Sturm-Liovulle

The Bessel equation of order $n$ is given by:

$$x^2 y'' + x y' + (x^2 - n^2) y = 0$$

For the Bessel function of the first kind of order $n=0$, the corresponding equation is:

$$x^2 y'' + x y' + x^2 y = 0$$

The solution to this differential equation is the Bessel function of the first kind of order 0, denoted by $J_0(x)$.

The Bessel equation as a Sturm-Liouville Eigenvalue Problem

The Bessel equation can be written as a Sturm-Liouville problem:

$$\frac{d}{dx} \left( x \frac{dy}{dx} \right) + \lambda x y = 0$$

with the boundary conditions that the solutions are finite at the origin, and the normalization condition:

$$\int_0^1 x J_n^2(x) dx = \frac{1}{2}$$

The eigenvalues are determined by the zeros of the Bessel function, and the corresponding eigenfunctions are given by:

$$y_n(x) = J_n(\alpha_n x)$$

where $\alpha_n$ are the positive zeros of the Bessel function.

Eigenvalues and Eigenfunctions

• Eigenvalues: The eigenvalues are given by $\lambda_n = \alpha_n^2$ where $\alpha_n$ are the positive zeros of the Bessel function $J_0(x)$.
• Eigenfunctions: The eigenfunctions corresponding to these eigenvalues are $y_n(x) = J_0(\alpha_n x)$.

Orthogonality

The eigenfunctions are orthogonal with respect to the weight function $x$ on the interval $[0,1]$:

$$\int_0^1 x J_0(\alpha_n x) J_0(\alpha_m x) dx = 0 \quad \text{for} n \neq m$$

This orthogonal property and the known eigenfunctions and eigenvalues provide a complete description of the Hilbert space corresponding to the Bessel function of the first kind of order 0. By employing the series expansion and Sturm-Liouville theory, you can analyze and represent functions in terms of this orthonormal basis, leveraging the well-known properties of Bessel functions.