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The Euler characteristic $chi =|V|-|E|$ and the total length $mathcal{L}$ are the most important topological and geometrical characteristics of a metric graph. Here, $|V|$ and $|E|$ denote the number of vertices and edges of a graph. The Euler characteristic determines the number $beta$ of independent cycles in a graph while the total length determines the asymptotic behavior of the energy eigenvalues via the Weyls law. We show theoretically and confirm experimentally that the Euler characteristic can be determined (heard) from a finite sequence of the lowest eigenenergies $lambda_1, ldots, lambda_N$ of a simple quantum graph, without any need to inspect the system visually. In the experiment quantum graphs are simulated by microwave networks. We demonstrate that the sequence of the lowest resonances of microwave networks with $beta leq 3$ can be directly used in determining whether a network is planar, i.e., can be embedded in the plane. Moreover, we show that the measured Euler characteristic $chi$ can be used as a sensitive revealer of the fully connected graphs.
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We evaluate the variance of coefficients of the characteristic polynomial for binary quantum graphs using a dynamical approach. This is the first example of a chaotic quantum system where a spectral statistic can be evaluated in terms of periodic orbits without taking the semiclassical limit, which is the limit of large graphs. The variance depends on the size of two classes of primitive pseudo orbits (sets of periodic orbits); pseudo orbits without self-intersections and those where all the self-intersections are 2-encounters at a single vertex. To show other pseudo orbits do not contribute we employ a parity argument for Lyndon word decompositions. For families of binary graphs with an increasing number of bonds, we show the periodic orbit formula approaches a universal constant independent of the coefficient of the polynomial. This constant is obtained by counting the total number of primitive pseudo orbits of a given length. To count periodic orbits and pseudo orbits we exploit further connections between orbits on binary graphs and Lyndon words.
By using the Hamilton-Jacobi [HJ] framework the topological theories associated with Euler and Pontryagin classes are analyzed. We report the construction of a fundamental $HJ$ differential where the characteristic equations and the symmetries of the theory are identified. Moreover, we work in both theories with the same phase space variables and we show that in spite of Pontryagin and Euler classes share the same equations of motion their symmetries are different. In addition, we report all HJ Hamiltonians and we compare our results with other formulations reported in the literature.
We compute Haar ensemble averages of ratios of random characteristic polynomials for the classical Lie groups K = O(N), SO(N), and USp(N). To that end, we start from the Clifford-Weyl algebera in its canonical realization on the complex of holomorphic differential forms for a C-vector space V. From it we construct the Fock representation of an orthosymplectic Lie superalgebra osp associated to V. Particular attention is paid to defining Howes oscillator semigroup and the representation that partially exponentiates the Lie algebra representation of sp in osp. In the process, by pushing the semigroup representation to its boundary and arguing by continuity, we provide a construction of the Shale-Weil-Segal representation of the metaplectic group. To deal with a product of n ratios of characteristic polynomials, we let V = C^n otimes C^N where C^N is equipped with its standard K-representation, and focus on the subspace of K-equivariant forms. By Howe duality, this is a highest-weight irreducible representation of the centralizer g of Lie(K) in osp. We identify the K-Haar expectation of n ratios with the character of this g-representation, which we show to be uniquely determined by analyticity, Weyl group invariance, certain weight constraints and a system of differential equations coming from the Laplace-Casimir invariants of g. We find an explicit solution to the problem posed by all these conditions. In this way we prove that the said Haar expectations are expressed by a Weyl-type character formula for all integers N ge 1. This completes earlier work by Conrey, Farmer, and Zirnbauer for the case of U(N).
We show that the averaged characteristic polynomial and the averaged inverse characteristic polynomial, associated with Hermitian matrices whose elements perform a random walk in the space of complex numbers, satisfy certain partial differential, diffusion-like, equations. These equations are valid for matrices of arbitrary size. Their solutions can be given an integral representation that allows for a simple study of their asymptotic behaviors for a broad range of initial conditions.
This paper contains a rigorous mathematical example of direct derivation of the system of Euler hydrodynamic equations from Hamiltonian equations for N point particle system as N tends to infinity. Direct means that the following standard tools are not used in the proof: stochastic dynamics, thermodynamics, Boltzmann kinetic equations, correlation functions approach by N. N. Bogolyubov.
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