Atkinson–Mingarelli theorem

In applied mathematics, the Atkinson–Mingarelli theorem, named after Frederick Valentine Atkinson and A. B. Mingarelli, concerns eigenvalues of certain Sturm–Liouville differential operators.

In the simplest of formulations let p, q, w be real-valued piecewise continuous functions defined on a closed bounded real interval, I = [a, b]. The function w(x), which is sometimes denoted by r(x), is called the "weight" or "density" function. Consider the Sturm–Liouville differential equation

${\displaystyle -{\frac {d}{dx}}\left[p(x){\frac {dy}{dx}}\right]+q(x)y=\lambda w(x)y,}$

(1)

where y is a function of the independent variable x. In this case, y is called a solution if it is continuously differentiable on (a,b) and (p y′)(x) is piecewise continuously differentiable and y satisfies the equation (1) at all except a finite number of points in (a,b). The unknown function y is typically required to satisfy some boundary conditions at a and b.

The boundary conditions under consideration here are usually called separated boundary conditions and they are of the form:

${\displaystyle \alpha _{1}y(a)+\alpha _{2}y'(a)=0\qquad (\alpha _{1}^{2}+\alpha _{2}^{2}>0),}$

(2)

${\displaystyle \beta _{1}y(b)+\beta _{2}y'(b)=0\qquad (\beta _{1}^{2}+\beta _{2}^{2}>0),}$

(3)

where the ${\displaystyle \{\alpha _{i},\beta _{i}\}}$, i = 1, 2 are real numbers. We define