A Laplacian eigenfunction on a two-dimensional Riemannian manifold provides a natural partition into Neumann domains (a.k.a. Morse-Smale complexes). This partition is generated by gradient flow lines of the eigenfunction-these bound the socalled Neumann domains. We prove that the Neumann Laplacian ∆ defined on a single Neumann domain is self-adjoint and possesses a purely discrete spectrum. In addition, we prove that the restriction of the eigenfunction to any one of its Neumann domains is an eigenfunction of ∆. As a comparison, similar statements for a nodal domain of an eigenfunction (with the Dirichlet Laplacian) are basic and well-known. The difficulty here is that the boundary of a Neumann domain may have cusps and cracks, and hence is not necessarily continuous, so standard results about Sobolev spaces are not available. Another very useful common fact is that the restricted eigenfunction on a nodal domain is the first eigenfunction of the Dirichlet Laplacian. This is no longer true for a Neumann domain. Our results enable the investigation of the resulting spectral position problem for Neumann domains, which is much more involved than its nodal analogue.

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We study a quantum Hamiltonian that is given by the (negative) Laplacian and an infinite chain of δ-like potentials with strength κ > 0 on the half line and which is equivalent to a one-parameter family of Laplacians on an infinite metric graph. This graph consists of an infinite chain of edges with the metric structure defined by assigning an interval In = [0, ln], , to each edge with length . We show that the one-parameter family of quantum graphs possesses a purely discrete and strictly positive spectrum for each κ > 0 and prove that the Dirichlet Laplacian is the limit of the one-parameter family in the strong resolvent sense. The spectrum of the resulting Dirichlet quantum graph is also purely discrete. The eigenvalues are given by λn = n2, , with multiplicities d(n), where d(n) denotes the divisor function. We can thus relate the spectral problem of this infinite quantum graph to Dirichlet's famous divisor problem and infer the non-standard Weyl asymptotics for the eigenvalue counting function. Based on an exact trace formula, the Voronoï summation formula, we derive explicit formulae for the trace of the wave group, the heat kernel, the resolvent and for various spectral zeta functions. These results enable us to establish a well-defined (renormalized) secular equation and a Selberg-like zeta function defined in terms of the classical periodic orbits of the graph for which we derive an exact functional equation and prove that the analogue of the Riemann hypothesis is true.

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