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should be chosen such that the resulting background scattering phase or cross-section is slowly depending on the scattering energy in the neighbourhood of the resonances (this is the so-called flat continuum hypothesis). If one succeeds in translating the flat continuum hypothesis in a mathematical
222:
are not defined within the
Feshbach–Fano method. This is its major power as well as its major weakness. On the one hand, this makes the method very general and, on the other hand, it introduces some arbitrariness which is difficult to control. Some authors define first the P space as an
450:
266:. It is often supposed that the solution of this problem is trivial or at least fulfilling some standard hypotheses which allow to skip its full resolution. Second by solving the resonant scattering problem corresponding to the effective complex (energy dependent) Hamiltonian
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space as an approximation to the resonance. This step relies always on some physical intuition which is not easy to quantify. In practice
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445:{\displaystyle H_{\mathrm {eff} }(E)=QHQ+\lim _{\varepsilon \to 0}QHP{1 \over E+i\varepsilon -PHP}PHQ=QHQ+\Delta (E)-i\Gamma (E)/2,\,}
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whose dimension is equal to the number of interacting resonances and depends parametrically on the scattering energy
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are interpreted as the projectors on the background and the resonant subspaces respectively.
262:) in two steps: First by solving the scattering problem ruled by the background Hamiltonian
878:{\displaystyle T_{\mathrm {total} }=T_{\mathrm {background} }+T_{\mathrm {resonances} }.\,}
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In general, the partitioning formalism is based on the definition of two complementary
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689:{\displaystyle z_{\mathrm {res} }=E_{\mathrm {res} }-i\Gamma _{\mathrm {res} }\,}
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460:
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146:, separates (partitions) the resonant and the background components of the
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project are sets of states obeying the continuum and the bound state
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to the background scattering but most authors define first the
158:. This approach allows us to define rigorously the concept of
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form, it is possible to generate a set of equations defining
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profile in the corresponding cross section. Both resulting
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matrix corresponding to the full scattering problem :
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are obtained by solving the so-called implicit equation
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The aim of the
Feshbach–Fano method is to solve the
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150:and therefore of the associated quantities like
255:governing a scattering process (defined by the
734:is close to the real axis it gives rise to a
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50:. Unsourced material may be challenged and
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114:Learn how and when to remove this message
900:Resonances in scattering from potentials
749:have to be added in order to obtain the
525:{\displaystyle \Gamma _{\mathrm {res} }}
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48:adding citations to reliable sources
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490:{\displaystyle E_{\mathrm {res} }}
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895:Resonance (particle physics)
63:"Feshbach–Fano partitioning"
248:on a less arbitrary basis.
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191:The subspaces onto which
594:{\displaystyle \det=0\,}
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162:in quantum mechanics.
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201:boundary conditions
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925:Scattering theory
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459:. The resonance
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152:cross sections
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104:December 2009
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65: –
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59:Find sources:
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29:This article
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736:Breit–Wigner
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42:Please help
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257:Hamiltonian
156:phase shift
461:parameters
176:such that
167:projectors
74:newspapers
669:Γ
662:−
576:−
506:Γ
419:Γ
413:−
401:Δ
362:−
359:ε
327:→
324:ε
160:resonance
31:does not
919:Category
889:See also
747:matrices
144:Ugo Fano
88:scholar
52:removed
37:sources
134:, the
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738:or a
95:JSTOR
81:books
740:Fano
604:for
497:and
244:and
218:and
207:and
195:and
187:= 1.
172:and
142:and
67:news
35:any
33:cite
543:det
320:lim
264:PHP
235:or
154:or
126:In
46:by
921::
183:+
872:.
866:s
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824:d
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797:b
792:T
788:=
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744:T
719:s
716:e
713:r
708:z
680:s
677:e
674:r
665:i
656:s
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650:r
645:E
641:=
635:s
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629:r
624:z
606:z
588:0
585:=
582:]
579:z
573:)
570:z
567:(
561:f
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555:e
550:H
546:[
517:s
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511:r
482:s
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471:E
457:E
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436:2
432:/
428:)
425:E
422:(
416:i
410:)
407:E
404:(
398:+
395:Q
392:H
389:Q
386:=
383:Q
380:H
377:P
371:P
368:H
365:P
356:i
353:+
350:E
346:1
341:P
338:H
335:Q
330:0
316:+
313:Q
310:H
307:Q
304:=
301:)
298:E
295:(
289:f
286:f
283:e
278:H
260:H
246:Q
242:P
237:Q
233:P
229:Q
220:Q
216:P
209:Q
205:P
197:Q
193:P
185:Q
181:P
174:Q
170:P
117:)
111:(
106:)
102:(
92:·
85:·
78:·
71:·
54:.
40:.
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