Reconstruction of the potential of the Sturm–Liouville operator from a finite set of eigenvalues and normalizing constants
- Авторлар: Savchuk A.1
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Мекемелер:
- Lomonosov Moscow State University
- Шығарылым: Том 99, № 5-6 (2016)
- Беттер: 715-728
- Бөлім: Article
- URL: https://journals.rcsi.science/0001-4346/article/view/149403
- DOI: https://doi.org/10.1134/S0001434616050102
- ID: 149403
Дәйексөз келтіру
Аннотация
It is well known that the potential q of the Sturm–Liouville operator Ly = −yʺ + q(x)y on the finite interval [0, π] can be uniquely reconstructed from the spectrum \(\left\{ {{\lambda _k}} \right\}_1^\infty \) and the normalizing numbers \(\left\{ {{\alpha _k}} \right\}_1^\infty \) of the operator LD with the Dirichlet conditions. For an arbitrary real-valued potential q lying in the Sobolev space \(W_2^\theta \left[ {0,\pi } \right],\theta > - 1\), we construct a function qN providing a 2N-approximation to the potential on the basis of the finite spectral data set \(\left\{ {{\lambda _k}} \right\}_1^N \cup \left\{ {{\alpha _k}} \right\}_1^N\). The main result is that, for arbitrary τ in the interval −1 ≤ τ < θ, the estimate \({\left\| {q - \left. {{q_N}} \right\|} \right._\tau } \leqslant C{N^{\tau - \theta }}\) is true, where \({\left\| {\left. \cdot \right\|} \right._\tau }\) is the norm on the Sobolev space \(W_2^\tau \). The constant C depends solely on \({\left\| {\left. q \right\|} \right._\theta }\).
Негізгі сөздер
Авторлар туралы
A. Savchuk
Lomonosov Moscow State University
Хат алмасуға жауапты Автор.
Email: artem_savchuk@mail.ru
Ресей, Moscow