[For the entire collection see Zbl 0665.00006.]
Real valued solutions \(\psi\) of Schrödinger equations \((-\Delta+V- E)\psi=0\) in a domain \(\Omega\subset\mathbb{R}^ n\), \(n\geq2\), are considered. Nodal properties of such solutions are studied under suitable assumptions on \(V\) and \(E\). Some theorems are proved, concerning the nodal set of \(\psi\) and the asymptotic behaviour of nodal lines.
Theorem 1 investigates various asymptotic properties of \(u=\psi/v\) (\(v\) is the solution of a radial comparison problem). Theorem 3 gives, under some assumptions, the asymptotic behaviour of the nodal lines.
Theorem 6: Let \(\psi\) satisfy: \(\Omega\subset\mathbb{R}^ n\) (\(n\geq2\)) a domain with \(x_0 \in \Omega\), \(V \in C^\infty (\Omega)\) and \(\psi\not\equiv0\), \(\psi \in C^\infty (\Omega)\) in \(\Omega\) and \(\psi(x_0)=0\), \(x_0=\mathcal O\), \(\displaystyle\lim_{r\to0-} r^{-M} \psi(ry) = Y_ M (y)\) \(\forall y\in S^{n-1}\), where \(Y_ M\) is a surface harmonic (\(M\geq1\)), \(y = x/r \in S^{n-1}\) (\(S^{n-1}\) the unit sphere in \(R^ n\)). Let \(D_{\varepsilon_0}\) denote a nodal domain of \(\psi\) with \(\mathcal O \in D_{\varepsilon_ 0}\), \(S(r) = \{y\in S^{n-1}\;| \;r_ y\in D_{\varepsilon_0}\}\), and denote \(\psi_0 = \left(\int_{S(r)} \psi^2 d\sigma\right)^{\frac{1}{2}}\) then \(\psi_0 r^{-M}\) and \(| S(r)|\) have for \(r\to0\) nonzero finite limits.