Summary
A class of non-Hermitian eigenvalue equations, which are aimed to determine positions and widths of resonant (decaying) levels, is derived quite generally through a projection operator procedure whereby the effects of the continuum states responsible for the decay is represented in an effective manner. The boundary conditions appropriate to these eigenvalue problems are discussed and the wave function renormalization is evaluated. A connection is drawn with the effective eigenvalue problems occurring in the many-body Green’s functions theory. Features like the energy dependence of the self-energy for the single-particle Green’s function and of the screened Coulomb interaction for the electron-hole Green’s function are accordingly interpreted in terms of elimination of continuum channels.
Riassunto
É noto da tempo, soprattutto dai lavori di Feshbach sul metodo delle hamiltoniane efficaci per reazioni nucleari, che tecniche di operatori di proiezione sono particolarmente adatte a costruire hamiltoniane efficaci allo scopo di determinare i parametri rilevanti delle risonanze nella teoria della diffusione. In questo articolo si pone in maggiore risalto l’aspetto quasi-legato degli stati risonanti e si considera un problema modello dove un insieme di stati discreti é accoppiato con un insieme di stati continui, i quali sono poi eliminati dalla trattazione esplicita mediante operatori di proiezione. In questo modo la sottomatrice dell’operatore risolvente nel sottospazio degli stati discreti risulta espressa in termini di un’hamiltoniana efficace che è non hermitiana e dipendente dall’energia. Tale procedimento è poi collegato con la risoluzione della equazione di Schrödinger soggetta alle condizioni al contorno del tipo Kapur-Peierls che sono appropriate agli stati quasi stazionari. Questo collegamento fornisce, tra l’altro, una relazione tra la rinormalizzazione della funzione d’onda e la dipendenza dall’energia dell’hamiltoniana efficace. I risultati generali così ottenuti sono poi applicati alla teoria delle funzioni di Green a molti corpi. In particolare, si mostra come sia l’equazione di Dyson che determina i livelli a quasi particella sia l’equazione agli autovalori per gli eccitoni con schermaggio dielettrico possano essere interpretate nel senso delle hamiltoniane efficaci.
Резюме
В общем виде выводится класс незрмитовых уравнений для собственных значений, которые преднаэначемы для определения положений и ширин резонансных (распадающих) уровней, используя процедуру операторов проектирования. В таком подходе влияние непрерывных состояний, ответственных за распад, учитывается эффективным образом. Обсуждаются граничные условия, соответствующие этим проблемам собственных значений, и вычисляется перенормировка волновой функции. Отмечается связь с проблемами эффективных собственных значений, которые имеют место в теории многочастичных гриновских функций. Такие особенности, как энергетическая зависимость собственной энергии для одночастичной функции Грина и экранированное кулоновское взаимодействие для электрон-дырочной гриновской функции, интерпретируются соответственно на основе исключения непрерывных каналов.
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Strinati, G. On the effective non-Hermitian eigenvalue problems for resonant levels. Il Nuovo Cimento D 4, 397–410 (1984). https://doi.org/10.1007/BF02451296
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DOI: https://doi.org/10.1007/BF02451296