The asymptotic form of the Titchmarsh–Weyl <i>m</i>-function associated with a second order differential equation with locally integrable coefficient

Harris, Bernard

doi:10.1017/s0308210500026329

Cited by 14 publications

(13 citation statements)

References 9 publications

(10 reference statements)

Supporting

Mentioning

Contrasting

Order By: Relevance

“…[4], [5], [6], [7], [8], and [9,10,11]. In Section 2 we present a simplified proof of the result of Atkinson [7], who derived the first two terms in the asymptotic expansion of m(A) as A-»°° in a sector in the upper half of the complex plane, and show that the error bound is slightly sharper than indicated by Atkinson.…”

Section: The Titchmarsh-weyl M-coefficientmentioning

confidence: 93%

Asymptotics of the Titchmarsh-Weyl m-coefficient for integrable potentials

Kaper

Kwong

1986

Proceedings of the Royal Society of Edinburgh: Section A Mathem

View full text Add to dashboard Cite

SynopsisThis article is concerned with the asymptotic behaviour of m(A), the Titchmarsh-Weyl m-coefficient, for the singular eigenvalue equation y" + (A -q{x))y = 0 on [0, °°), as A-» °° in a sector in the upper half of the complex plane. It is assumed that the potential function q is integrable near 0. A simplified proof is given of a result of Atkinson [7], who derived the first two terms in the asymptotic expansion of m(A), and a sharper error bound is obtained. The proof is then generalised to derive subsequent terms in the asymptotic expansion. It is shown that the Titchmarsh-Weyl m-coefficient admits an asymptotic power series expansion if the potential function satisfies some smoothness condition. A simple method to compute the expansion coefficients is presented. The results for the first few coefficients agree with those given by Harris [9]. The Titchmarsh-Weyl m-coefficientConsider the singular eigenvalue equationwhere the (real-valued) potential function q is integrable near the origin: q e L\0, X) for some X > 0; A is a complex constant, Im A =£ 0. Let and 6 be the solutions to (1) that satisfy the initial conditions l, 4>'(0) = 0,These functions, which also depend on A, define a family of functions Each function M X {K, •) maps the real line onto a circle C x {k). If Im A > 0, the interior of C X (X) corresponds to the lower half of the z-plane. The closed disc D X (X) with boundary C X (X) is known as the Weyl disc; its diameter is diam D x {k) = Im (X)\ X \{t)\ 2 dt (4)

show abstract

Section: The Titchmarsh-weyl M-coefficientmentioning

confidence: 93%

Asymptotics of the Titchmarsh-Weyl m-coefficient for integrable potentials

Kaper

Kwong

1986

Proceedings of the Royal Society of Edinburgh: Section A Mathem

View full text Add to dashboard Cite

show abstract

“…This is the correct boundary condition forR(x, −∞). Let us assume that the solution h of equation (4.3) satisfies the same boundary condition as (4.11), 12) for any ξ and µ. Integrating (4.6) from z = −∞ to z = x, and using (4.12), we obtain…”

Section: Solution Of the Inhomogeneous Equationmentioning

confidence: 99%

“…[4][5][6][7][8][9][10][11][12][13][14][15][16] Since the Green function can be expressed solely in terms of R r (x, −∞; k) and R l (∞, x; k) as shown in (1.11), all the information about the behavior of solutions can be obtained through the analysis of the reflection coefficients. The study of the reflection coefficients is also important in spectral theory and inverse problems.…”

Section: Introductionmentioning

confidence: 99%

Formulation of a unified method for low- and high-energy expansions in the analysis of reflection coefficients for one-dimensional Schrödinger equation

Miyazawa

2015

Journal of Mathematical Physics

View full text Add to dashboard Cite

We study low-energy expansion and high-energy expansion of reflection coefficients for one-dimensional Schrödinger equation, from which expansions of the Green function can be obtained. Making use of the equivalent Fokker-Planck equation, we develop a generalized formulation of a method for deriving these expansions in a unified manner. In this formalism, the underlying algebraic structure of the problem can be clearly understood, and the basic formulas necessary for the expansions can be derived in a natural way. We also examine the validity of the expansions for various asymptotic behaviors of the potential at spatial infinity.

show abstract

“…To get more terms in the masymptotics, as well as higher-order error bounds, therefore requires that a better estimate for the first error term in (3.8) be obtained, although the estimation of V2(Tp) in (4.2) seems to be optimal. To this end, an alternative iterative approach along the lines of [12,[14][15][16] can be expected to provide a suitable replacement for theorem 3.2, which would yield refinements to (3.13), (3.15) and (3.16). At the same time, improvements for the higher-order terms in (3.13) require that more terms in the expansions (1.13) and (1.16) of lemma 1.1 be obtained and that a less restrictive definition of a(x) be employed so that the higher-order error terms can be exploited.…”

Section: Examplesmentioning

confidence: 99%

“…A further term in the approximation was obtained by Atkinson [1]. The topic has been substantially developed in a series of papers by Harris [12][13][14][15][16], in which higher-order approximations are found, depending on the degree of regularity of q. Simplified proofs of the results in [1,12] have been given by Kaper and Kwong [23]. The case of (1.2) or (1.…”

Section: Introductionmentioning

confidence: 99%

Asymptotics of the Titchmarsh—Weyl m-coefficient for non-integrable potentials

Atkinson

Fulton

1999

Proceedings of the Royal Society of Edinburgh: Section A Mathem

View full text Add to dashboard Cite

Asymptotic formulae for the Titchmarsh-Weyl m-coefficient on rays in the complex A-plane for the equation -y" + qy = \y when the potential is limit circle and non-oscillatory at x = 0 are obtained under assumptions slightly more general than xq(x) S Li(0,c). The behaviour of q at the right end-point is arbitrary and may fall in either the limit-point or limit-circle case. A method of regularization of the equation is given that can be made to depend either on a solution of the equation for A = 0 or more directly on an approximation to the solution in terms of q. This enables equivalent definitions of the m-coefficient to be given for the singular Sturm-Liouville problem associated with a singular limit-circle boundary condition, and its associated regular Sturm-Liouville problem. As a consequence, it becomes possible to apply asymptotic results obtained by Atkinson for the regular problem in order to give asymptotic results for the singular problem. Potentials of the form q{x) = C/xi, 1 $J j < 2, are included. In the case j = 1, an independent calculation of the limit-point m-coefficient over the range (0, oo), relying on Whittaker functions, verifies the main result.

show abstract

The asymptotic form of the Titchmarsh–Weyl m-function associated with a second order differential equation with locally integrable coefficient

Abstract: SynopsisWe derive an asymptotic expansion for the Titchmarsh–Weyl m-function associated with the second order linear differential equationin the case where the only restriction on the real-valued function q is

Cited by 14 publications

References 9 publications

Asymptotics of the Titchmarsh-Weyl m-coefficient for integrable potentials

Asymptotics of the Titchmarsh-Weyl m-coefficient for integrable potentials

Formulation of a unified method for low- and high-energy expansions in the analysis of reflection coefficients for one-dimensional Schrödinger equation

Asymptotics of the Titchmarsh—Weyl m-coefficient for non-integrable potentials

Contact Info

Product

Resources

About