By Vladimir A. Marchenko

The spectral concept of Sturm-Liouville operators is a classical area of study, comprising a wide selection of difficulties. along with the fundamental effects at the constitution of the spectrum and the eigenfunction growth of standard and singular Sturm-Liouville difficulties, it's during this area that one-dimensional quantum scattering thought, inverse spectral difficulties, and the fabulous connections of the speculation with nonlinear evolution equations first develop into similar. the most objective of this publication is to teach what should be accomplished by using transformation operators in spectral conception in addition to of their functions. the most tools and ends up in this zone (many of that are credited to the writer) are for the 1st time tested from a unified perspective. The direct and inverse difficulties of spectral research and the inverse scattering challenge are solved with assistance from the transformation operators in either self-adjoint and nonself-adjoint situations. The asymptotic formulae for spectral capabilities, hint formulae, and the precise relation (in either instructions) among the smoothness of capability and the asymptotics of eigenvalues (or the lengths of gaps within the spectrum) are got. additionally, the purposes of transformation operators and their generalizations to soliton idea (i.e., fixing nonlinear equations of Korteweg-de Vries style) are thought of. the recent bankruptcy five is dedicated to the soundness of the inverse challenge suggestions. The estimation of the accuracy with which the possibility of the Sturm-Liouville operator might be restored from the scattering info or the spectral functionality, in the event that they are just identified on a finite period of a spectral parameter (i.e., on a finite period of energy), is acquired.

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K0 (AT ∗ M,π ) следующей C ∗ -алгебры (ср. ]) AT ∗ M,π = (u, v) u ∈ C0 (T (M \U )) , v ∈ C0 T ∗ Y , Endπ! 1. ], так как при Y = ∅ соответствующая K-группа есть Kc0 (T ∗ M )) и определена точная последовательность Kc1 (T ∗ (M \U )) → Kc0 (T ∗ Y ) → K0 (AT ∗ M,π ) → Kc0 (T ∗ (M \U )) → Kc1 (T ∗ Y ) . (9) — последовательность пары, отвечающая идеалу {(0, v) |v ∈ C0 (T Y, Endπ!

1 A−1 y = − (E + N1 )−1 P −1 (y ′ (x) + ay(x)), δ (4) 1 A(0, t)(E + N1 )−1 P −1 (y ′ (t) + ay(t))dt, y(0) = (5) 0 где P f (x) = (α1 −α2 )f (x)+(α3 −α4 )f (1−x), N1 – интегральный оператор, a – комплексная константа. ) к виду: γ1 y(0) + γ2 y(1) = (y, ϕ), (y, ϕ) – скалярное произведение, ϕ – непрерывная функция, N – такой оператор, что E + N = (E + N1 )−1 , E – единичный оператор. Построение резольвенты R0,λ сводится к исследованию краевой задачи для дифференциальной системы первого порядка в пространстве вектор-функций размерности 2.

20) 0 0 . 0 −i Тогда для регулярной струны с условием в правом конце вида f1 (l + 0) = 0 получим Положим B1 = dom(LS ) = dom(LB1 ) = {f ∈ dom(L∗ ) : Γ1 f = B1 Γ0 f }. (21) 28 Section 1. Spectral Problems Очевидно, что вектор–функция f (x, λ) является решением уравнения L∗0 y − λy = 0 в том и только том случае, когда она имеет вид c1 ψ(x, λ2 ) + c2 ϕ(x, λ2 ) i (c ψ ′ (x, λ2 ) + c2 ϕ′m (x, λ2 )) λ 1 m f (x, λ) = (22) . Пусть функция Вейля, соответствующая граничной тройке Π1 имеет вид M11 (λ) M12 (λ) M (λ) = .

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