Reconstruction of the potential of the Sturm–Liouville operator from a finite set of eigenvalues and normalizing constantsстатья
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Дата последнего поиска статьи во внешних источниках: 2 ноября 2016 г.
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 {λk}∞1 and the normalizing numbers {αk}∞1 of the operator LD with the Dirichlet conditions. For an arbitrary real-valued potential q lying in the Sobolev space W^θ_2[0,π],θ>−1, we construct a function qN providing a 2N-approximation to the potential on the basis of the finite spectral data set {λ_k}N1∪{α_k}N1. The main result is that, for arbitrary τ in the interval −1 ≤ τ < θ, the estimate ∥q−q_N∥τ⩽CN^{τ−θ} is true, where ∥⋅∥τ is the norm on the Sobolev space W^τ_2. The constant C depends solely on ∥q∥θ.