Eigenvalue Counting Function Research Articles

Edge currents and eigenvalue estimates for magnetic barrier Schrödinger operators

We study two-dimensional magnetic Schrodinger operators with a magnetic field that is equal to b>0 for x > 0 and (-b) for x < 0. This magnetic Schrodinger operator exhibits a magnetic barrier at x=0. The unperturbed system is invariant with respect to translations in the y-direction. As a result, the Schrodinger operator admits a direct integral decomposition. We analyze the band functions of the fiber operators as functions of the wave number and establish their asymptotic behavior. Because the fiber operators are reflection symmetric, the band functions may be classified as odd or even. The odd band functions have a unique absolute minimum. We calculate the effective mass at the minimum and prove that it is positive. The even band functions are monotone decreasing. We prove that the eigenvalues of an Airy operator, respectively, harmonic oscillator operator, describe the asymptotic behavior of the band functions for large negative, respectively positive, wave numbers. We prove a Mourre estimate for perturbations of the magnetic Schrodinger operator and establish the existence of absolutely continuous spectrum in certain energy intervals. We prove lower bounds on magnetic edge currents for states with energies in the same intervals. We also prove that these lower bounds imply stable lower bounds for the asymptotic currents. We study the perturbation by slowly decaying negative potentials and establish the asymptotic behavior of the eigenvalue counting function for the infinitely-many eigenvalues below the bottom of the essential spectrum.

Uniform Existence of the Integrated Density of States on Metric Cayley Graphs

Given a finitely generated amenable group we consider ergodic random Schr\"odinger operators on a Cayley graph with random potentials and random boundary conditions. We show that the normalised eigenvalue counting functions of finite volume parts converge uniformly. The integrated density of states as the limit can be expressed by a Pastur-Shubin formula. The spectrum supports the corresponding measure and discontinuities correspond to the existence of compactly supported eigenfunctions.

Improvement of eigenfunction estimates on manifolds of nonpositive curvature

Abstract Let (M,g) be a compact, boundaryless manifold of dimension n with the property that either (i) n = 2 and (M,g) has no conjugate points, or (ii) the sectional curvatures of (M,g) are nonpositive. Let Δ be the positive Laplacian on M determined by g. We study the L 2 → Lp mapping properties of a spectral cluster of (Δ)1/2 of width 1/log λ. Under the geometric assumptions above, Bérard [Math. Z. 155 (1977), 249–276] obtained a logarithmic improvement for the remainder term of the eigenvalue counting function which directly leads to a (log λ)1/2 improvement for Hörmander's estimate on the L ∞ norms of eigenfunctions. In this paper we extend this improvement to the Lp estimates for all p &gt; 2(n+1)/(n-1).

Analyzing self-similar and fractal properties of the C. elegans neural network.

The brain is one of the most studied and highly complex systems in the biological world. While much research has concentrated on studying the brain directly, our focus is the structure of the brain itself: at its core an interconnected network of nodes (neurons). A better understanding of the structural connectivity of the brain should elucidate some of its functional properties. In this paper we analyze the connectome of the nematode Caenorhabditis elegans. Consisting of only 302 neurons, it is one of the better-understood neural networks. Using a Laplacian Matrix of the 279-neuron “giant component” of the network, we use an eigenvalue counting function to look for fractal-like self similarity. This matrix representation is also used to plot visualizations of the neural network in eigenfunction coordinates. Small-world properties of the system are examined, including average path length and clustering coefficient. We test for localization of eigenfunctions, using graph energy and spacial variance on these functions. To better understand results, all calculations are also performed on random networks, branching trees, and known fractals, as well as fractals which have been “rewired” to have small-world properties. We propose algorithms for generating Laplacian matrices of each of these graphs.

Bounds on positive interior transmission eigenvalues

This paper contains lower bounds on the counting function of the positive eigenvalues of the interior transmission problem when the latter is elliptic. In particular, these bounds justify the existence of an infinite set of interior transmission eigenvalues and provide asymptotic estimates from above on the counting function for the large values of the wave number. They also lead to certain important upper estimates on the first few interior transmission eigenvalues. We consider the classical transmission problem as well as the case when the inhomogeneous medium contains an obstacle.

The asymptotic distribution of a single eigenvalue gap of a Wigner matrix

We show that the distribution of (a suitable rescaling of) a single eigenvalue gap \(\lambda _{i+1}(M_n)-\lambda _i(M_n)\) of a random Wigner matrix ensemble in the bulk is asymptotically given by the Gaudin–Mehta distribution, if the Wigner ensemble obeys a finite moment condition and matches moments with the GUE ensemble to fourth order. This is new even in the GUE case, as prior results establishing the Gaudin–Mehta law required either an averaging in the eigenvalue index parameter \(i\), or fixing the energy level \(u\) instead of the eigenvalue index. The extension from the GUE case to the Wigner case is a routine application of the Four Moment Theorem. The main difficulty is to establish the approximate independence of the eigenvalue counting function \(N_{(-\infty ,x)}(\tilde{M}_n)\) (where \(\tilde{M}_n\) is a suitably rescaled version of \(M_n\)) with the event that there is no spectrum in an interval \([x,x+s]\), in the case of a GUE matrix. This will be done through some general considerations regarding determinantal processes given by a projection kernel.

