n -Laplacians on Metric Graphs and Almost Periodic Functions: I
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Annales Henri Poincar´ e
n-Laplacians on Metric Graphs and Almost Periodic Functions: I Pavel Kurasov
and Jacob Muller
Abstract. The spectra of n-Laplacian operators (−Δ)n on finite metric graphs are studied. An effective secular equation is derived and the spectral asymptotics are analysed, exploiting the fact that the secular function is close to a trigonometric polynomial. The notion of the quasispectrum is introduced, and its uniqueness is proved using the theory of almost periodic functions. To achieve this, new results concerning roots of functions close to almost periodic functions are proved. The results obtained on almost periodic functions are of general interest outside the theory of differential operators. Mathematics Subject Classification. 30D99, 34L20, 81U20.
Contents 1. Introduction 2. Differential Operators on Metric Graphs 3. n-Laplacians on Metric Graphs 4. Transmission Matrices and the Secular Equation for n-Laplacians 5. Scaling-Invariant Vertex Conditions 6. The Secular Equation for Scaling-Invariant Vertex Conditions 7. Almost Periodic Functions and Holomorphic Perturbations 8. Asymptotic Approximation of the Spectrum 9. Conclusion Acknowledgements Appendix A. Proofs for Section 3 Appendix B. Further Properties of the Vertex Transmission and Scattering Matrices References
P. Kurasov, J. Muller
Ann. Henri Poincar´e
1. Introduction 1.1. Motivation Quantum graphs have proved to be an important area of research in both physics and mathematics. By quantum graphs, one understands ordinary differential equations on metric graphs, coupled by matching conditions at the vertices. Most works on the subject consider second-order (Schr¨ odinger) differential operators (see, for example, [8,20,21,25,28,34,36,40,41,45,51]), but the methods developed can be generalised to differential expressions of arbitrary order. This has already been done in the case of first-order (Dirac and momentum) and to some extent fourth-order operators (e.g. [11,13,22,27,32]). The recent status of research in this area is well reflected in the monographs [8,41]. Recent development in spectral theory of Schr¨ odinger operators on metric graphs has seen a connection with trigonometric polynomials and the classical theory of almost periodic functions (see [12,47]). These studies were based on the Gutkin–Kottos–Smilansky formula for the secular equation for the Laplacian [28,35] det [Se (k)Sv (k) − I] = 0,
(1.1)
where Se and Sv are the edge and vertex scattering matrices (see Sect. 5), respectively. For scaling-invariant vertex conditions, Sv is independent of the energy, whilst the entries of Se are given by exponentials with real frequencies; hence, the secular function is a trigonometric polynomial of the form aj eirj k , (1.2) p(k) = j∈J
where aj ∈ C and rj ∈ R, not necessarily rationally dependent. For general vertex conditions and Schr¨ odinger operators with nonzero potentials, the eigenvalues are not given by zeros of trigonometric polynomials, but are asymptotically close to such zeros, leading to the notion of asymptoti
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