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.

@ __References__: Mosini SHPMP(00)
[history]; Bohm & Sato PRD(05)
[general theory, properties]; de la Madrid AIP(07)qp/06 [rigged-Hilbert-space description]; Rosas-Ortiz et al AIP(08)-a0902 [primer];
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 resonances 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 8
jan 2018