Calculating the branch points of the eigenvalues of the Coulomb spheroidal wave equation

A method for computing the eigenvalues λ _mn (b, c) and the eigenfunctions of the Coulomb spheroidal wave equation is proposed in the case of complex parameters b and c. The solution is represented as a combination of power series expansions that are then matched at a single point. An extensive numerical analysis shows that certain b _s and c _s are second-order branch points for λ _mn (b, c) with different indices n ₁ and n ₂, so that the eigenvalues at these points are double. Padé approximants, quadratic Hermite-Padé approximants, the finite element method, and the generalized Newton method are used to compute the branch points b _s and c _s and the double eigenvalues to high accuracy. A large number of these singular points are calculated.     

Power series expansions for spheroidal wave functions with small arguments

Power series expansions for the angular spheroidal wave functions of the first kind S_mn(c,η), with small arguments c, are derived for general integer values of m and n. The various evaluated expansion coefficients can also be used in the calculation of the corresponding angular functions of the second kind, as well as for the radial functions of any kind. Only the prolate functions are considered explicitly, but corresponding formulas for the oblate ones are obtained immediately.     

Convergence of ray sequences of Padé approximants for 2 f 1(a, 1; c; z), (c > a > 0)

The Padé table of ₂ F ₁(a, 1; c; z) is normal for c > a > 0 (cf. [4]). For m ≥ n - 1 and c ? Z^-, the denominator polynomial Q _mn (z) in the [m/n] Padé approximant P _mn (z)/Q _mn (z) for ₂ F ₁(a, 1; c; z) and the remainder term Q _mn (z)₂ F ₁(a, 1; c; z)-P_mn (z) were explicitly evaluated by Padé (cf. [2], [6] or [9]). We show that for c > a > 0 and m ≥ n - 1, the poles of P_mn (z)/Q_mn (z) lie on the cut (1,∞). We deduce that the sequence of approximants P_mn (z)/Q_mn (z) converges to ₂ F ₁(a, 1; c; z) as m → ∞, n/m → ρ with 0 < ρ ≤ 1, uniformly on compact subsets of the unit disc |z| < 1 for c > a > 0.     

Numerical analysis of the spectrum of the Orr-Sommerfeld problem

A high-accuracy method for computing the eigenvalues λ _n and the eigenfunctions of the Orr-Sommerfeld operator is developed. The solution is represented as a combination of power series expansions, and the latter are then matched. The convergence rate of the expansions is analyzed by applying the theory of recurrence equations. For the Couette and Poiseuille flows in a channel, the behavior of the spectrum as the Reynolds number R increases is studied in detail. For the Couette flow, it is shown that the eigenvalues λ _n regarded as functions of R have a countable set of branch points R _k > 0 at which the eigenvalues have a multiplicity of 2. The first ten of these points are presented within ten decimals.     

The Eigenvalues of the Angular Spheroidal Wave Equation

Approximate relations are obtained between the eigenvalues λ and the ellipticity parameter c² of the angular spheroidal wave equation. Although based on WKBJ methods and the assumption that λ is large, the relations are useful throughout the complex c²-plane. They are exact at c² = 0, and reproduce the standard asymptotic formulas for λ when c² is large. At intermediate values of c², they provide approximations for the square-root branch points of the multivalued function λ(c²) in the complex c²-plane at which adjacent eigenvalues of the same class become equal in pairs. These branch points lie on an infinite sequence of distorted circular rings. Their exact locations have been computed for the first four rings for angular wavenumbers m = 0,…,4.     

On balanced claw designs of complete multi-partite graphs

In this paper, it is shown that a necessary and sufficient condition for the existence of a balanced claw design BCD(m, n, c, λ) of a complete m-partite graph λK_m(n, n,…,n) is λ(m - 1)n ≡ 0 (mod 2c) and (m - 1)n ? c.     

Eigenvalues of the Manifold Sp（n）/U（n）

In this paper, we give the eigenvalues of the manifold Sp(n)/U(n). We prove that an eigenvalue λ _s (f ₂, f ₂, …, f _n ) of the Lie group Sp(n), corresponding to the representation with label (f ₁, f ₂, ..., f _n ), is an eigenvalue of the manifold Sp(n)/U(n), if and only if f ₁, f ₂, …, f _n are all even.     

