Geometric aspects of Sturm-Liouville problems. III. Level surfaces of the \(n\)th eigenvalue (Q2381610)

Consider spectral problems associated with a regular Sturm-Liouville equation \[ -(py')' + qy = \lambda \omega y\text{ on }(a, b), \tag{SLE} \] where \(- \infty \leq a < b \leq \infty\), \(p, \omega > 0\) a. e. on \((a, b)\), \(1/p, q, \omega\) are Lebesgue integrable real valued functions on \((a, b)\). Assume \(\lambda_n\) to be real-valued on the space of all self-adjoint BC's. The purpose of this article is to determine the level hypersurfaces of each \(\lambda_n\). Let \(\varphi_{11} (\cdot, \lambda), \varphi_{12} (\cdot, \lambda)\) be the solutions of the SLE determined by the initial conditions: \(\varphi_{11} (a, \lambda) = 1, p \varphi_{11}' (a, \lambda) = 0; \varphi_{12} (a, \lambda) = 0, p \varphi_{12}' (a, \lambda) = 1\). Then the matrix \(\Phi (t, \lambda)\) with the first row being \(\varphi_{11} (t, \lambda), \varphi_{12} (t, \lambda)\) and the second row being \(p \varphi_{11}' (t, \lambda), p \varphi_{12}'(t, \lambda)\) is called the transfer matrix of the SLE. For \(\lambda = \kappa\), the constant matrix \(C\) is defined to be \(\Phi (b, \kappa)\). The authors describe the hypersurfaces associated with \(\lambda = \kappa\) by the entries of \(C\). Also the shapes of these surfaces are determined.