Summary
The spectral decomposition of certain unbounded operators can, in some cases, be simplified by a method called «generalized spectral decomposition». The generalized spectrum is not necessarily equal to the spectrum, but several useful containment theorems are proved. This technique is applied to the spectral decomposition of the reduced transport operator in one space dimension. In particular, under the assumption that the scattering kernel is degenerate and each element square integrable, it is shown that the generalized spectral decomposition is identical with the ordinary spectral decomposition.
Riassunto
In certi casi si può semplificare la scomposizione spettrale di certi operatori illimitati con un metodo chiamato «decomposizione spettrale generalizzata». Non necessariamente lo spettro generalizzato risulta uguale allo spettro, ma si dimostrano vari utili teoremi di appartenenza. Si applica questa tecnica alla decomposizione spettrale dell’operatore di trasporto ridotto in una dimensione spaziale. In particolare, a condizione che il nocciolo di scattering sia degenere e che ogni elemento sia quadrato integrabile, si fa vedere che la decomposizione spettrale generalizzata è identica alla decomposizione spettrale ordinaria.
Резюме
В некоторых случаях спектральное разложение определенных неограниченных операторов может быть упрощено с помощью метода, называемого «обобщенным спектральным разложением». Обобщенный спектр не обязательно равен рассматриваемому спектру. Доказываются некоторые полезные теоремы. Эта техника применяется к спектральному разложению приведенного транспортного оператора в одномерном пространстве. В частности, предполагая, что ядро рассеяния является вырожденным и каждый элемент квадратично интегрируемым, показывается, что обобщенное спектральное разложение идентично обычному спектральному разложению.
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The elaborate definition of ϕ n thus accomplishes the same effect as did eq. (36), which yielded a simplified operator (eq. (38)) because of a vanishing integral (eq. (37)).
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Baird, L.C., Zweifel, P.F. Generalized spectral decomposition, with application to the reduced transport operator. Nuovo Cim B 23, 402–416 (1974). https://doi.org/10.1007/BF02723647
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DOI: https://doi.org/10.1007/BF02723647