Spectral Theory of Differential Operators with Complex Coefficients

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The success of Weyls theory as well as problems from physics soon led to attempts to extend this theory into several directions. These were higher order symmetric operators, spectral operators and Sturm-Liouville operators with complex coefficients. The key problem with spectral differential operators is to show that a given operator is spectral. Thus the theory hardly developed beyond constant coefficient operators. Huige [7], extending earlier work of Schwartz, used the Fourier transform to prove the spectrality of constant coefficient differential operators on the half line and he extended this by allowing perturbations with rapidly decaying coefficients. The Sturm-Liouville operators with complex coefficients studied by the Russian school, Neumark et al [8] and Sims [9] and a few others were just of this type. However these operators were mostly analyzed in respect to an eigenfunction expansion by using Fourier analysis as well complex contour integrals.The difficulties stem mostly from the higher order singularities with its nilpotent summands and the spectral singularities in the essential spectrum. Of these nothing is known. The role of the m-function is likewise not clear, in particular the behavior of m near the singularities. Compared to a wealth of examples of Sturm-Liouville operators, there are no (!) examples known, which exhibit this strange behaviour. Thus this direction of study has been dormant ever since. Ultimately it is the absence of the spectral theorem, which makes the non self-adjoint operator theory so much more complicated. However, if the form of the eigenfunctions is approximately known, more can be said. Behncke extended the theory of asymptotic integration, introduced by Levinson and made it applicable to spectral problems [1]. The m-matrix as well as asymptotic integration is thus ideal tools to analyze the spectral theory of symmetric differential operators. In altogether 8 papers Hinton and Behncke have studied many aspects of the spectral theory of higher order differential operators. Our main result states that Hamiltonian systems with coefficients, which are not too oscillatory have only absolutely continuous essential spectrum with eigenvalues accumulating at the double roots of the characteristic polynomial at most [4]. For symmetric Hamiltonian systems the essential spectrum will in general be a sequence of intervals for which the boundary is determined by the discriminant of the characteristic polynomial. For general C-symmetric Hamiltonians the spectrum will be a rather general algebraic curve.