On the spectrum of semi-classical Witten-Laplacians and Schrödinger operators in large dimension

We investigate the low-lying spectrum of Witten–Laplacians on forms of arbitrary degree in the semi-classical limit and uniformly in the space dimension. We show that under suitable assumptions implying that the phase function has a unique local minimum one obtains a number of clusters of discrete eigenvalues at the bottom of the spectrum. Moreover, we are able to count the number of eigenvalues in each cluster. We apply our results to certain sequences of Schrödinger operators having strictly convex potentials and show that some well-known results of semi-classical analysis hold also uniformly in the dimension.     

单位球面中具有3个不同 Blaschke 特征值的 Blaschke 平行子流形

Blaschke张量A是单位球面S~n中子流形的M?bius微分几何的一个基本不变量,而A的特征值称为Blaschke特征值.作者研究了S~n中具有平行Blaschke张量的子流形(简称为Blaschke平行子流形).主要结果是对S~n中具有3个不同Blaschke特征值的Blaschke平行子流形进行了完全的分类.     

A global characterization of the Fu?ík spectrum for a system of ordinary differential equations

The Fu?ík spectrum for systems of second order ordinary differential equations with Dirichlet or Neumann boundary values is considered: it is proved that the Fu?ík spectrum consists of global C¹ surfaces, and that through each eigenvalue of the linear system pass exactly two of these surfaces. Further qualitative, asymptotic and symmetry properties of these spectral surfaces are given. Finally, related problems with nonlinearities which cross asymptotically some eigenvalues, as well as linear-superlinear systems are studied.     

Quasimodes and Bohr-Sommerfeld Conditions for the Toeplitz Operators

Abstract

This article is devoted to the quantization of the Lagrangian submanifolds in the context of geometric quantization. The objects we define are similar to the Lagrangian distributions of the cotangent phase space theory. We apply this to construct quasimodes for the Toeplitz operators and we state the Bohr-Sommerfeld conditions under the usual regularity assumption. To compare with the Bohr-Sommerfeld conditions for a pseudodifferential operator with small parameter, the Maslov index, defined from the vertical polarization, is replaced with a curvature integral, defined from the complex polarization. We also consider the quantization of the symplectomorphisms, the realization of semi-classical equivalence between two different quantizations of a symplectic manifold and the microlocal equivalences.     

Asymptotically Holomorphic Families of Symplectic Submanifolds

We construct a wide range of symplectic submanifolds in a compact symplectic manifold as the zero sets of asymptotically holomorphic sections of vector bundles obtained by tensoring an arbitrary vector bundle by large powers of the complex line bundle whose first Chern class is the symplectic form. We also show that, asymptotically, all sequences of submanifolds constructed from a given vector bundle are isotopic. Furthermore, we prove a result analogous to the Lefschetz hyperplane theorem for the constructed submanifolds. Submitted: December 1996, final version: October 1997     

CONVERGENCE OF LAPLACIAN SPECTRA FROM RANDOM SAMPLES

Eigenvectors and eigenvalues of discrete Laplacians are often used for manifold learning and nonlinear dimensionality reduction. Graph Laplacian is one widely used discrete laplacian on point cloud. It was previously proved by Belkin and Niyogithat the eigenvectors and eigenvalues of the graph Laplacian converge to the eigenfunctions and eigenvalues of the Laplace-Beltrami operator of the manifold in the limit of infinitely many data points sampled independently from the uniform distribution over the manifold. Recently, we introduced Point Integral method (PIM) to solve elliptic equations and corresponding eigenvalue problem on point clouds. In this paper, we prove that the eigenvectors and eigenvalues obtained by PIM converge in the limit of infinitely many random samples. Moreover, estimation of the convergence rate is also given.     

Sandpile groups and the coeulerian property for random directed graphs

We consider random directed graphs, and calculate the distribution of the cokernels of their laplacian, following the methods used by Wood. As a corollary, we show that the probability that a random digraph is coeulerian is asymptotically upper bounded by a constant around 0.43.     

QUANTUM PHENOMENON OF THE ENERGY DENSITY OF A HARMONIC MAP TO A SPHERE

This paper proves that if the energy density of a harmonic map to a unit sphere varies between two successive half eigenvalues, then it must be one of them. Applying this result to the Gaussian maps of some submanifolds, the quantum phenomena of the square length of the second fundamental forms of these submanifolds is obtained. Some related topics are discussed in this note.     

关于Finsler子流形的平均曲率

通过使用由射影球丛诱导的体积元来研究Finsler子流形几何,推导了体积泛函的第一变分公式。给出了Finsler子流形的平均曲率形式和第二基本形式的定义,该定义在Riemannian情形下与通常的概念一致．此外,通过推导射影球丛纤维上的散度公式。给出了平均曲率形式的一种非常简洁的等价表示,并得到一些关于Minkowski空间中Finsler子流形的有趣的结果．     

Harmonic projection methods for large non-symmetric eigenvalue problems

The problem of finding interior eigenvalues of a large nonsymmetric matrix is examined. A procedure for extracting approximate eigenpairs from a subspace is discussed. It is related to the Rayleigh–Ritz procedure, but is designed for finding interior eigenvalues. Harmonic Ritz values and other approximate eigenvalues are generated. This procedure can be applied to the Arnoldi method, to preconditioning methods, and to other methods for nonsymmetric eigenvalue problems that use the Rayleigh–Ritz procedure. The subject of estimating the boundary of the entire spectrum is briefly discussed, and the importance of preconditioning for interior eigenvalue problems is mentioned. © 1998 John Wiley & Sons, Ltd.     

