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

Assume that p(x) has a finite number of sign changes and that the positive (resp. negative) part of the function p(x)/w(x) defined by ${\displaystyle (w/p)_{+}(x)=\max\{w(x)/p(x),0\}}$, (resp. ${\displaystyle (w/p)_{-}(x)=\max\{-w(x)/p(x),0\})}$ are not identically zero functions over I. Then the eigenvalue problem (1), (2)–(3) has an infinite number of real positive eigenvalues ${\displaystyle {\lambda _{i}}^{+}}$, $${\displaystyle 0<{\lambda _{1}}^{+}<{\lambda _{2}}^{+}<{\lambda _{3}}^{+}<\cdots <{\lambda _{n}}^{+}<\cdots \to \infty ;}$$ and an infinite number of negative eigenvalues ${\displaystyle {\lambda _{i}}^{-}}$, $${\displaystyle 0>{\lambda _{1}}^{-}>{\lambda _{2}}^{-}>{\lambda _{3}}^{-}>\cdots >{\lambda _{n}}^{-}>\cdots \to -\infty ;}$$ whose spectral asymptotics are given by their solution [2] of Jörgens' Conjecture [3]: $${\displaystyle {\lambda _{n}}^{+}\sim {\frac {n^{2}\pi ^{2}}{\left(\int _{a}^{b}{\sqrt {(w/p)_{+}(x)}}\,dx\right)^{2}}},\quad n\to \infty ,}$$ and $${\displaystyle {\lambda _{n}}^{-}\sim {\frac {-n^{2}\pi ^{2}}{\left(\int _{a}^{b}{\sqrt {(w/p)_{-}(x)}}\,dx\right)^{2}}},\quad n\to \infty .}$$

For more information on the general theory behind (1) see the article on Sturm–Liouville theory. The stated theorem is actually valid more generally for coefficient functions ${\displaystyle 1/p,\,q,\,w}$ that are Lebesgue integrable over I.

1. F. V. Atkinson, A. B. Mingarelli, Multiparameter Eigenvalue Problems – Sturm–Liouville Theory, CRC Press, Taylor and Francis, 2010. ISBN 978-1-4398-1622-6
2. F. V. Atkinson, A. B. Mingarelli, Asymptotics of the number of zeros and of the eigenvalues of general weighted Sturm–Liouville problems, J. für die Reine und Ang. Math. (Crelle), 375/376 (1987), 380–393. See also free download of the original paper.
3. K. Jörgens, Spectral theory of second-order ordinary differential operators, Lectures delivered at Aarhus Universitet, 1962/63.