The band spectrum of the periodic Airy–Schrödinger operator on the real line

Abstract

We introduce the periodic Airy-Schrödinger operator and we study its band spectrum. This is an example of an explicitly solvable model with a periodic potential which is not differentiable at its minima and maxima. We define a semiclassical regime in which the results are stated for a fixed value of the semiclassical parameter and are thus estimates instead of asymptotic results. We prove that there exists a sequence of explicit constants, which are zeroes of classical functions, giving upper bounds of the semiclassical parameter for which the spectral bands are in the semiclassical regime. We completely determine the behaviour of the edges of the first spectral band with respect to the semiclassical parameter. Then, we investigate the spectral bands and gaps situated in the range of the potential. We prove precise estimates on the widths of these spectral bands and these spectral gaps and we determine an upper bound on the integrated spectral density in this range. Finally, in the semiclassical regime, we get estimates of the edges of every spectral bands and thus of the widths of every spectral bands and spectral gaps.1991 Mathematics Subject Classification. 34C25, 34E05, 34L40, 81Q80.

Integrated density of states: From the finite range to the periodic Airy–Schrödinger operator

“…Hence all the eigenvalues of H 2N +1 are in the band spectrum of H. In particular there is no eigenvalue of H 2N +1 in the interior of the spectral gaps of H. Recall also that from [2,Theorem 2.4], all the spectral bands E i min , E i max in (10) are included in the interval [−V 0 , 0]. These results remain true for an even number of wells as explained in Appendix C.…”

“…In [2], we obtained the band spectrum of the Airy-Schrödinger operator −h 2 d 2 dx 2 + V where V is periodic of period 2, even, continuous, piecewise affine with maximum value 0 and minimal value some reference potential −V 0 with V 0 ∈ R * + . However, from our analysis, the question of what could be, physically, the contribution to the behavior of the scattered solution for each value of an energy in a band is not obtained by the classical method of bands analysis.…”

“…whereṼ is the 2L 0 -periodic function equal to V on [−L 0 , L 0 ]. The description of the spectrum of H is already given in [2] but for reader's convenience, let us recall it here.…”

“…As the results of [2] for the periodic Airy-Schrödinger operator, Theorem 1 is not stated in the semiclassical limit, but for the semiclassical parameter c larger than an explicit constant. More precisely, one can have access to p bands in a very precise way when c ≥ c p .…”

“…In the upper bound of (101), the dominant term is, for every j ≥ 1, 3 . Note that the notations in [2] and in the present paper correspond through c = h − 2 3 . The constant K 2j is defined in the proof of [2, Proposition 6.1], but it can be much improved for our purpose.…”