The Morse and Maslov indices for multidimensional Schrödinger operators with matrix-valued potentials (Q2821685)

The authors consider the spectral problem NEWLINE\[NEWLINE L_{G}u=-\Delta u +V(x)u=\lambda u \;(x\in \Omega\subset {\mathbb R}^{n}),\;Tr(u)=\Bigl(u|_{\partial \Omega},\frac{\partial u}{\partial n}|_{\partial \Omega}\Bigr)\in G. \eqno{(1)} NEWLINE\]NEWLINE where \(n\) is the interior unit normal to \(\partial \Omega\), \(G\) is a closed subspace of the space \(W_{2}^{1/2}(\partial \Omega)\times W_{2}^{-1/2}(\partial \Omega)\), and \(\Omega\) is a bounded star-shaped domain with sufficiently smooth boundary. The function \(u\) is a vector-function, respectively \(V\) is a continuous matrix-function. The subspace \(G\) is chosen so that the corresponding problem (1) is selfadjoint. Given \(t\in (0,1)\), define the domains \(\Omega_{t}=\{\tau y:\;y\in \partial \Omega,\;\tau\in [0,t)\}\) and the rescaling problem \(L_{t,G}u=-\Delta u +t^{2}V(tx)u=t^{2}\lambda u\), \(Tr_{t}(u)=\Bigl(u|_{\partial \Omega},t^{-1}\frac{\partial u}{\partial n}|_{\partial \Omega}\Bigr)\in G\). Let \(K_{t}\) be the family of weak solutions (from the space \(W_{2}^{1}(\Omega)\)) to the equation \(L_{t,G}u=0\) (\(x\in \Omega\)). It is possible to define the Maslov index of the family \(\gamma(t)=Tr_{t}K_{t}\) (these subspaces form a smooth path) with respect to \(G\). The main results are connections between the Morse index of \(L_{G}\) (the number of negative eigenvalues of \(L_{G}\)), the Maslov index, the number of negative eigenvalues of the matrix \(V(0)\) (\(0\in \Omega\)), and the number of negative eigenvalues of some bilinear form.