution approach];
Bohm RPMP(11),
Gadella & Kielanowski RPMP(11) [formalism].

@ __Decay__: Rotter a0710 [Feshbach-projector description];
de la Madrid NPA(15)-a1505 [decay widths, constants and branching fractions];
Baungärtel a1706 [mathematical];
Wyrzykowski a1801 [transition to non-exponential decay].

**Gamow States / Vectors** > s.a. quantum state evolution
[decay of unstable states]; types of quantum states [unstable].

* __Idea__: Generalized eigenvectors
of a quantum Hamiltonian with complex eigenvalues that describe exponentially decaying
(or growing) states and can be used to model irreversibility in quantum theory.

@ __General references__: Bohm et al AJP(89)dec;
Bollini et al PLB(96);
Gaioli et al IJTP(99) [and time asymmetry];
de la Madrid & Gadella AJP(02)jun-qp [intro];
Castagnino et al JPA(01)qp/02,
PLA(01)qp/02;
Civitarese & Gadella PRP(04);
Kaldass ht/05-conf;
de la Madrid JMP(12)-a1210 [rigged-Hilbert-space approach];
> s.a. Friedrichs Model.

@ __Relativistic__: Antoniou et al JMP(98);
Kielanowski IJTP(03).

@ __Special systems__: Antoniou et al JMP(98) [degenerate scattering resonances],
JPA(03) [models],
IJTP(03) [exactly solvable].

**Special Types and Related Topics**

* __Parametric resonance__: A resonance
that arises when the parameters on which an oscillating system depends are varied
periodically, and the driving frequency goes through special values;
__Example__: An LC circuit with characteristic frequency *ω*
= (*LC*)^{−1/2} in which the capacitance
*C* is varied periodically.

* __Stochastic resonance__:
The amplification of a periodic signal applied to a non-linear system
obtained by adding noise.

* __Feshbach resonance__: A scattering resonance
that occurs when the energy of an unbound state of a two-body system matches the energy of
an excited state of the compound system; Recognized long ago as an important feature in
nuclear, atomic, and molecular scattering, and in photoionization and photodissociation,
they have assumed new importance in ultracold atomic systems.

@ __Parametric__: Weigert JPA(02)qp/01 [quantum];
Berges & Serreau PRL(03)hp/02 [in quantum field theory];
Leroy et al EJP(06) [Hamiltonian approach].

@ __Parametric wave excitation__:
Bechhoefer & Johnson AJP(96)dec [Faraday waves].

@ __Stochastic resonance__: Marchesoni Phy(09) [in mechanical system and Bose-Einstein condensates].

@ __Related topics__: Bohm & Harshman NPB(00)hp,
Bohm et al ht/01 [mass and width];
Kleefeld ht/03-conf [formulation];
Stefanov mp/04 [Complex Absorbing Potential method];
de la Madrid et al CzJP(05)qp [resonance expansions];
Chin et al RMP(10) [Feshbach resonances in ultracold gases].

**Applications** > see particle effects.

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send feedback and suggestions to bombelli at olemiss.edu – modified 21 nov 2018