The first eigenvalue for a quasilinear Schrödinger operator and its application

In this paper, we study the first eigenvalue for a quasilinear Schrödinger operator, which is greater than the first eigenvalue of the usual laplacian operator. As an application we treat a quasilinear resonance problem involving a subcritical growth perturbation.     

Asymptotic distribution of Wishart matrix for block-wise dispersion of population eigenvalues

This paper deals with the asymptotic distribution of Wishart matrix and its application to the estimation of the population matrix parameter when the population eigenvalues are block-wise infinitely dispersed. We show that the appropriately normalized eigenvectors and eigenvalues asymptotically generate two Wishart matrices and one normally distributed random matrix, which are mutually independent. For a family of orthogonally equivariant estimators, we calculate the asymptotic risks with respect to the entropy or the quadratic loss function and derive the asymptotically best estimator among the family. We numerically show (1) the convergence in both the distributions and the risks are quick enough for a practical use, (2) the asymptotically best estimator is robust against the deviation of the population eigenvalues from the block-wise infinite dispersion.     

Lagrangean heuristics combined with reoptimization for the 0-1 bidimensional knapsack problem

First, this paper deals with lagrangean heuristics for the 0-1 bidimensional knapsack problem. A projected subgradient algorithm is performed for solving a lagrangean dual of the problem, to improve the convergence of the classical subgradient algorithm. Secondly, a local search is introduced to improve the lower bound on the value of the biknapsack produced by lagrangean heuristics. Thirdly, a variable fixing phase is embedded in the process. Finally, the sequence of 0-1 one-dimensional knapsack instances obtained from the algorithm are solved by using reoptimization techniques in order to reduce the total computational time effort. Computational results are presented.     

一类逼近l1精确罚函数的罚函数

本文对可微非线性规划问题提出了一个渐近算法,它是基于一类逼近l1精确罚函数的罚函数而提出的,我们证明了算法所得的极小点列的聚点均为原问题的最优解,并在Mangasarian-Fromovitz约束条件下,证明了有限次迭代之后,所有迭代均为可行的,即迭代所得的极小点为可行点.     

A Generalization of a Levitin and Parnovski Universal Inequality for Eigenvalues

In this paper, we derive “universal” inequalities for the sums of eigenvalues of the Hodge de Rham Laplacian on Euclidean closed submanifolds and of eigenvalues of the Kohn Laplacian on the Heisenberg group. These inequalities generalize the Levitin–Parnovski inequality obtained for the sums of eigenvalues of the Dirichlet Laplacian of a bounded Euclidean domain.     

Pseudo-parallel Lagrangian submanifolds in complex space forms

In this work we study pseudo-parallel Lagrangian submanifolds in a complex space form. We give several general properties of pseudo-parallel submanifolds. For the 2-dimensional case, we show that any minimal Lagrangian surface is pseudo-parallel. We also give examples of non-minimal pseudo-parallel Lagrangian surfaces. Here we prove a local classification of the pseudo-parallel Lagrangian surfaces. In particular, semi-parallel Lagrangian surfaces are totally geodesic or flat. Finally, we give examples of pseudo-parallel Lagrangian surfaces which are not semi-parallel.     

The theory of regularized traces of Sturm-Liouville operators as applied to approximate calculation of eigenvalues and eigenfunctions of certain singular operators

The effectiveness of the theory of regularized traces as applied to the approximate calculation of eigenvalues and eigenfunctions is demonstrated for certain singular differential operators. Singular operators of the Bessel type and operators from fluid dynamics and mathematical physics are considered.     

N-body quantum scattering theory in two Hilbert spaces. III. Theory of approximations

A rigorous mathematical theory of approximations is developed for abstract nonrelativistic quantum scattering systems within the two-Hilbert-space framework. An approximate space of asymptotic states and an approximate asymptotic Hamiltonian must be specified initially. An approximate N-particle Hamiltonian is then constructed and proved to be self-adjoint. Approximate wave operators are shown to exist and, in certain interesting cases, to be asymptotically complete. Certain sequences of the approximate wave operators are proved to converge to the exact wave operators in an appropriate limit. Thus approximate scattering operators are unitary and converge to the exact scattering operator.     

Classification of the Blaschke Isoparametric Hypersurfaces with Three Distinct Blaschke Eigenvalues

An immersed umbilic-free submanifold in the unit sphere is called Blaschke isoparametric if its Möbius form vanishes identically and all of its Blaschke eigenvalues are constant. Then the classification of Blaschke isoparametric hypersurfaces is natural and interesting in the Möbius geometry of submanifolds. In this paper, we give a classification of the Blaschke isoparametric hypersurfaces with three distinct Blaschke eigenvalues one of which is simple.     

Classification results for biharmonic submanifolds in spheres

We study biharmonic submanifolds of the Euclidean sphere that satisfy certain geometric properties. We classify: (i) the biharmonic hypersurfaces with at most two distinct principal curvatures; (ii) the conformally flat biharmonic hypersurfaces. We obtain some rigidity results for pseudoumbilical biharmonic submanifolds of codimension 2 and for biharmonic surfaces with parallel mean curvature vector field. We also study the type, in the sense of B-Y. Chen, of compact proper biharmonic submanifolds with constant mean curvature in spheres. Dedicated to Professor Vasile Oproiu on his 65th birthday The first author was supported by a INdAM doctoral fellowship, Italy. The second author was supported by PRIN 2005, Italy. The third author was supported by Grant CEEX ET 5871/2006, Romania