The Computation of the Titchmarsh-Weyl m-Function

The classical limit-point, limit-circle theory of Weyl [20] is concerned with the L² (0, ∞) solutions of the differential equation 1 https://www.w3.org/1998/Math/MathML"> M y = λ y https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429332623/d03ac290-bd50-452e-ba31-4edd6e078f1a/content/eqn1.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> over [0, ∞), where 2 https://www.w3.org/1998/Math/MathML"> M y = − p y ′ ′ + q y , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429332623/d03ac290-bd50-452e-ba31-4edd6e078f1a/content/eqn2.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> λ is a strictly complex parameter and p, q are real-valued functions which satisfy p^-1,https://www.w3.org/1998/Math/MathML">q∈Lloc1(0,∞)https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429332623/d03ac290-bd50-452e-ba31-4edd6e078f1a/content/ieq0367.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/>. Weyl showed that the mapping z → F_x(λ,z), X > 0 defined by https://www.w3.org/1998/Math/MathML"> F X ( λ , z ) = − θ ( X , λ ) z − θ ′ ( X , λ ) ϕ ( X , λ ) z − ϕ ′ ( X , λ )   ( λ ∈ C ± ) , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429332623/d03ac290-bd50-452e-ba31-4edd6e078f1a/content/eqn0450.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> maps the real line onto the boundary of a circle C_X ≡ C_X(λ) in C, where θ(x,λ) and ϕ(x,λ) are the solutions of (1) which satisfy 3 https://www.w3.org/1998/Math/MathML"> θ ( 0 ) = 0 ,   θ ′ ( 0 ) = 1 ,   ϕ ( 0 ) = = 1 ,   ϕ ′ ( 0 ) = 0 , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429332623/d03ac290-bd50-452e-ba31-4edd6e078f1a/content/eqn3.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> 198and the diameter d_x (λ) of the circle C_x (λ) is 4 https://www.w3.org/1998/Math/MathML"> d X ( λ ) = | im λ ∫ 0 X ϕ 2 d x | − 1 ,         im  λ ≠ 0 . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429332623/d03ac290-bd50-452e-ba31-4edd6e078f1a/content/eqn4.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> The circles d_x(λ) are nested in the sense that C_x(λ) lies within C_y (λ) when X > Y and so as X→ ∞, C_x(λ) tends either to a point m(λ) (the so-called limit-point) or a circle (the limit-circle). The classification is independent of λ ∈ C\R. We shall be concerned only with the limit-point case. In this case there is, up to constant multiples, precisely one L² (0, ∞) solution of (1) for λ ∈ C \ R, and this is 5 https://www.w3.org/1998/Math/MathML"> ψ ( x , λ ) = θ ( x , λ ) + m ( λ ) ϕ ( x , λ ) . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429332623/d03ac290-bd50-452e-ba31-4edd6e078f1a/content/eqn5.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> The m(-) is the Titchmarsh-Weyl function.