Asymptotics of the Discrete Spectrum of the Three-Particle Schrödinger Difference Operator on a Lattice

We consider the Hamiltonian H (K) of a system consisting of three bosons that interact through attractive pair contact potentials on a three-dimensional integer lattice. We obtain an asymptotic value for the number N(K,z) of eigenvalues of the operator H₀(K) lying below z0 with respect to the total quasimomentum K0 and the spectral parameter z–0.     

Threshold Phenomena in the Spectrum of the Two-Particle Schrödinger Operator on a Lattice

Theoretical and Mathematical Physics - For a broad class of short-range pairwise attraction potentials, we study threshold phenomena in the spectrum of the two-particle Schrödinger operator...     

Absolute Continuity of the Spectrum of a Periodic 3D Magnetic Schrödinger Operator with Singular Electric Potential

Mathematical Notes - We prove that the spectrum of a periodic 3D magnetic Schrödinger operator whose electric potential $$V=d\mu/dx$$ is the derivative of a measure is absolutely continuous...     

Absolute Continuity of the Spectrum of a Periodic Schrödinger Operator

We prove the absolute continuity of the spectrum of the Schrödinger operator in , , with periodic (with a common period lattice ) scalar and vector potentials for which either , , or the Fourier series of the vector potential converges absolutely, , where is an elementary cell of the lattice , for , and for , and the value of is sufficiently small, where and otherwise, , and .     

On a Multiple Singular Integral Operator

ＯｎａＭｕｌｔｉｐｌｅＳｉｎｇｕｌａｒＩｎｔｅｇｒａｌＯｐｅｒａｔｏｒＨｕＧｕｏｅｎ（胡国恩）（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓ，ＩｎｓｔｉｔｕｔｅｏｆＩｎｆｏｒｍａｔｉｏｎＥｎｇｉｎｅｅｒｉｎｇ，Ｚｈｅｎｇｚｈｏｕ，４５０００２）Ａｂ...     

Spectrum and Spectral Singularities of a Quadratic Pencil of a Schr?dinger Operator with Boundary Conditions Dependent on the Eigenparameter

In this paper, we consider the following quadratic pencil of Schr?dinger operators L(λ)generated in L²(R⁺) by the equation ■ with the boundary condition ■ where p(x) and q(x) are complex valued functions and α₀, α₁, β₀, β₁ are complex numbers with α₀β₁-α₁β₀≠0. It is proved that L(λ) has a finite number of eigenvalues and spectral singularities,and each of them is of a finite multiplicity...     

Bounds on the vertex–edge domination number of a tree

A vertex–edge dominating set of a graph G is a set D of vertices of G such that every edge of G is incident with a vertex of D or a vertex adjacent to a vertex of D. The vertex–edge domination number of a graph G  , denoted by γ_ve(T)${\gamma }_{\mathrm{ve}}(T)$, is the minimum cardinality of a vertex–edge dominating set of G. We prove that for every tree T   of order n?3$n?3$ with l leaves and s   support vertices, we have (n−l−s+3)/4?γ_ve(T)?n/3$(n-l-s+3)/4?{\gamma }_{\mathrm{ve}}(T)?n/3$, and we characterize the trees attaining each of the bounds.     

Spectrum of the Kerzman–Stein Operator for Model Domains

For a domain , the Kerzman-Stein operator is the skew-hermitian part of the Cauchy operator acting on L²(b), which is defined with respect to Euclidean measure. In this paper we compute the spectrum of the Kerzman-Stein operator for three domains whose boundaries consist of two circular arcs: a strip, a wedge, and an annulus. We also treat the case of a domain bounded by two logarithmic spirals.     

Some New Combinatorial Conditions on the Singular Fibre of a Fibration

Ｓｏｍｅ　Ｎｅｗ　Ｃｏｍｂｉｎａｔｏｒｉａｌ　Ｃｏｎｄｉｔｉｏｎｓ　ｏｎ　ｔｈｅ　Ｓｉｎｇｕｌａｒ　Ｆｉｂｒｅ　ｏｆ　ａ　ＦｉｂｒａｔｉｏｎＴａｎｇＭｉｎｇｙｙｕａｎ（唐明元）（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉｃｓ，ＳｈａｎｇｈａｉＮｏｒｍ...     

