A Novel Spectral Density Function Validation for Bessel’s Equation in L-N Form

Introduction

Background

The Titchmarsh-Weyl m-Function

Consider the general Sturm-Liouville equation:

$$-y^{″}(x)+q(x)y(x)=\lambda y(x),x\in (a,\infty ),q(x)\in L_1(a,\infty ),\mathrm{and}\underset{x\to \infty }{\lim }q(x)=0.$$

(1)

Suppose x = a is a singular endpoint of the Limit Point (LP) classification as given in [13]. Let {u(x,λ),ν(x,λ)} be the fundamental system of (1) normalized so as to have a Wronskian determinant, w_a(u(x,λ),ν(x,λ)) = 1, and satisfying the boundary conditions,

The Titchmarsh-Weyl m-function, m(λ) is defined by the requirement

As (1) is a linear differential equation, ψ(x,λ), being a linear combination of solutions to (1) near x = a, satisfies (1) near x = a. Moreover, due to x = a being an LP singular endpoint, ψ(x,λ) and m(λ) are uniquely defined in (3) by this square-integrability requirement given in [11, p. 86].

The spectral density function f(λ) is then characterized by the Titchmarsh-Kodaira formula,

where

See [11, p. 43, eq. 3.3.1]. In the next section, we define the Appell system.

The Appell System

In 1880, M. Appell in [2] gave a companion system to the Sturm-Liouville equation (1) as

Some fundamental properties pertaining to system (6) are now given below (See also [7], p. 6-7).

(i)

If y(x) is any solution of the Sturm-Liouville equation (1), then

 is a solution to the Appell system (6).

(ii)

Let {u(x,λ),ν(x,λ)} be the fundamental system of (1), where (2) holds.

A fundamental system of three linearly independent solutions to (6) is

(iii)

For any solution to Appell system (6), let the indefinite inner product be defined by: It follows that:

,(See [7], pages 6-7).

(iv)

Letbe any two solutions to Appell system (6) where

Under these assumptions,

(v)

Let y(x) be any solution to the Sturm-Liouville equation (1) and let be a solution to Appell system (6). It follows that:

(vi)

Near x = ∞, x₀ > 0, if either q(x) ∈ L₁(x₀, ∞) or q’(x) ∈ L₁(x₀, ∞), q(x) ∈ AC_loc(x₀, ∞), and , then the terminal value problem below has a unique solution.

(vii)

Let be the unique solution to the terminal value problem (7). It follows that . Furthermore, when x = 0 is either regular or a RSP of LC/N or LP/N type, x = ∞ is LP/O-N with cut-off ∧ = 0 and q(x) is absolutely integrable near x = ∞ the spectral density function, f(λ) for λ ∈ (0, ∞), is characterized by:

where f(λ) is absolutely continuous for λ ∈ (0, ∞) (See [8, p. 40].).

The proofs of these properties of the Appell system are well-documented and generally require only algebraic manipulations with no special assumptions on the potential q(x) according to Fulton in [7, p.6]. For this reason, the proofs of the properties (i)-(vii) are omitted in this paper. In the next section, we’ll utilize these properties and demonstrate the viability of the spectral density function characterization given in (vii) by providing the first nontrivial example of a spectral density function calculation by use of (8), as applied to the Bessel equation in L-N form.

Calculation of the SDF for Bessel’s Equation in Liouville-Normal Form

Consider the Bessel Equation in L-N form

$-y^{″}(x)+\left({\displaystyle \frac{{\nu }^2-1/4}{x^2}}\right)y(x)=\lambda y(x),$ for a < x < ∞, a > 0 and with ν ≠ 0,1,2….(9)

Here observe that x = ∞ is a LP/O-N singular endpoint with cutoff λ = 0. Let {u(x,λ),ν(x,λ)} be the fundamental system to (9) such that , for all λ ∈ (0, ∞). Here the potential function and thus the Appell system property (vi), from above, holds. The corresponding Appell system terminal value problem for (9) is then

Let U₁(x) be the unique solution to (10) where are defined by the relation

The fundamental system to ODE (9), {u(x,λ),ν(x,λ)}, which satisfies the Wronskian requirement, w_a({u(x,λ),ν(x,λ)}) = 1, is uniquely determined with

where the constants C₁, C₂, C₃, and C₄ are given by

The definitions for these suitably normalized solutions, u(x,λ), ν(x,λ), along with the indicated values for C₁, C₂, C₃, and C₄ in (14), ensure that the Wronskian determinant, w_a({u(x,λ),ν(x,λ)}) = 1, as required. The pertinent Wronskian relations for the Bessel functions J_ν(x) and Y_ν(x) can be found in [12, p. 76]. Now that the fundamental system to (9) has been given explicitly, we can use the characterization of solution as given in (11) and impose the terminal condition in (10) to yield

where are uniquely defined by (15) as guaranteed by Appell system property (vi). Now for further progress towards obtaining explicit representations of , and then f(λ) as characterized by (8), we next make use of well-known asymptotic relations for the Bessel functions J_ν(x) and Y_ν(x) as x → ∞. The asymptotic relations for the Bessel functions near infinity, given below as (16)-(19), are in many books (See for instance the extensive work of G.N. Watson, A Treatise on the Theory of Bessel Functions, [12], p. 199). As x → ∞,

