The \(2L_0\)-periodic Airy-Schrödinger operator on the Sobolev space \(H^2(\mathbb{R})\) is defined by \[ H=-\frac{\hbar ^2}{2m}\,\frac{\mathrm{d}^2}{\mathrm{d}z^2}+V, \] where \(V\) is a \(2L_0\)-periodic function of the form \[ \forall z\in [-L_0,L_0],\;V(z)=V_0\left( \frac{|z|}{L_0}-1 \right) \] with a positive number \(V_0\). The operator \(H\) has purely absolutely continuous spectrum consisting of spectral bands. Rescaling leads to an operator \[ \mathbf{H} = -h^2 \frac{\mathrm{d}^2}{\mathrm{d}x^2}+\mathbf{V},\quad h>0 \] with the \(2\)-periodic potential \(\mathbf{V}\) given by \(\mathbf{V}(x)=|x|-1\) on \([-1,1]\). The spectrum of \(\mathbf{H}\) is the union of spectral bands, the \(p\)-th spectral band is denoted by \([\mathbf{E}^p_{\min},\mathbf{E}^p_{\max}]\), \(p\in \mathbb{N}\). The \(p\)-th spectral gap is \([\mathbf{E}^p_{\max},\mathbf{E}^{p+1}_{\min}]\). The location of the spectral bands and spectral gaps of \(\mathbf{H}\) is investigated. The potential \(\mathbf{V}\) is not differentiable at its extrema, and therefore known results on the location of spectral bands are not applicable. A crucial role is played by the zeros of the Airy function Ai and the zeros of the solutions \(u\) and \(v\) of the Airy equation \(y''=xy\) which satisfy \(u(0)=1\), \(u'(0)=0\), \(v(0)=0\), \(v'(0)=1\). All zeros of these entire functions and their derivatives are real and non-positive.
It is shown that for \(h \geq c_0^{-\frac 32}\) there is no spectral gap of \(\mathbf{H}\) in \([-1, 0]\) and that the first spectral gap intersects \([-1, 0)\) as soon as \(0<h< c_0^{-\frac32}\) , where \(-c_0\) is the largest zero of \(v'\). An estimate of the minimum of the spectrum is \[ -1<\mathbf{E}^0_{\min}<\min\left( -\frac 12,-1+\widetilde a_1h^{\frac 23} \right), \] where \(-\widetilde a_1\) is the largest zero of the derivative of Ai. If \(h<c_0^{-\frac 32}\), then there is an explicitly given integer \(k_0\) for which an upper bound for the length of the \(p\)-th spectral bound, \(p=2,\dots,k_0\), and positive lower and upper bounds for the length of the \(p\)-th spectral gap, \(p=2,\dots,k_0-1\), are derived. All bounds only depend on \(h\) and \(p\). Further estimates for the lengths and locations of the spectral bands are also obtained. The results are too technical to be stated here.