On the Existence of WKB-Type Asymptotics for the Generalized Eigenvectors of Discrete String Operators

Let J be a Jacobi real symmetric matrix on l₂ with zero diagonaland non-diagonal entries of the form {1 + p_n}. If p_n–1± p_n = O(n^–) with some 2/3, then the existenceof bounded solutions of Ju = u is proved for almost every (–2, 2) with the WKB-type asymptotic behavior. 2000 MathematicsSubject Classification 47B36, 47B37, 47B39.     

The Discrete Spectrum Asymptotics with Respect to a Large Coupling Constant under Strong Nonnegative Perturbations

Let A be a self-adjoint operator, let (, ) be an inner gap in the spectrum of the operator A, and let B(t) = A + tW ^* W, where the operator W(A – iI)^-1 is not necessarily bounded. Conditions are obtained under which the spectrum of B(t) in (, ) is discrete. Let N(, A, W, ), (, ), > 0, be the number of eigenvalues of the operator B(t) passing the point (, ) as t increases from 0 to . The asymptotics of N(, A, W, ) as + is obtained in terms of the spectral asymptotics of a certain self-adjoint compact operator. Bibliography: 5 titles.     

Asymptotics of the Discrete Chebyshev Polynomials

The discrete Chebyshev polynomials are orthogonal with respect to a distribution, which is a step function with jumps one unit at the points , N being a fixed positive integer. By using a double integral representation, we have recently obtained asymptotic expansions for in the double scaling limit, namely, and , where and ; see [8]. In this paper, we continue to investigate the behavior of these polynomials when the parameter b approaches the endpoints of the interval (0, 1). While the case is relatively simple (because it is very much like the case when b is fixed), the case is quite complicated. The discussion of the latter case is divided into several subcases, depending on the quantities n, x, and , and different special functions have been used as approximants, including Airy, Bessel, and Kummer functions.     

Discrete String Simulating the Musical Scale

A discrete string with fixed endpoints carrying a finite number of beads is determined by the masses of beads and the distances between them. The string possesses a set of simple eigenfrequencies corresponding to harmonic eigenmodes. In this paper, the following problem is studied: to find a discrete string carrying seven beads such that its eigenfrequencies coincide with the freqiencies of the notes of the first octave of the musical scale. The problem is solved in two steps. First, the spectral inverse problem is considered, i.e., we recover the string from its spectrum and a set of constants related to the normalized eigenmodes. A procedure of solving this problem is described. One of the main results of the paper is a necessary and sufficient condition for the solvability of the spectral inverse problem. The second step is numerical realization of the procedure. Bibliography: 3 titles.     

Prices and Asymptotics for Discrete Variance Swaps

Abstract

We study the fair strike of a discrete variance swap for a general time-homogeneous stochastic volatility model. In the special cases of Heston, Hull–White and Schöbel–Zhu stochastic volatility models, we give simple explicit expressions (improving Broadie and Jain (2008a). The effect of jumps and discrete sampling on volatility and variance swaps. International Journal of Theoretical and Applied Finance, 11(8), 761–797) in the case of the Heston model). We give conditions on parameters under which the fair strike of a discrete variance swap is higher or lower than that of the continuous variance swap. The interest rate and the correlation between the underlying price and its volatility are key elements in this analysis. We derive asymptotics for the discrete variance swaps and compare our results with those of Broadie and Jain (2008a. The effect of jumps and discrete sampling on volatility and variance swaps. International Journal of Theoretical and Applied Finance, 11(8), 761–797), Jarrow et al. (2013. Discretely sampled variance and volatility swaps versus their continuous approximations. Finance and Stochastics, 17(2), 305–324) and Keller-Ressel and Griessler (2012. Convex order of discrete, continuous and predictable quadratic variation and applications to options on variance. Working paper. Retrieved from http://arxiv.org/abs/1103.2310).     

Asymptotics of Maxima of Discrete Random Variables

If X₁, X₂,..., X_n are independent and identically distributed discrete random variables and M_n=max (X₁,..., X_n) we examine the limiting behavior of (M_n–b(n))/a(n) as n . It is well known that for discrete distributions such as Poisson and geometric the limiting distribution is not non-degenerate. However, by tuning the parameters of the discrete distribution to vary as n , it is possible to obtain non-degenerate limits for (M_n–b(n))/a(n). We consider four families of discrete distributions and show how this can be done.     

