Band functions in the presence of magnetic steps (Q2788508)

The authors study two-dimensional Schrödinger operators with piecewise constant magnetic fields NEWLINE\[NEWLINEH=-\frac{\partial^2}{\partial x^2}+\left(-i\frac{\partial}{\partial y}-b(x)\right)^2NEWLINE\]NEWLINE in \(L^2({\mathbb R}^2)\) with \(b(x)=b_1 x \chi_{{\mathbb R}_-}(x)+b_2 x \chi_{{\mathbb R}_+}(x)\), where and \(\chi_X (x)\) is the characteristic function on \(X \subset {\mathbb R}\). The operator \(H\) has translation invariance in the \(y\)-direction and thus we have the direct integral decomposition associated with the Fourier transform \({\mathcal F}_y\) with respect to \(y\) NEWLINE\[NEWLINE{\mathcal F}_yH{\mathcal F}_y^{-1}=\int\limits_{k\in{\mathbb R}}^\otimes h(k)dk.NEWLINE\]NEWLINE The \textit{fiber} operators \(h(k)\) act on \(L^2({\mathbb R})\). Denote by \(\lambda_n(k)\) the \(n\)th Rayleigh quotient of \(h(k)\) (\(n\)th \textit{band} function).NEWLINENEWLINEThe authors consider two cases unstudied in the previous publications. {\parindent=6mm \begin{itemize} \item[1)] Let \((b_1, b_2) = (0, 1)\). It is proved that if \(k < 0\), then the fiber operator \(h(k)\) has only a purely absolutely continuous spectrum \([k^2,\infty)\). Consequently, the spectrum of \(H\) is \([0,\infty)\). If \(k > 0\), the essential spectrum of \(h(k)\) is \([k^2,\infty)\). The band functions \(\lambda_n\) are non-decreasing functions with \(E_{n-1} < \lambda_n(k) < E_n\) and \(\lim\limits_{k\to\infty} \lambda_n(k) = E_n\), where \(E_0 = 0\) and for \(n \geq 1, E_n = 2n - 1\). \item [2)] Let \(-1<b_1<0, b_2=1\). It is proved that for \(k \in {\mathbb R}\) the fiber operator \(h(k)\) has only a purely discrete spectrum \(\lambda_n(k)\) with simple eigenvalues and \(\lim\limits_{k\to-\infty} \lambda_n(k) = +\infty\). The set of limit points of the band functions \(\lambda_n(k)\) and asymptotics expansion of the band functions at infinity are obtained.NEWLINENEWLINE\end{itemize}}