On the riesz basisness of the root functions of the nonself-adjoint sturm-liouville operator

On the nonself-adjoint ordinary differential operators with periodic boundary conditions

“…Let P and A be the operators generated in L 2 [0, 1] by the periodic (1) y (k) (1) = y (k) (0), k = 0, 1, 2, . .…”

On the nonself-adjoint differential operators with the quasiperiodic boundary conditions

“…In the same way we do the second iteration of (8). Namely, first in (26) we replace ( 2 ( the validity of this replacement can be obtained from (23), replacing e (2kπi+it)x in the lefthand side of (23) by e (2πi(k−k 1 )+it)x ). Then isolate the terms containing the multiplicand Ψ k,t (x), e (2kπi+it)x (i.e., case…”

“…The case n = 2, that is, the case of the SturmLiouville operator is investigated in [2,8,9]. In the classical investigations ( see [1,4,5]) in order to obtain the asymptotic formulas of order O(1/ρ m+1 ) for the solutions y 1 (x, ρ), .…”

“…Here and in forthcoming relations by N we denote a large integer, that is, N 1, and by c m , for m = 1, 2, ..., the positive, independent on N constant, whose exact value is inessential. Note that (2kπi + it) n for k ∈ Z is the eigenvalue of the operator L t (0) corresponding to the eigenfunction e (2kπi+it)x , where L t (0) is the operator generated by the expression y (n) and the boundary conditions (2). The formula (3) shows that the eigenvalue λ k (t) of the operator L t (p) is close to the eigenvalue (2kπi + it) n of the operator L t (0) and far from the other eigenvalues (2mπi + it)…”

Convergence of spectral decompositions of Hill operators with trigonometric polynomial potentials

“…In this case properly chosen two-dimensional block-decompositions do converge as it has been shown by Shkalikov [21][22][23] (even in a more general context of ordinary differential operators of higher order). For certain classes of potentials, there have been given sufficient and necessary conditions on whether blocks could be split into (one-dimensional) eigenfunction decompositions [2,13,14,26]. Maybe, in 2006 Makin [12] and the authors [3,Theorem 71] gave first examples of such potentials that SEAF for periodic or antiperiodic boundary conditions is NOT a basis in L 2 ([0, π]) even though all but finitely many eigenvalues are simple.…”