Absence of positive eigenvalues for a class of subelliptic operators (Q1914725)

Consider the degenerate elliptic operator \({\mathcal L}= \Delta_z+ |z|^{2\alpha} \Delta_t\) in \(\mathbb{R}^{n+ m}\), where \(\alpha> 0\), \(z\in \mathbb{R}^n\) and \(t\in \mathbb{R}^m\). We show that, under suitable conditions on the potential \(V= V(z, t)\), the operator \(- {\mathcal L}+ V\) has no positive eigenvalue with an \(L^2\)-eigenfunction in any unbounded domain which contains \(\{(z, t)\in \mathbb{R}^{n+ m}: |z|+ |t|\geq R\}\) for some \(R> 0\).