Estimates for fundamental solutions and spectral bounds for a class of Schrödinger operators

We study a class of Schrödinger operators of the form , where is a nonnegative function singular at 0, that is V(0)=0. Under suitable assumptions on the potential V, we derive sharp lower and upper bounds for the fundamental solution hε. Moreover, we obtain information on the spectrum of the self-adjoint operator defined by Lε in L²(R). In particular, we give a lower bound for the eigenvalues.     

On bounding Hecke-Siegel eigenvalues

We use the action of the Hecke operators (1≤j≤n) on the Fourier coefficients of Siegel modular forms to bound the eigenvalues of these Hecke operators. This extends work of Duke-Howe-Li and of Kohnen, who provided bounds on the eigenvalues of the operator T(p).     

Lower bounds for the energy of digraphs

The energy of a digraph D is defined as , where z₁,…,z_n are the eigenvalues of D. In this article we find lower bounds for the energy of digraphs in terms of the number of closed walks of length 2, extending in this way the result obtained by Caporossi et al. [G. Caporossi, D. Cvetkovi?, I. Gutman, P. Hansen, Variable neighborhood search for extremal graphs. 2. Finding graphs with extremal energy, J. Chem. Inf. Comput. Sci. 39 (1999) 984-996]: for all graphs G with m edges. Also, we study digraphs with three eigenvalues.     

Singular-value-like decomposition for complex matrix triples

The classical singular value decomposition for a matrix A∈C^m×n is a canonical form for A that also displays the eigenvalues of the Hermitian matrices AA^∗ and A^∗A. In this paper, we develop a corresponding decomposition for A that provides the Jordan canonical forms for the complex symmetric matrices and . More generally, we consider the matrix triple , where are invertible and either complex symmetric or complex skew-symmetric, and we provide a canonical form under transformations of the form , where X,Y are nonsingular.     

Some relations between rank, chromatic number and energy of graphs

The energy of a graph G, denoted by E(G), is defined as the sum of the absolute values of all eigenvalues of G. Let G be a graph of order n and be the rank of the adjacency matrix of G. In this paper we characterize all graphs with . Among other results we show that apart from a few families of graphs, , where n is the number of vertices of G, and χ(G) are the complement and the chromatic number of G, respectively. Moreover some new lower bounds for E(G) in terms of are given.     

Reduced Gutzwiller formula with symmetry: Case of a Lie group

We consider a classical Hamiltonian H on R2d, invariant by a Lie group of symmetry G, whose Weyl quantization is a selfadjoint operator on L²(Rd). If χ is an irreducible character of G, we investigate the spectrum of its restriction to the symmetry subspace of L²(Rd) coming from the decomposition of Peter-Weyl. We give semi-classical Weyl asymptotics for the eigenvalues counting function of in an interval of R, and interpret it geometrically in terms of dynamics in the reduced space R2d/G. Besides, oscillations of the spectral density of are described by a Gutzwiller trace formula involving periodic orbits of the reduced space, corresponding to quasi-periodic orbits of R2d.     

Spectrum of a non-self-adjoint operator associated with the periodic heat equation

We study the spectrum of the linear operator L=−θ_∂−?θ_∂(sinθθ_∂) subject to the periodic boundary conditions on θ∈[−π,π]. We prove that the operator is closed in with the domain in for |?|<2, its spectrum consists of an infinite sequence of isolated eigenvalues and the set of corresponding eigenfunctions is complete. By using numerical approximations of eigenvalues and eigenfunctions, we show that all eigenvalues are simple, located on the imaginary axis and the angle between two subsequent eigenfunctions tends to zero for larger eigenvalues. As a result, the complete set of linearly independent eigenfunctions does not form a basis in .     

Positive integral operators in unbounded domains

We study positive integral operators in with continuous kernel k(x,y). We show that if the operator is compact and Hilbert-Schmidt. If in addition k(x,x)→0 as |x|→∞, k is represented by an absolutely and uniformly convergent bilinear series of uniformly continuous eigenfunctions and is trace class. Replacing the first assumption by the stronger then and the bilinear series converges also in L¹. Sharp norm bounds are obtained and Mercer's theorem is derived as a special case.     

On maximal functions for Mikhlin-Hörmander multipliers

Given Mikhlin-Hörmander multipliers , with uniform estimates we prove an optimal bound in L^p for the maximal function and related bounds for maximal functions generated by dilations. These improve the results in [M. Christ, L. Grafakos, P. Honzík, A. Seeger, Maximal functions associated with multipliers of Mikhlin-Hörmander type, Math. Z. 249 (2005) 223-240].     

Further results on iterative methods for computing generalized inverses

In this paper, we reconsider the iterative method X_k=X_k−1+βY(I−AX_k−1), k=1,2,…,β∈C?{0} for computing the generalized inverse over Banach spaces or the generalized Drazin inverse a^d of a Banach algebra element a, reveal the intrinsic relationship between the convergence of such iterations and the existence of or a^d, and present the error bounds of the iterative methods for approximating or a^d. Moreover, we deduce some necessary and sufficient conditions for iterative convergence to or a^d.     

Mixed lower bounds for quantum transport

Let H be a self-adjoint operator on a separable Hilbert space , . Given an orthonormal basis of , we consider the time-averaged moments 〈|X|_ψ^p〉(T) of the position operator associated to . We derive lower bounds for the moments in terms of both spectral measure μ_ψ and generalized eigenfunctions u_ψ(n,x) of the state ψ. As a particular corollary, we generalize the recently obtained lower bound in terms of multifractal dimensions of μ_ψ and give some equivalent forms of it which can be useful in applications. We establish, in particular, the relations between the L^q-norms (q>1/2) of the imaginary part of Borel transform of probability measures and the corresponding multifractal dimensions.     

The case of equality in the Dobrushin-Deutsch-Zenger bound

Suppose that A=(a_i,j) is an n×n real matrix with constant row sums μ. Then the Dobrushin-Deutsch-Zenger (DDZ) bound on the eigenvalues of A other than μ is given by . When A a transition matrix of a finite homogeneous Markov chain so that μ=1,Z(A) is called the coefficient of ergodicity of the chain as it bounds the asymptotic rate of convergence, namely, , of the iteration , to the stationary distribution vector of the chain.In this paper we study the structure of real matrices for which the DDZ bound is sharp. We apply our results to the study of the class of graphs for which the transition matrix arising from a random walk on the graph attains the bound. We also characterize the eigenvalues λ of A for which |λ|=Z(A) for some stochastic matrix A.     

Monoids with sub-log-exponential free spectra

A classic result from the 1960s states that the asymptotic growth of the free spectrum of a finite group is sub-log-exponential if and only if is nilpotent. Thus a monoid is sub-log-exponential implies , the pseudovariety of semigroups with nilpotent subgroups. Unfortunately, little more is known about the boundary between the sub-log-exponential and log-exponential monoids.The pseudovariety consists of those finite semigroups satisfying (x^ωy^ω)^ω(y^ωx^ω)^ω(x^ωy^ω)^ω≈(x^ωy^ω)^ω. Here it is shown that a monoid is sub-log-exponential implies . A quick application: a regular sub-log-exponential monoid is orthodox. It is conjectured that a finite monoid is sub-log-exponential if and only if it is , the finite monoids in having nilpotent subgroups. The forward direction of the conjecture is proved; moreover, the conjecture is proved for when is completely (0)-simple. In particular, the six-element Brandt monoid (the Perkins semigroup) is sub-log-exponential.