Detailed asymptotic of eigenvalues on time scales

Abstract

Let T ¼ {a n } n < {0} be a time scale with zero Minkowski (or box) dimension, where {a n } n is a monotonically decreasing sequence converging to zero, and a 1 ¼ 1. In this paper, we find an upper bound for the eigenvalue counting function of the linear problem 2 u DD ¼ lu s , with Dirichlet boundary conditions. We obtain that the ntheigenvalue is bounded below by 4a 22 n21 . We show that the bound is optimal for the qdifference equations arising in quantum calculus.

Parameter-Dependent Dirac Systems on Time Scales

“…The proof is clear. (5) and (6) are disconjugate for each fixed ∈ ℝ which satisfies ( ( , ) − ) > 0 on к .…”

Sturm–Liouville operators on time scales

“…L = 0), the leading asymptotics will change and will no longer be captured by the above theorem. See [2], [3] for some results in this direction.…”

Boundary‐value problem for a class of second‐order parameter‐dependent dynamic equations on a time scale

“…Sturm-Liouville theory on time scales was studied first by Erbe and Hilger 2 in 1993. The most important results on the properties of eigenvalues and eigenfunctions of the classical Sturm-Liouville problem on time scales were given in various publications (see, eg, [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17] and the references therein).…”