Spectral Asymptotics Revisited

We discuss two heuristic ideas concerning the spectrum of a Laplacian, and we give theorems and conjectures from the realms of manifolds, graphs and fractals that validate these heuristics. The first heuristic concerns Laplacians that do not have discrete spectra: here we discuss the notion of “spectral mass” closely related to the “integrated density of states”, an average of the diagonal of the kernel of the spectral projection operator, and show that this can serve as a substitute for the eigenvalue counting function. The second heuristic is an “asymptotic splitting law” that describes the proportions of the spectrum that transforms according to the irreducible representations of a finite group that acts as a symmetry group of the Laplacian. For this to be valid we require the existence of a fundamental domain with relatively small boundary. We also give a version in the case that the symmetry group is a compact Lie group. Many of our results are reformulations of known results, and some are merely conjectures, but there is something to be gained by looking at them together with a unified perspective.

Spectral analysis of 1D nearest-neighbor random walks and applications to subdiffusive trap and barrier models

We consider a family X^{(n)}, n \in \bbN_+, of continuous-time nearest-neighbor random walks on the one dimensional lattice Z. We reduce the spectral analysis of the Markov generator of X^{(n)} with Dirichlet conditions outside (0,n) to the analogous problem for a suitable generalized second order differential operator -D_{m_n} D_x, with Dirichlet conditions outside a given interval. If the measures dm_n weakly converge to some measure dm_*, we prove a limit theorem for the eigenvalues and eigenfunctions of -D_{m_n}D_x to the corresponding spectral quantities of -D_{m_*} D_x. As second result, we prove the Dirichlet-Neumann bracketing for the operators -D_m D_x and, as a consequence, we establish lower and upper bounds for the asymptotic annealed eigenvalue counting functions in the case that m is a self--similar stochastic process. Finally, we apply the above results to investigate the spectral structure of some classes of subdiffusive random trap and barrier models coming from one-dimensional physics.

The Weyl-type asymptotic formula for biharmonic Steklov eigenvalues on Riemannian manifolds

Let Ω be a bounded domain with C 2 -smooth boundary in an n-dimensional oriented Riemannian manifold. It is well known that for the biharmonic equation Δ 2 u = 0 in Ω with the condition u = 0 on ∂ Ω, there exists an infinite set { u k } of biharmonic functions in Ω with positive eigenvalues { λ k } satisfying Δ u k + λ k ϱ ∂ u k ∂ ν = 0 on ∂ Ω. In this paper, by a new method we establish the Weyl-type asymptotic formula for the counting function of the biharmonic Steklov eigenvalues λ k .

An exact trace formula and zeta functions for an infinite quantum graph with a non-standard Weyl asymptotics

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.

Asymptotics for Functions Associated with Heat Flow on the Sierpinski Carpet

Abstract We establish the asymptotic behaviour of the partition function, the heat content, the integrated eigenvalue counting function, and, for certain points, the on-diagonal heat kernel of generalized Sierpinski carpets. For all these functions the leading term is of the form xγΦ (log x) for a suitable exponent γ and Φ a periodic function. We also discuss similar results for the heat content of affine nested fractals.

Spectral theory for nonconservative transmission line networks

The global theory of transmission line networks withnonconservative junction conditions is developed from a spectraltheoretic viewpoint. The rather general junction conditionslead to spectral problems for nonnormal operators.The theory of analytic functions which are almost periodic in a stripis used to establish the existence of an infinite sequence of eigenvaluesand the completeness of generalized eigenfunctions. Simple eigenvalues are generic.The asymptotic behavior of an eigenvalue counting function is determined.Specialized results are developed for rational graphs.

A note on the Central Limit Theorem for the Eigenvalue Counting Function of Wigner Matrices

The purpose of this note is to establish a Central Limit Theorem for the number of eigenvalues of a Wigner matrix in an interval. The proof relies on the correct asymptotics of the variance of the eigenvalue counting function of GUE matrices due to Gustavsson, and its extension to large families of Wigner matrices by means of the Tao and Vu Four Moment Theorem and recent localization results by Erd?s, Yau and Yin.

L p -Approximation of the Integrated Density of States for Schrödinger Operators with Finite Local Complexity

We study spectral properties of Schrödinger operators on $${\mathbb R^d}$$ . The electromagnetic potential is assumed to be determined locally by a colouring of the lattice points in $${\mathbb Z^d}$$ , with the property that frequencies of finite patterns are well defined. We prove that the integrated density of states (spectral distribution function) is approximated by its finite volume analogues, i.e. the normalised eigenvalue counting functions. The convergence holds in the space L p (I) where I is any finite energy interval and 1 ≤ p < ∞ is arbitrary.