Spectrum of the two-particle Schrödinger operator on a lattice

We consider the family of two-particle discrete Schrödinger operators H(k) associated with the Hamiltonian of a system of two fermions on a ν-dimensional lattice ?, ν ≥, 1, where k ∈ \(\mathbb{T}^\nu \) ≡ (? π, π]^ν is a two-particle quasimomentum. We prove that the operator H(k), k ∈ \(\mathbb{T}^\nu \), k ≠ 0, has an eigenvalue to the left of the essential spectrum for any dimension ν = 1, 2, ... if the operator H(0) has a virtual level (ν = 1, 2) or an eigenvalue (ν ≥ 3) at the bottom of the essential spectrum (of the two-particle continuum).     

Virtual Bound Levels in a Gap of the Essential Spectrum of the Weakly Perturbed Periodic Schrödinger Operator

In the space \({L_{2}(\mathbf{R}^{d}) (d \le 3)}\) we consider the Schrödinger operator \({H_{\gamma}=-{\Delta}+ V(\mathbf{x})\cdot+\gamma W(\mathbf{x})\cdot}\), where \({V(\mathbf{x})=V(x_{1}, x_{2}, \dots, x_{d})}\) is a periodic function with respect to all the variables, \({\gamma}\) is a small real coupling constant and the perturbation \({W(\mathbf{x})}\) tends to zero sufficiently fast as \({|\mathbf{x}|\rightarrow\infty}\). We study so called virtual bound levels of the operator \({H_\gamma}\), i.e., those eigenvalues of \({H_\gamma}\) which are born at the moment \({\gamma=0}\) in a gap \({(\lambda_-,\,\lambda_+)}\) of the spectrum of the unperturbed operator \({H_0=-\Delta+ V(\mathbf{x})\cdot}\) from an edge of this gap while \({\gamma}\) increases or decreases. We assume that the dispersion function of H₀, branching from an edge of \({(\lambda_-,\lambda_+)}\), is non-degenerate in the Morse sense at its extremal set. For a definite perturbation \({(W(\mathbf{x})\ge 0)}\) we show that if d ≤ 2, then in the gap there exist virtual eigenvalues which are born from this edge. We investigate their number and an asymptotic behavior of them and of the corresponding eigenfunctions as \({\gamma\rightarrow 0}\). For an indefinite perturbation we estimate the multiplicity of virtual bound levels. In particular, we show that if d = 3 and both edges of the gap \({(\lambda_-,\,\lambda_+)}\) are non-degenerate, then under additional conditions there is a threshold for the birth of the impurity spectrum in the gap, i.e., \({\sigma(H_\gamma)\cap(\lambda_-,\,\lambda_+)=\emptyset}\) for a small enough \({|\gamma|}\).     

Bounds on the clique-transversal number of regular graphs

A clique-transversal set D of a graph G is a set of vertices of G such that D meets all cliques of G.The clique-transversal number,denoted Tc(G),is the minimum cardinality of a clique- transversal set in G.In this paper we present the bounds on the clique-transversal number for regular graphs and characterize the extremal graphs achieving the lower bound.Also,we give the sharp bounds on the clique-transversal number for claw-free cubic graphs and we characterize the extremal graphs achieving the lower bound.     

Upper Bounds on the First Eigenvalue for a Diffusion Operator via Bakry–Émery Ricci Curvature II

Let ${L=\Delta-\nabla\varphi\cdot\nabla}$ be a symmetric diffusion operator with an invariant measure ${d\mu=e^{-\varphi}dx}$ on a complete Riemannian manifold. In this paper we prove Li–Yau gradient estimates for weighted elliptic equations on the complete manifold with ${|\nabla \varphi| \leq \theta}$ and ∞-dimensional Bakry–Émery Ricci curvature bounded below by some negative constant. Based on this, we give an upper bound on the first eigenvalue of the diffusion operator L on this kind manifold, and thereby generalize a Cheng’s result on the Laplacian case (Math Z, 143:289–297, 1975).     

