Spectral theory for periodic Schrödinger operators with reflection symmetries
Let \(H=-\Delta+V,\) be defined on \({\mathbb R}^d\) with smooth potential \(V\), such that \(V(x)=V(x+n)\), for all \(n\in {\mathbb Z}^d\), so Dom\((H)=W^{2,2}({\mathbb R}^d) \). In addition the authors assume that \(H\) has reflection symmetries, i.e. \(V(T_jx)=V(x)\), \(j=1,\dots,d\), where \(T_j(x_1,\dots,x_{j-1},x_j,x_{j+1},\dots,x_d)=(x_1,\dots,x_{j-1},-x_j,x_{j+1 },\dots,x_d)\). For any \(q\in {\mathbb R}^d\) the space \(W_q({\mathbb R}^d)\) is the space of functions \(u\in W_{\text{loc}}^{2,2}({\mathbb R}^d)\) such that \(u(x-n)=e^{2\pi i\langle n,q\rangle }u(x), \forall n\in{\mathbb Z}^d\). The associated Floquet operators \(H^q\) is restriction of \(H\) to \(W_q\). It is shown that the associated lowest eigenvalues \(\lambda_q\) are simple if \(q=(q_1,q_2,\dots,q_d)\in [0,1/2)^d\). Moreover, if \(q_i=1/2\) for \(i\in I\), where \(I\) is a subset of \(\{1,2,\dots,d\}\) and \(q_i\in [0,1/2)\) for \(i\not\in I\), then the \(m(q)\) multiplicity of \(\lambda_q\) satisfies \(m(q)\leq 2^{| I| }\).
scientific article; zbMATH DE number 2051478