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Register a new userIsospectral discrete and quantum graphs with the same flip counts and nodal counts
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The existence of non-isomorphic graphs which share the same Laplace spectrum (to be referred to as isospectral graphs) leads naturally to the following question: What additional information is required in order to resolve isospectral graphs? It was suggested by Band, Shapira and Smilansky that this might be achieved by either counting the number of nodal domains or the number of times the eigenfunctions change sign (the so-called flip count). Recently examples of (discrete) isospectral graphs with the same flip count and nodal count have been constructed by K. Ammann by utilising Godsil-McKay switching. Here we provide a simple alternative mechanism that produces systematic examples of both discrete and quantum isospectral graphs with the same flip and nodal counts.
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We establish the Lagrangian nature of the discrete isospectral and isomonodromic dynamical systems corresponding to the re-factorization transformations of the rational matrix functions on the Riemann sphere. Specifically, in the isospectral case we generalize the Moser-Veselov approach to integrability of discrete systems via the re-factorization of matrix polynomials to a more general class of matrix rational functions that have a simple divisor and, in the quadratic case, explicitly write the Lagrangian function for such systems. Next we show that if we let certain parameters in this Lagrangian to be time-dependent, the resulting Euler-Lagrange equations describe the isomonodromic transformations of systems of linear difference equations. It is known that in some special cases such equations reduce to the difference Painleve equation. As an example, we show how to obtain the difference Painlev`e V equation in this way, and hence we establish that this equation can be written in the Lagrangian form.
It has been suggested that the distribution of the suitably normalized number of zeros of Laplacian eigenfunctions contains information about the geometry of the underlying domain. We study this distribution (more precisely, the distribution of the nodal surplus) for Laplacian eigenfunctions of a metric graph. The existence of the distribution is established, along with its symmetry. One consequence of the symmetry is that the graphs first Betti number can be recovered as twice the average nodal surplus of its eigenfunctions. Furthermore, for graphs with disjoint cycles it is proven that the distribution has a universal form --- it is binomial over the allowed range of values of the surplus. To prove the latter result, we introduce the notion of a local nodal surplus and study its symmetry and dependence properties, establishing that the local nodal surpluses of disjoint cycles behave like independent Bernoulli variables.
We prove an analogue of the magnetic nodal theorem on quantum graphs: the number of zeros $phi$ of the $n$-th eigenfunction of the Schrodinger operator on a quantum graph is related to the stability of the $n$-th eigenvalue of the perturbation of the operator by magnetic potential. More precisely, we consider the $n$-th eigenvalue as a function of the magnetic perturbation and show that its Morse index at zero magnetic field is equal to $phi - (n-1)$.
We associate an explicit equivalent descendent insertion to any relative insertion in quantum K-theory of Nakajima varieties. This also serves as an explicit formula for off-shell Bethe eigenfunctions for general quantum loop algebras associated to quivers and gives the general integral solution to the corresponding quantum Knizhnik-Zamolodchikov and dynamical q-difference equations.
According to a well-know theorem by Sturm, a vibrating string is divided into exactly N nodal intervals by zeros of its N-th eigenfunction. Courant showed that one half of Sturms theorem for the strings applies to the theory of membranes: N-th eigenfunction cannot have more than N domains. He also gave an example of a eigenfunction high in the spectrum with a minimal number of nodal domains, thus excluding the existence of a non-trivial lower bound. An analogue of Sturms result for discretizations of the interval was discussed by Gantmacher and Krein. The discretization of an interval is a graph of a simple form, a chain-graph. But what can be said about more complicated graphs? It has been known since the early 90s that the nodal count for a generic eigenfunction of the Schrodinger operator on quantum trees (where each edge is identified with an interval of the real line and some matching conditions are enforced on the vertices) is exact too: zeros of the N-th eigenfunction divide the tree into exactly N subtrees. We discuss two extensions of this result in two directions. One deals with the same continuous Schrodinger operator but on general graphs (i.e. non-trees) and another deals with discrete Schrodinger operator on combinatorial graphs (both trees and non-trees). The result that we derive applies to both types of graphs: the number of nodal domains of the N-th eigenfunction is bounded below by N-L, where L is the number of links that distinguish the graph from a tree (defined as the dimension of the cycle space or the rank of the fundamental group of the graph). We also show that if it the genericity condition is dropped, the nodal count can fall arbitrarily far below the number of the corresponding eigenfunction.
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