Bounds of Singular Integrals on Weighted Hardy Spaces and Discrete Littlewood–Paley Analysis

We apply the discrete version of Calderón??s reproducing formula and Littlewood?CPaley theory with weights to establish the $H^{p}_{w} \to H^{p}_{w}$ (0<p<??) and $H^{p}_{w}\to L^{p}_{w}$ (0<p??1) boundedness for singular integral operators and derive some explicit bounds for the operator norms of singular integrals acting on these weighted Hardy spaces when we only assume w??A _??. The bounds will be expressed in terms of the A _q constant of w if q>q _w =inf?{s:w??A _s }. Our results can be regarded as a natural extension of the results about the growth of the A _p constant of singular integral operators on classical weighted Lebesgue spaces $L^{p}_{w}$ in Hytonen et al. (arXiv:1006.2530, 2010; arXiv:0911.0713, 2009), Lerner (Ill.?J.?Math. 52:653?C666, 2008; Proc. Am. Math. Soc. 136(8):2829?C2833, 2008), Lerner et?al. (Int.?Math. Res. Notes 2008:rnm 126, 2008; Math. Res. Lett. 16:149?C156, 2009), Lacey et?al. (arXiv:0905.3839v2, 2009; arXiv:0906.1941, 2009), Petermichl (Am. J. Math. 129(5):1355?C1375, 2007; Proc. Am. Math. Soc. 136(4):1237?C1249, 2008), and Petermichl and Volberg (Duke Math. J. 112(2):281?C305, 2002). Our main result is stated in Theorem?1.1. Our method avoids the atomic decomposition which was usually used in proving boundedness of singular integral operators on Hardy spaces.     

On the Singular Spectrum for Adiabatic Quasi-Periodic Schrödinger Operators on the Real Line

In this paper, we study spectral properties of a family of quasi-periodic Schrödinger operators on the real line in the adiabatic limit. We assume that the adiabatic iso-energetic curves are extended along the momentum direction. In the energy intervals where this happens, we obtain an asymptotic formula for the Lyapunov exponent, and show that the spectrum is purely singular.Résumé. Cet article est consacré à létude du spectre dune famille dopérateurs de Schrödinger quasi-périodiques sur laxe réel lorsque les courbes iso-énergétiques adiabatiques sont non bornées dans la direction des moments. Dans des intervalles dénergies où cette propriété est vérifiée, nous obtenons une formule asymptotique pour lexposant de Lyapounoff, et nous démontrons que le spectre est purement singulier.Communicated by Bernard Helffersubmitted 17/06/03, accepted 05/03/04     

Strongly Singular Calderon-Zygmund Operator and Commutator on Morrey Type Spaces

In this paper, the author considers the boundedness of strongly singular Calderdn Zygmund operator and commutator generated by this operator and Lipschitz function on the classical Morrey space and generalized Morrey space. Moreover, the boundedness of strongly singular Calderón- Zygmund operator on the predual of Morrey space is discussed.     

Spectrum of the Neumann–Poincaré Operator for Ellipsoids and Tunability

We consider tunability of the eigenvalues of the Neumann–Poincaré operator on ellipsoids. We show in particular that for any number \({\rm \lambda}\) with \({-1/2 < {\rm \lambda} < 1/2}\) there is an ellipsoid (a prolate spheroid or an oblate spheroid) on which \({\rm \lambda}\) is an eigenvalue of the Neumann–Poincaré operator. As a byproduct, we find that there is a domain in three dimensions, actually an oblate spheroid, on which 0 is an eigenvalue of the Neumann–Poincaré operator.     

Spectrum of the three-particle Schr?dinger operator on a one-dimensional lattice

We consider a system of three arbitrary quantum particles on a one-dimensional lattice interacting pairwise via attractive contact potentials. We prove that the discrete spectrum of the corresponding Schr?dinger operator is finite for all values of the total quasimomentum in the case where the masses of two particles are finite. We show that the discrete spectrum of the Schr?dinger operator is infinite in the case where the masses of two particles in a three-particle system are infinite.     

The Classification of Equiaffine Indefinite Flat Homogeneous Surfaces in ℝ4

In this paper a complete classification of equiaffine homogeneous surfaces in ⁴ with indefinite flat affine metric is given.