Spectral theory of higher order differential operators by examples (Q361725)

In recent years, the first author and co-authors have developed the method of asymptotic integration in order to investigate deficiency indices, essential spectra and absolutely continuous spectra of higher order symmetric differential operators on the semi-axis \((0,\infty )\). With the usual techniques, these results can be extended to such differential equations on \(\mathbb R\) and to singular differential equations on bounded intervals. The methods and proofs are, however, very technical, and hence, in this overview article the authors present the general methods without proofs, explaining the main steps of the method: transformation into Levinson form, diagonalization, dichotomy condition and \(M\)-function.NEWLINENEWLINETwo further sections provide examples with bounded and unbounded coefficients, respectively. The examples for bounded coefficients include differential operators with step potentials, bump potentials and square well potentials, which are of the form NEWLINE\[NEWLINELy=(-1)^ny^{(2n)}+a\chi_Iy NEWLINE\]NEWLINE on \(\mathbb R\) with \(I=[0,\infty )\) and \(I=[-1,1]\), respectively, the Everitt-Markett 6th order Bessel type equation on \([0,\infty )\), and operators with the Wigner--von Neumann potential and rapidly oscillating potentials.NEWLINENEWLINEFor operators with unbounded coefficients, lower order terms may have a more significant influence on the spectral quantities since, in general, they can no longer be estimated by higher order terms. A few general results, some with proofs, are presented. Again, several examples for unbounded coefficients are discussed, showing for example that all deficiency indices \((d,d)\) with \(n\leq d\leq 2n\) can be achieved.