Semiclassical asymptotic behavior of orthogonal polynomials
- PDF / 547,508 Bytes
- 35 Pages / 439.37 x 666.142 pts Page_size
- 83 Downloads / 223 Views
Semiclassical asymptotic behavior of orthogonal polynomials D. R. Yafaev1,2 Received: 16 February 2020 / Revised: 21 June 2020 / Accepted: 3 July 2020 © Springer Nature B.V. 2020
Abstract Our goal is to find asymptotic formulas for orthonormal polynomials Pn (z) with the recurrence coefficients slowly stabilizing as n → ∞. To that end, we develop scattering theory of Jacobi operators with long-range coefficients and study the corresponding second-order difference equation. We introduce the Jost solutions f n (z) of this equation by a condition for n → ∞ and suggest an Ansatz for them playing the role of the semiclassical Liouville–Green Ansatz for the corresponding solutions of the Schrödinger equation. This allows us to study Jacobi operators and their eigenfunctions Pn (z) by traditional methods of spectral theory developed for differential equations. In particular, we express all coefficients in asymptotic formulas for Pn (z) as → ∞ in terms of the Wronskian of the solutions {Pn (z)} and { f n (z)}. Keywords Jacobi matrices · Long-range perturbations · Difference equations · Orthogonal polynomials · Asymptotics for large numbers Mathematics Subject Classification 33C45 · 39A70 · 47A40 · 47B39
1 Introduction 1.1 Jacobi and orthogonal polynomials As is well known, the theories of Jacobi operators given by three-diagonal matrices
Supported by project Russian Science Foundation 17-11-01126.
B
D. R. Yafaev [email protected]
1
CNRS, IRMAR-UMR 6625, Univ Rennes, 35000 Rennes, France
2
SPGU, Univ. Nab. 7/9, Saint Petersburg, Russia 199034
123
D. R. Yafaev
⎛
b0 ⎜a0 ⎜ ⎜ J =⎜0 ⎜0 ⎝ .. .
a0 b1 a1 0 .. .
0 a1 b2 a2 .. .
0 0 a2 b3 .. .
0 0 0 a3 .. .
⎞ ··· · · ·⎟ ⎟ · · ·⎟ ⎟ · · ·⎟ ⎠ .. .
(1.1)
in the space 2 (Z+ ) and of differential Schrödinger operators H = Dp(x)D + q(x) (with, for example, the boundary condition u(0) = 0) in the space L 2 (R+ ) are to a large extent similar. For Jacobi operators, n ∈ Z+ plays the role of x ∈ R+ and the coefficients an , bn , play the roles of the functions p(x), q(x), respectively. In our opinion, a consistent analogy between Jacobi and Schrödinger operators sheds a new light on some aspects of the orthogonal polynomials theory. Of course, this point of view is not new; for example, it was advocated long ago by Case [6]. In this paper, the sequences an > 0 and bn = b¯n in (1.1) are assumed to be bounded, so that J is a bounded self-adjoint operator in the space 2 (Z+ ). Its spectral family will be denoted E(λ). The spectrum of J is simple with e0 = (1, 0, 0, . . .) being a generating vector. It is natural to define the spectral measure of J by the relation dρ(λ) = d(E(λ)e0 , e0 ). Orthogonal polynomials Pn (z) associated with the Jacobi matrix (1.1) are defined by the recurrence relation an−1 Pn−1 (z) + bn Pn (z) + an Pn+1 (z) = z Pn (z), n ∈ Z+ ,
(1.2)
and the boundary conditions P−1 (z) = 0, P0 (z) = 1. Obviously, Pn (z) is a polynomial of degree n and the vector P(z) = {Pn (z)}∞ n=−1 formally satisfies the equation J P(z) = z P(z), that is, it is an “eigenv
Data Loading...
