applied mathematics Applied mathematics is the application of mathematics, mathematical methods by different fields such as physics, engineering, medicine, biology, finance, business, computer science, and Industrial sector, industry. Thus, applied mathematics is a ...

, the Atkinson–Mingarelli theorem, named after

Frederick Valentine Atkinson Frederick Valentine "Derick" Atkinson (25 January 1916 – 13 November 2002) was a British mathematician, formerly of the University of Toronto, Canada, where he spent most of his career. