Nodal counting on quantum graphs

We consider the real eigenfunctions of the Schrödinger operator on graphs, and count their nodal domains. The number of nodal domains fluctuates within an interval whose size equals the number of bonds B. For well connected graphs, with incommensurate bond lengths, the distribution of the number of nodal domains in the interval mentioned above approaches a Gaussian distribution in the limit when the number of vertices is large. The approach to this limit is not simple, and we discuss it in detail. At the same time we define a random wave model for graphs, and compare the predictions of this model with analytic and numerical computations.     

A Lower Bound for Nodal Count on Discrete and Metric Graphs

We study the number of nodal domains (maximal connected regions on which a function has constant sign) of the eigenfunctions of Schrödinger operators on graphs. Under a certain genericity condition, we show that the number of nodal domains of the n ^th eigenfunction is bounded below by n  ?  ?, where ? is the number of links that distinguish the graph from a tree.Our results apply to operators on both discrete (combinatorial) and metric (quantum) graphs. They complement already known analogues of a result by Courant who proved the upper bound n for the number of nodal domains.To illustrate that the genericity condition is essential we show that if it is dropped, the nodal count can fall arbitrarily far below the number of the corresponding eigenfunction.In the Appendix we review the proof of the case ?  =  0 on metric trees which has been obtained by other authors.     

On the Nodal Count for Flat Tori

We give an explicit formula for the number of nodal domains of certain eigenfunctions on a flat torus. We apply this to an isospectral but not isometric family of pairs of flat four-dimensional tori constructed by Conway and Sloane, and we show that corresponding eigenfunctions have the same number of nodal domains. This disproves a conjecture by Brüning, Gnutzmann, and Smilansky.     

Courant-sharp eigenvalues of Neumann 2-rep-tiles

We find the Courant-sharp Neumann eigenvalues of the Laplacian on some 2-rep-tile domains. In \(\mathbb {R}^{2}\), the domains we consider are the isosceles right triangle and the rectangle with edge ratio \(\sqrt{2}\) (also known as the A4 paper). In \(\mathbb {R}^{n}\), the domains are boxes which generalize the mentioned planar rectangle. The symmetries of those domains reveal a special structure of their eigenfunctions, which we call folding\unfolding. This structure affects the nodal set of the eigenfunctions, which, in turn, allows to derive necessary conditions for Courant-sharpness. In addition, the eigenvalues of these domains are arranged as a lattice which allows for a comparison between the nodal count and the spectral position. The Courant-sharpness of most eigenvalues is ruled out using those methods. In addition, this analysis allows to estimate the nodal deficiency—the difference between the spectral position and the nodal count.     

Can one count the shape of a drum?

Sequences of nodal counts store information on the geometry (metric) of the domain where the wave equation is considered. To demonstrate this statement, we consider the eigenfunctions of the Laplace-Beltrami operator on surfaces of revolution. Arranging the wave functions by increasing values of the eigenvalues, and counting the number of their nodal domains, we obtain the nodal sequence whose properties we study. This sequence is expressed as a trace formula, which consists of a smooth (Weyl-like) part which depends on global geometrical parameters, and a fluctuating part, which involves the classical periodic orbits on the torus and their actions (lengths). The geometrical content of the nodal sequence is thus explicitly revealed.     

On the nodal behaviour of eigenfunctions

The generic behaviour of the nodal pattern of the eigenfunctions of the Schrödinger equation depending on parameters is discussed. Numerical examples are presented for the case of a hard-walled parallelogram.     

Resolving the isospectrality of the dihedral graphs by counting nodal domains

We discuss isospectral quantum graphs which are not isometric. These graphs are the analogues of the isospectral domains in R² which were introduced recently in [1–5] all based on Sunada's construction of isospectral domains [6]. After discussing some of the properties of these graphs, we present an example which support the conjecture that by counting the nodal domains of the corresponding eigenfunctions one can resolve the isospectral ambiguity.     

The Number of Nodal Domains on Quantum Graphs as a Stability Index of Graph Partitions

The Courant theorem provides an upper bound for the number of nodal domains of eigenfunctions of a wide class of Laplacian-type operators. In particular, it holds for generic eigenfunctions of a quantum graph. The theorem stipulates that, after ordering the eigenvalues as a non decreasing sequence, the number of nodal domains ν _n of the n ^th eigenfunction satisfies n ≥ ν _n . Here, we provide a new interpretation for the Courant nodal deficiency d _n = n − ν _n in the case of quantum graphs. It equals the Morse index — at a critical point — of an energy functional on a suitably defined space of graph partitions. Thus, the nodal deficiency assumes a previously unknown and profound meaning — it is the number of unstable directions in the vicinity of the critical point corresponding to the n ^th eigenfunction. To demonstrate this connection, the space of graph partitions and the energy functional are defined and the corresponding critical partitions are studied in detail.     

Nyström Method for the Coulomb and Screened Coulomb Potentials

In this paper, we show that the simple Nyström method can yield very accurate eigenvalues and eigenfunctions not only for large principal quantum number but also for large angular momentum quantum number. We demonstrate that the furcation phenomenon emerging in the calculated eigenfunctions can be regarded as an indicator for the bad behavior of the integral equation and the unreliability of the obtained results.     

R-matrix arising from affine Hecke algebras and its application to Macdonald's difference operators

We shall give a certain trigonometric R-matrix associated with each root system by using affine Hecke algebras. From this R-matrix, we derive a quantum Knizhnik-Zamolodchikov equation after Cherednik, and show that the solutions of this KZ equation yield eigenfunctions of Macdonald's difference operators.     