Differentiable Eigenvalues of Perturbed Quadratic Matrix Functions

Let T(λ, ε ) = λ² + λC + λεD + K be a perturbed quadratic matrix polynomial, where C, D, and K are n × n hermitian matrices. Let λ₀ be an eigenvalue of the unperturbed matrix polynomial T(λ, 0). With the falling part of the Newton diagram of det T(λ, ε), we find the number of differentiable eigenvalues. Some results are extended to the general case L(λ, ε) = λ² + λD(ε) + K, where D(ε) is an analytic hermitian matrix function. We show that if K is negative definite on Ker L(λ₀, 0), then every eigenvalue λ(ε) of L(λ, ε) near λ₀ is analytic.     

Ruzsa’s constant on additive functions

A function f : N → R is called additive if f(mn)= f(m)+f(n)for all m, n with(m, n)= 1. Let μ(x)= max n≤x(f(n)f(n + 1))and ν(x)= max n≤x(f(n + 1)f(n)). In 1979, Ruzsa proved that there exists a constant c such that for any additive function f , μ(x)≤ cν(x 2 )+ c f , where c f is a constant depending only on f . Denote by R af the least such constant c. We call R af Ruzsa's constant on additive functions. In this paper, we prove that R af ≤ 20.     

The existence and convergence of subsequences of Padé approximants

We prove the existence of an infinite number of Padé approximants, and thereby remedy a defect in Nuttall's theorem. It is proved that the sequences of Padé approximants shown by Perron, Gammel, and Wallin to be everywhere divergent contain subsequences which are everywhere convergent. It is further proved that there always exist, for entire functions, everywhere convergent [1, N] and [2, N] subsequences of Padé approximants. There must exist subsequences of [m, N] Padé approximants (N → ∞) which converge almost everywhere in $\text{¦z¦ ? ? < R}$ to functions f(z) which are regular except for a finite number (n ? m) of poles in $\text{¦z¦ < R}$. We prove convergence of the [N, N + j] Padé approximants in the mean on the Riemann sphere for meromorphic functions.     

A New Algorithm for the Symmetric Tridiagonal Eigenvalue Problem

We apply a novel approach to approximate within ϵ to all the eigenvalues of an n × n symmetric tridiagonal matrix A using at most n²([3 log₂(625n⁶)] + (83n − 34)[log₂ (log₂((λ₁ − λ_n)/(2ϵ))/log₂(25_n))]) arithmetic operations where λ₁ and λ_n denote the extremal eigenvalues of A. The algorithm can be modified to compute any fixed numbers of the largest and the smallest eigenvalues of A and may also be applied to the band symmetric matrices without their reduction to the tridiagonal form.     

A centerless representation of the Virasoro algebra associated with the unitary circular ensemble

We consider the 2-dimensional Toda lattice tau functions τ_n(t,s;η,θ) deforming the probabilities τ_n(η,θ) that a randomly chosen matrix from the unitary group U(n), for the Haar measure, has no eigenvalues within an arc (η,θ) of the unit circle. We show that these tau functions satisfy a centerless Virasoro algebra of constraints, with a boundary part in the sense of Adler, Shiota and van Moerbeke. As an application, we obtain a new derivation of a differential equation due to Tracy and Widom, satisfied by these probabilities, linking it to the Painlevé VI equation.     

Trace Formulas for Some Singular Differential Operators and Applications

We characterize the convergence of the series ∑ λ^–1_n, where λⁿ are the non‐zero eigenvalues of some boundary value problems for degenerate second order ordinary differential operators and we prove a formula for the above sum when the coefficient of the zero‐order term vanishes. We study these operators both in weighted Hilbert spaces and in spaces of continuous functions. After investigating the boundary behaviour of the eigenfunctions, we give applications to the regularity of the generated semigroups.     

Perturbation of resonances in quantum mechanics

Schrödinger operators on L²($\text{R}$³) of the form ?Δ + V_λ with potentials V_λ real-analytic in λ are discussed. The analytic structure in V_λ and k (with k² the energy variable) of the resolvent kernel, the eigenvalues and resonances is exhibited and we obtain in particular convergent perturbation expansions for the resonances and the corresponding resonance functions. The lower order expansion coefficients are computed explicitly. The resonances and the corresponding functions are also computed for a particle moving under the action of n point interactions. This gives asymptotic low energy information about Schrödinger Hamiltonians with short range potentials. The perturbation theory of resonances, eigenvalues and of the corresponding functions for Hamiltonians describing n point interactions perturbed by a potential is also given.     

