On the Neumann problem for the Sturm-Liouville equation with Cantor-type self-similar weight

On Spectral Asymptotics of the Neumann Problem for the Sturm–Liouville Equation with Self-Similar Weight of Generalized Cantor Type

“…In [2], the conjecture was confirmed for "even" ladders (see Definition 3 below). For such ladders, the following theorem was proved.…”

“…If we set γ (1) = max k 2 |a k+1 −b k | , then the intersections become close to the edges of I k , and the function remains undefined in the middle of intermediate segments. 2 We define it by zero on all remaining intervals. The signs of nonzero parameters c k are defined so that on each intermediate segment, the function z has a change of sign.…”

“…At the step j, the function z is composed of eigenfunctions of boundary-value problems on subsections I k with certain boundary conditions of the form so that the eigenvalues, which correspond to the functions on segments I k , remain equal, z remains smooth on the intervals, where the smoothness has already been achieved, and β (k) j and α (k+1) j decrease at the ends of 1 Henceforce, "smoothly" means C 1 -smoothly. 2 The distance to the intersection point near I k is equal to |I k | · (γ (1) ) −1 , which is less than half the length of any intermediate segment by the choice of γ (1) . 817 the remaining segments.…”

“…Note that the constructed function z has exactly m(n + 1) − 1 roots and hence it is an eigenfunction corresponding to the eigenvalue μ (2) m(n+1)−1 of the boundary-value problem…”

“…n } ∞ n=0 , and {μ (2) n } ∞ n=0 be sequences of eigenvalues of Eq. (1) numbered in ascending order and corresponding to the boundary-value problems λ n : y (0) = y (1) = 0,…”