Nodal densities of planar gaussian random waves

The nodal densities of gaussian random functions, modelling various physical systems including chaotic quantum eigenfunctions and optical speckle patterns, are reviewed. The nodal domains of isotropically random real and complex functions are formulated in terms of their Minkowski functionals, and their correlations and spectra are discussed. The results on the statistical densities of the zeros of the real and complex functions, and their derivatives, in two dimensions are reviewed. New results are derived on the nodal domains of the hessian determinant (gaussian curvature) of two-dimensional random surfaces.     

A nodal domain theorem for integrable billiards in two dimensions

Eigenfunctions of integrable planar billiards are studied — in particular, the number of nodal domains, ν$\nu$ of the eigenfunctions with Dirichlet boundary conditions are considered. The billiards for which the time-independent Schrödinger equation (Helmholtz equation) is separable admit trivial expressions for the number of domains. Here, we discover that for all separable and non-separable integrable billiards, ν$\nu$ satisfies certain difference equations. This has been possible because the eigenfunctions can be classified in families labelled by the same value of mmodkn$mmodkn$, given a particular k$k$, for a set of quantum numbers, m,n$m,n$. Further, we observe that the patterns in a family are similar and the algebraic representation of the geometrical nodal patterns is found. Instances of this representation are explained in detail to understand the beauty of the patterns. This paper therefore presents a mathematical connection between integrable systems and difference equations.     

Arithmetic and Equidistribution of Measures on the Sphere

Motivated by problems of mathematical physics (quantum chaos) questions of equidistribution of eigenfunctions of the Laplace operator on a Riemannian manifold have been studied by several authors. We consider here, in analogy with arithmetic hyperbolic surfaces, orthonormal bases of eigenfunctions of the Laplace operator on the two dimensional unit sphere which are also eigenfunctions of an algebra of Hecke operators which act on these spherical harmonics. We formulate an analogue of the equidistribution of mass conjecture for these eigenfunctions as well as of the conjecture that their moments tend to moments of the Gaussian as the eigenvalue increases. For such orthonormal bases we show that these conjectures are related to the analytic properties of degree eight arithmetic L-functions associated to triples of eigenfunctions. Moreover we establish the conjecture for the third moments and give a conditional (on standard analytic conjectures about these arithmetic L-functions) proof of the equidistribution of mass conjecture. Electronic Supplementary Material: Supplementary material is available in the online version of this article at http://dx.doi.org/10.1007/s00220-003-0922-5Part of this work was done at the Institute for Advanced Study, Princeton, NJ.     

Higgs Bundles,Gauge Theories and Quantum Groups

The appearance of the Bethe Ansatz equation for the Nonlinear Schrödinger equation in the equivariant integration over the moduli space of Higgs bundles is revisited. We argue that the wave functions of the corresponding two-dimensional topological U(N) gauge theory reproduce quantum wave functions of the Nonlinear Schrödinger equation in the N-particle sector. This implies the full equivalence between the above gauge theory and the N-particle sub-sector of the quantum theory of the Nonlinear Schrödinger equation. This also implies the explicit correspondence between the gauge theory and the representation theory of the degenerate double affine Hecke algebra. We propose a similar construction based on the G/G gauged WZW model leading to the representation theory of the double affine Hecke algebra.     

The Modulus of Continuity¶for Γ0(m)\? Semi-Classical Limits

We study the behavior of a large-eigenvalue limit of eigenfunctions for the hyperbolic Laplacian for the modular quotient SL(2;ℤ)\?. Féjer summation and results of S. Zelditch are used to show that the microlocal lifts of eigenfunctions have large-eigenvalue limit a geodesic flow invariant measure for the modular unit cotangent bundle. The limit is studied for Hecke–Maass forms, joint eigenfunctions of the Hecke operators and the hyperbolic Laplacian. The first modulus of continuity result is presented for the limit. The singular concentration set of the limit cannot be a compact union of closed geodesics and measured geodesic laminations. Received: 10 March 2000 / Accepted: 26 July 2000     

Periodic Integrable Systems with Delta-Potentials

In this paper we study root system generalizations of the quantum Bose-gas on the circle with pair-wise delta-function interactions. The underlying symmetry structures are shown to be governed by the associated graded algebra of Cherednik's (suitably filtered) degenerate double affine Hecke algebra, acting by Dunkl-type differential-reflection operators. We use Gutkin's generalization of the equivalence between the impenetrable Bose-gas and the free Fermi-gas to derive the Bethe ansatz equations and the Bethe ansatz eigenfunctions.     

Eigenfunctions and eigenvalues of the Wigner operator

It can be proved, that the Wigner operator, which results from the Quantum-Mechanical foundation of Bopp, accepts as eigenvalues the differences of the eigenvalues of two equivalent Schrödinger equations. The eigenfunctions result with the help of a Fourier transform in the phase space of the corresponding eigenfunctions of the Schrödinger equations.     

Cantor Spectrum and KDS Eigenstates

In this note we consider KDS eigenstates of one-dimensional Schrödinger operators with ergodic potential, which are a class of generalized eigenfunctions including Bloch eigenstates. We show that if the spectrum, restricted to an interval, has zero Lyapunov exponents and is a Cantor set, then for a residual subset of energies, KDS eigenstates do not exist. In particular, we show that the quasi-periodic Schrödinger operators whose Schrödinger quasi-periodic cocycles are reducible for all energies have a limit band-type spectrum.     

Semiclassical Eigenfunctions of the Schrödinger Operator on a Graph That Are Localized Near a Subgraph

We study semiclassical eigenvalues and eigenfunctions of the Schrödinger operator on a geometric graph. We show that nontrivial boundary conditions at vertices lead to the existence of eigenfunctions, concentrated near a single vertex. We also construct semiclassical eigenfunctions, localized near edges and discuss general construction of spectral series which correspond to a general subgraph.