Для цитирования:
Kuznetsova M. A. On recovering non-local perturbation of non-self-adjoint Sturm – Liouville operator [Кузнецова М. А. Восстановление нелокального возмущения несамосопряженного оператора Штурма – Лиувилля] // Известия Саратовского университета. Новая серия. Серия: Математика. Механика. Информатика. 2024. Т. 24, вып. 4. С. 488-497. DOI: 10.18500/1816-9791-2024-24-4-488-497, EDN: GRSGAI
On recovering non-local perturbation of non-self-adjoint Sturm – Liouville operator
[Восстановление нелокального возмущения несамосопряженного оператора Штурма – Лиувилля]
В последнее время возник значительный интерес к обратным спектральным задачам для нелокальных операторов, возникающих во многих приложениях. В настоящей работе рассматривается оператор с замороженным аргументом $ly = -y''(x) + p(x)y(x) + q(x)y(a)$, который является нелокальным возмущением несамосопряженного оператора Штурма – Лиувилля. Исследуется обратная задача восстановления потенциала $q \in L_2(0, \pi)$ по спектру при известном коэффициенте $p\in L_2(0, \pi)$. В то время как предыдущие работы были сосредоточены только на случае $p=0$, здесь исследуется более сложный несамосопряженный случай, требующий учета кратностей собственных значений. Мы развиваем подход, основанный на связи между характеристической функцией и коэффициентами $\{ \xi_n\}_{n \ge 1}$ потенциала $q$ по некоторому базису. Получены необходимые и достаточные условия для спектра, которые являются асимптотическими формулами особого вида. Из них следует, что часть спектра не зависит от $q$, т. е. является неинформативной. Для однозначной разрешимости обратной задачи кроме спектра необходимо задать часть коэффициентов $\xi_n$, которые являются минимальными дополнительными данными. Для обратной задачи по спектру и дополнительным данным получены теорема единственности и алгоритм.
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