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1.
We consider asymptotics of orthogonal polynomials with respect to weights w(x)dx = e−Q(x) dx on the real line, where Q(x) = Σ qk xk, q2m > 0, denotes a polynomial of even order with positive leading coefficient. The orthogonal polynomial problem is formulated as a Riemann‐Hilbert problem following [22, 23]. We employ the steepest‐descent‐type method introduced in [18] and further developed in [17, 19] in order to obtain uniform Plancherel‐Rotach‐type asymptotics in the entire complex plane, as well as asymptotic formulae for the zeros, the leading coefficients, and the recurrence coefficients of the orthogonal polynomials. © 1999 John Wiley & Sons, Inc. 相似文献
2.
O. A. Shveikina 《Differential Equations》2014,50(5):623-632
We consider the Sturm-Liouville operator L(y) = ?d 2 y/dx 2 + q(x)y in the space L 2[0, π], where the potential q(x) is a complex-valued distribution of the first order of singularity; i.e., q(x) = u′(x), u ∈ L 2[0, π]. (Here the derivative is understood in the sense of distributions.) We obtain asymptotic formulas for the eigenvalues and eigenfunctions of the operator in the case of the Neumann-Dirichlet conditions [y [1](0) = 0, y(π) = 0] and Neumann conditions [y [1](0) = 0, y [1](π) = 0] and refine similar formulas for all types of boundary conditions. The leading and second terms of asymptotics are found in closed form. 相似文献
3.
In this paper, we study the planar Hamiltonian system = J (A(θ)x + ▽f(x, θ)), θ = ω, x ∈ R2 , θ∈ Td , where f is real analytic in x and θ, A(θ) is a 2 × 2 real analytic symmetric matrix, J = (1-1 ) and ω is a Diophantine vector. Under the assumption that the unperturbed system = JA(θ)x, θ = ω is reducible and stable, we obtain a series of criteria for the stability and instability of the equilibrium of the perturbed system. 相似文献
4.
Julio Alcántara-Bode 《Integral Equations and Operator Theory》2005,53(3):301-309
It is proven that the set of eigenvectors and generalized eigenvectors associated to the non-zero eigenvalues of the Hilbert-Schmidt
(non nuclear, non normal) integral operator on L2(0, 1)
[Ar (a)f](q) = ò01 r( \fracaq x )f(x)dx [A_{\rho } (\alpha )f](\theta ) = {\int_0^1 {\rho {\left( {\frac{{\alpha \theta }} {x}} \right)}f(x)dx} } 相似文献
5.
Erhard Heinz Ralf Beyerstedt 《Calculus of Variations and Partial Differential Equations》1994,2(2):241-247
Letz=z(x, y) be a real-valued twice continuously differentiable solution of the elliptic Monge-Ampère equationAr+2Bs+Ct+rt – s
2=E in the punctured disk 0<(x–x
0)2+(y–y
0)2<2. Assume thatq is continuous at (x0, y0). Our aim is to give sufficient conditions on the coefficientsA,..., E which ensure that the singularity (x
0,y
0) is removable. This generalizes an earlier result of Jörgens (Math. Ann. 129 (1955), 330–344). 相似文献
6.
Consider the Hill's operator Q = ?d2/dx2 + q(x) in which q(x), 0 ≤ x ≤ 1, is a white noise. Denote by f(μ) the probability density function of ?λ0(q), the negative of the ground state eigenvalue, at μ. We prove the detailed asymptotics as μ → + ∞. This result is based on a precise Laplace analysis of a functional integral representation for f(μ) established by S. Cambronero and H. P. McKean in 5 . © 2005 Wiley Periodicals, Inc. 相似文献
7.
It is well known that every x ∈ (0, 1] can be expanded to an infinite Lüroth series in the form of
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