Convergence of rows of the Padé table

We prove that at least an infinite subsequence of [$\text{l}\text{2}$] Padé approximants converge to f(z) if f(z) is holomorphic. We speculate that convergence of the [$\text{L ?}\text{m}\text{μ + m}$] approximants to c(z) is associated with convergence of [$\text{L}\text{μ}$] approximants to h(z) where c(z) is meromorphic with μ poles and σ(z) is the polynomial of degree μ which renders g(z) = σ(z)c(z) and h(z) = σ(z)g(z) holomorphic. We formulate this conjecture precisely and prove it for (i) m = 1 and (ii) m = 2 and h(z) a holomorphic function of order less than 1.     

Orthogonal polynomials with complex-valued weight function,I

In this paper we investigate the asymptotic behavior of polynomialsQ _mn(z), m, n ∈ N, of degree ≤n that satisfy the orthogonal relation $$\oint_c {\zeta ^l Q_{mn} (\zeta )} \frac{{f(\zeta )d\zeta }}{{\omega _{m + n} (\zeta )}} = 0,l = 0,...,n - 1,$$ where/tf(z) is a function, which is supposed to be analytic on a continuum \(V \subseteq \hat C\) and all its singularities are supposed to be contained in a set \(E \subseteq \hat C\) of capacity zero, ω _m+n (z) is a polynomial of degreem+n+1 with all its zeros contained inV, andC is a curve separatingV from the setE. We show that if the zeros of ω _m+n have a certain asymptotic distribution form+n → ∞ and ifm/n ar 1, then the zeros of the polynomialsQ _mn have a unique asymptotic distribution, which is closely related with the extremal domainD for single-valued analytic continuation of the functionf(z). The results are essential for the investigation of Padé and best rational approximants to the functionf(z).     

On the eigenvalues of the fredholm operator

We prove that if ω(t, x, K ₂ ^(m) )?c(x)ω(t) for allxε[a, b] andx ε [0,b-a] wherec ∈L ¹(a, b) and ω is a modulus of continuity, then λ _n =O(n ^?m-1/2ω(1/n)) and this estimate is unimprovable.     

The combinatorial structure of (m,n)-convex sets

LetS be a closed subset of a Hausdorff linear topological space,S having no isolated points, and letc _s (m) denote the largest integern for whichS is (m,n)-convex. Ifc _s (k)=0 andc _s (k+1)=1, then $$ c_s \left( m \right) = \sum\limits_{i = 1}^k {\left( {\begin{array}{*{20}c} {\left[ {\frac{{m + k - i}} {k}} \right]} \\ 2 \\ \end{array} } \right)} $$ . Moreover, ifT is a minimalm subset ofS, the combinatorial structure ofT is revealed.     

Some limit theorems on the eigenvectors of large dimensional sample covariance matrices

Let {v_ij} i,j = 1, 2,…, be i.i.d. standardized random variables. For each n, let V_n = (v_ij) i = 1, 2,…, n; j = 1, 2,…, s = s(n), where $\text{(}\text{n}\text{s}\text{) → y > 0}$ as n → ∞, and let $\text{M}{}_{\mathrm{n}}\text{= (}\text{1}\text{s}\text{)V}{}_{\mathrm{n}}\text{V}{}_{\mathrm{n}}{}^{\mathrm{T}}$. Previous results [7, 8] have shown the eigenvectors of M_n to display behavior, for n large, similar to those of the corresponding Wishart matrix. A certain stochastic process X_n on [0, 1], constructed from the eigenvectors of M_n, is known to converge weakly, as n → ∞, on D[0, 1] to Brownian bridge when v₁₁ is N(0, 1), but it is not known whether this property holds for any other distribution. The present paper provides evidence that this property may hold in the non-Wishart case in the form of limit theorems on the convergence in distribution of random variables constructed from integrating analytic function w.r.t. X_n(F_n(x)), where F_n is the empirical distribution function of the eigenvalues of M_n. The theorems assume certain conditions on the moments of v₁₁ including E(v₁₁⁴) = 3, the latter being necessary for the theorems to hold.     

Recovery of a function from the coefficients of its Dirichlet series

Let L(λ) be an entire function of exponential type, letγ(t) be the function associated with L(λ) in the sense of Borel, let \(\bar D\) be the smallest closed convex set containing all the singular points ofγ(t), let λ₀, λ₁, ..., λ_n, ... be the simple zeros of L(λ), and let A \(\bar D\) be the space of functions analytic on \(\bar D\) with the topology of the inductive limit. With an arbitraryf (z) ∈ A( \(\bar D\) ) we can associate the series whereC is a closed contour containing \(\bar D\) , on and inside of whichf (z) is analytic. We give a method of recoveringf (z) from the Dirichlet coefficientsa _n.