Identification of the absent spectral gaps in a class of generalized Kronig-Penney Hamiltonians (Q2472517)

The author considers a one-dimensional Schrödinger operator with periodic point interaction given as follows. Let \(\kappa \in(0,2\pi)\), \(\Gamma _1=2\pi \mathbb Z\), \(\Gamma _2=\{\kappa \}+2\pi \mathbb Z\), \(\Gamma =\Gamma_1 \cup \Gamma _2\), \(\theta _1,\theta _2\in [-\pi/2,\pi/2)\), and \(A_1,A_2\in \text{SO}(2)\setminus \{\pm I\}\). Then the operator \(H\) in \(L^2(\mathbb R)\) is defined by \[ (Hy)(x)=\frac {d^2}{dx^2}y(x), \;x\in \mathbb R\setminus \Gamma , \] where \[ \text{Dom\,}H =\left\{y\in H^2(\mathbb R\setminus \Gamma)\;\Bigg|\;\binom{y(x+0)}{y'(x+0)}=e^{i\theta _j}A_j \binom{y(x-0)}{y'(x-0)},\;x\in \Gamma _j,\;j=1,2\right\}. \] The operator \(H\) is unitarily equivalent to an infinite direct product of translations \(H_\mu\) of the the Schrödinger operator on the Hilbert space \[ \mathcal H_\mu=\{u\in L^2_{\text{loc}}(\mathbb R)\mid u(x+2\pi)= e^{i\mu}u(x)\;\text{a.e.}\;x\in \mathbb R\} \] with the Lebesgue integral over \((0,2\pi)\) as inner product. It is shown that \[ \sigma (H)\bigcup_{\mu\in[\theta _1+\theta _2,\theta _1+\theta _2+\pi]} \sigma (H_\mu)=\bigcup_{j=1}^\infty \bigcup_{\mu\in[\theta _1+\theta _2,\theta _1+\theta _2+\pi]} \{\lambda _j(\mu)\}, \] where \(\lambda _j(\mu)\) is the \(j\)-th eigenvalue of \(H_\mu\), counted with multiplicity. Let \(\alpha _j\in(-\pi,0)\cup(0,\pi)\) be the rotation angle in \(A_j\in \text{SO}(2)\), \(j=1,2\). The existence and location of gaps in the spectrum of \(H\) is investigated. The results depend on the values of \(\alpha _j\). For example, if \(\alpha _1\neq\pm\alpha _2\), then all spectral gaps are present.