Low Energy Asymptotics of the Spectral Shift Function for Pauli Operators with Nonconstant Magnetic Fields

We consider the 3D Pauli operator with nonconstant magnetic field \mathbf B of constant direction, perturbed by a symmetric matrix-valued electric potential V whose coefficients decay fast enough at infinity. We investigate the low-energy asymptotics of the corresponding spectral shift function. As a corollary, for generic negative V , we obtain a generalized Levinson formula, relating the low-energy asymptotics of the eigenvalue counting function and of the scattering phase of the perturbed operator.

Unicity of the Integrated Density of States for Relativistic Schrödinger Operators with Regular Magnetic Fields and Singular Electric Potentials

We show coincidence of the two definitions of the integrated density of states (IDS) for a class of relativistic Schrödinger operators with magnetic fields and scalar potentials introduced in Iftimie et al. (Publ Res Inst Math Sci 43(3):585–623, 2007; Topics in applied mathematics and mathematical physics, Editura Academiei Române, 2008), the first one relying on the eigenvalue counting function of operators induced on open bounded sets with Dirichlet boundary conditions, the other one involving the spectral projections of the operator defined on the entire space. In this way one generalizes the results of Doi et al. (Math Z 237:335–371, 2001) and Iftimie (Publ Res Inst Math Sci 41(2):307–327, 2005) for non-relativistic operators. The proofs needs the magnetic pseudodifferential calculus developed in Iftimie et al. (Publ Res Inst Math Sci 43(3):585–623, 2007), as well as a Feynman-Kac-Itô formula for Lévy processes (Ichinose and Tamura, Commun Math Phys 105(2):239–257, 1986; Iftimie et al. Topics in applied mathematics and mathematical physics, Editura Academiei Române, 2008). In addition, in case when both the magnetic field and the scalar potential are periodic, one also proves the existence of the IDS.

The Berry–Keating operator on and on compact quantum graphs with general self-adjoint realizations

The Berry–Keating operator (Berry and Keating 1999 SIAM Rev. 41 236) governing the Schrödinger dynamics is discussed in the Hilbert space and on compact quantum graphs. It has been proved that the spectrum of HBK defined on is purely continuous and thus this quantization of HBK cannot yield the hypothetical Hilbert–Polya operator possessing as eigenvalues the nontrivial zeros of the Riemann zeta function. A complete classification of all self-adjoint extensions of HBK acting on compact quantum graphs is given together with the corresponding secular equation in form of a determinant whose zeros determine the discrete spectrum of HBK. In addition, an exact trace formula and the Weyl asymptotics of the eigenvalue counting function are derived. Furthermore, we introduce the ‘squared’ Berry–Keating operator which is a special case of the Black–Scholes operator used in financial theory of option pricing. Again, all self-adjoint extensions, the corresponding secular equation, the trace formula and the Weyl asymptotics are derived for H2BK on compact quantum graphs. While the spectra of both HBK and H2BK on any compact quantum graph are discrete, their Weyl asymptotics demonstrate that neither HBK nor H2BK can yield as eigenvalues the nontrivial Riemann zeros. Some simple examples are worked out in detail.

Spectral Asymptotics for Stable Trees

We calculate the mean and almost-sure leading order behaviour of the high frequency asymptotics of the eigenvalue counting function associated with the natural Dirichlet form on $\alpha$-stable trees, which lead in turn to short-time heat kernel asymptotics for these random structures. In particular, the conclusions we obtain demonstrate that the spectral dimension of an $\alpha$-stable tree is almost-surely equal to $2\alpha/(2\alpha-1)$, matching that of certain related discrete models. We also show that the exponent for the second term in the asymptotic expansion of the eigenvalue counting function is no greater than $1/(2\alpha-1)$. To prove our results, we adapt a self-similar fractal argument previously applied to the continuum random tree, replacing the decomposition of the continuum tree at the branch point of three suitably chosen vertices with a recently developed spinal decomposition for $\alpha$-stable trees

Spectral asymptotics for Laplacians on self-similar sets

Given a self-similar Dirichlet form on a self-similar set, we first give an estimate on the asymptotic order of the associated eigenvalue counting function in terms of a ‘geometric counting function’ defined through a family of coverings of the self-similar set naturally associated with the Dirichlet space. Secondly, under (sub-)Gaussian heat kernel upper bound, we prove a detailed short time asymptotic behavior of the partition function, which is the Laplace–Stieltjes transform of the eigenvalue counting function associated with the Dirichlet form. This result can be applicable to a class of infinitely ramified self-similar sets including generalized Sierpinski carpets, and is an extension of the result given recently by B.M. Hambly for the Brownian motion on generalized Sierpinski carpets. Moreover, we also provide a sharp remainder estimate for the short time asymptotic behavior of the partition function.

Detailed asymptotic of eigenvalues on time scales

Let 𝕋 = {a n } n ∪{0} be a time scale with zero Minkowski (or box) dimension, where {a n } n is a monotonically decreasing sequence converging to zero, and a 1 = 1. In this paper, we find an upper bound for the eigenvalue counting function of the linear problem − u ΔΔ = λu σ, with Dirichlet boundary conditions. We obtain that the nth-eigenvalue is bounded below by . We show that the bound is optimal for the q-difference equations arising in quantum calculus.