By letting $w(x,\lambda )=w(x):=\sqrt{\lambda }x-{\displaystyle \frac{\nu \pi }{2}}-{\displaystyle \frac{\pi }{4}}$ and applying (16)-(19) to (15), we obtain

Now to satisfy relations (20)-(22), nine equations emerge, three of which are independent,

or in a matrix form

The solution to (24) is computed via Mathematica to be

Inserting the definitions for C₁, C₂, C₃, and C₄ as defined in (14) now reveals that

Finally, we apply the characterization of the spectral density function given in (8), obtaining (29) . This work, culminating in equation (29), constitutes an original determination of the spectral density function for Bessel’s Equation in L-N form, obtained using the new method proposed by Fulton, Pearson, and Pruess in [7]. In the next brief section, we’ll validate this result obtained using independent checks.

Validation of Results (A)

To confirm the validity of the representations obtained above for and f(λ), eq.’s (26) and (29), we compare the spectral density function obtained in (29) with the classical spectral density function result for the Bessel Equation (6), obtained by E.C. Titchmarsh via the Titchmarsh-Kodaira formula (4) (See [11, p. 86]). Here we find full agreement in the f(λ) representations and so the f(λ) in equation (29) is validated. As f(λ), given in equation (29), is computed directly by use of in (26) (which was obtained by solving the Appell system terminal value problem (10)), we may conclude that the representation in (26) for is validated.

(B) To validate the representation of $\tilde{b}$, in eq. (27), the theory of C. Fulton, D. Pearson, and S. Pruess in [7, eq. 4.22] is employed, whereby the Titchmarsh-Weyl m-function has the form, (30) , According to classical theory of Titchmarsh in [11], the m-function for Bessel Equation (9) is (31) . By calculation of m(λ) via (30) using (16)-(19), (26), and (27), after minor algebraic manipulations, we indeed obtain the classical m-function result given in (31) and thus we may conclude that the calculation of $\tilde{b}$, as was determined in (27), is validated.

(C) To validate the representation of $\tilde{c}$ obtained in (28), according to the Appell system theory in property (vii), necessarily . While this calculation to validate $\tilde{c}$ is rather tedious, we will give some details of it here to conclude our investigation. Inserting the values for $\tilde{a}$, $\tilde{b}$, and $\tilde{c}$, as given in (26)-(28), into the equation , and expansion yields twenty-eight terms on the LHS, all of which are products of the Bessel functions J_ν, Y_ν, and their derivatives, and all of which are having the argument $a\sqrt{\lambda }$. Multiple pairs of these terms (and sometimes triples) combine, are opposites, and cancel leaving just three terms that do immediately cancel by trivial algebraic manipulation. At this stage, the LHS of takes the form (32) $4\tilde{a}\tilde{c}-{\tilde{b}}^2=(a^2{\pi }^2\lambda )(J_{\nu }^2{Y^{\prime }}_{\nu }^2+{J^{\prime }}_{\nu }^2{Y}_{\nu }^2-2J_{\nu }{J^{\prime }}_v{Y}_{\nu }{Y^{\prime }}_{\nu })$, with the arguments of these Bessel functions all being $a\sqrt{\lambda }$. To make further progress, we find that the RHS of (32) factors giving (33) $4\tilde{a}\tilde{c}-{\tilde{b}}^2=(a^2{\pi }^2\lambda ){(J_{\nu }{Y^{\prime }}_{\nu }-{J^{\prime }}_{\nu }{Y}_{\nu })}^2$. Finally, to obtain the value of 4 on the RHS of (33), we employ a Wronskian identity from Watson’s [12, p. 76], with his z-argument replaced by $a\sqrt{\lambda }$. This identity takes the form of (34) $J_v(a\sqrt{\lambda })\cdot {Y^{\prime }}_{\nu }(a\sqrt{\lambda })-{J^{\prime }}_{\nu }(a\sqrt{\lambda })\cdot Y_{\nu }(a\sqrt{\lambda })={\displaystyle \frac{2}{\pi a\sqrt{\lambda }}}$. Upon insertion of (34) into (33), we complete our validation of $\tilde{c}$.

Conclusion

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