Relative Discrete Spectrum and Joinings

 A joining characterization of ergodic isometric extensions is given. We also give a simple joining proof of a relative version of the Halmos-von Neumann theorem.     

Relative Discrete Spectrum and Joinings

 A joining characterization of ergodic isometric extensions is given. We also give a simple joining proof of a relative version of the Halmos-von Neumann theorem. Research partly supported by KBN grant 2 P03A 002 14 (1998). Received June 5, 2001; in revised form March 4, 2002     

Semiclassical Asymptotics of the Spectrum of the Subcritical Harper Operator

Mathematical Notes -     

一类自伴微分算子谱的离散性

本文研究了２ｎ阶实系数Ｅｕｌｅｒ微分算式生成的对称微分算子,得到了自伴Ｅｕｌｅｒ微分算子的谱是离散的充分必要条件．     

On the Discrete Spectrum of a Family of Differential Operators

We consider a family A of differential operators in L ²(²) depending on a parameter 0. The operator A formally corresponds to the quadratic form

Asymptotics of Discrete Riesz d-Polarization on Subsets of d-Dimensional Manifolds

We prove a conjecture of T. Erdélyi and E.B. Saff, concerning the form of the dominant term (as N?→?∞) of the N-point Riesz d-polarization constant for an infinite compact subset A of a d-dimensional C ¹-manifold embedded in ? ^m (d?≤?m). Moreover, if we assume further that the d-dimensional Hausdorff measure of A is positive, we show that any asymptotically optimal sequence of N-point configurations for the N-point d-polarization problem on A is asymptotically uniformly distributed with respect to \(\mathcal H_d|_A\) . These results also hold for finite unions of such sets A provided that their pairwise intersections have \(\mathcal H_d\) -measure zero.     

Boundary Area Growth and the Spectrum of Discrete Laplacian

We introduce the boundary area growth as a new quantity for an infinite graph. Using this, we give some upper bounds for the bottom of the spectrum of the discrete Laplacian which relates closely to the transition operator. We also give some applications and examples.     

Asymptotics of Eigenvalues for a Class of Jacobi Matrices with Limiting Point Spectrum

In this paper, we study a class of Jacobi matrices with very rapidly decreasing weights. It is shown that the Weyl function (the matrix element of the resolvent of the operator) for the class under study can be expressed as the ratio of two entire transcendental functions of order zero. It is shown that the coefficients in the expansion of these functions in Taylor series are proportional to the generating functions of the number of integral solutions defined by certain Diophantine equations. An asymptotic estimate for the eigenvalues is obtained.     

Asymptotics and Blow-up for Mass Critical Nonlinear Dispersive Equations

The author considers mass critical nonlinear Schr（o）dinger and Korteweg-de Vries equations.A review on results related to the blow-up of solution of these equations is given.     

A Discrete Leading Symbol and Spectral Asymptotics for Natural Differential Operators

We initiate a systematic study of natural differential operators in Riemannian geometry whose leading symbols are not of Laplace type. In particular, we define a discrete leading symbol for such operators which may be computed pointwise, or from spectral asymptotics. We indicate how this can be applied to the computation of another kind of spectral asymptotics, namely asymptotic expansions of fundamental solutions, and to the computation of conformally covariant operators.     

On the Discrete Spectrum of Generalized Quantum Tubes

The spectrum (of the Dirichlet Laplacian) of non-compact, non-complete Riemannian manifolds is much less understood than their compact counterparts. In particular it is often not even known whether such a manifold has any discrete spectra. In this article, we will prove that a certain type of non-compact, non-complete manifold called the quantum tube has non-empty discrete spectrum. The quantum tube is a tubular neighborhood built about an immersed complete manifold in Euclidean space. The terminology of “quantum” implies that the geometry of the underlying complete manifold can induce discrete spectra – hence quantization. We will show how the Weyl tube invariants appear in determining the existence of discrete spectra. This is an extension and generalization, on the geometric side, of the previous work of the author on the “quantum layer.”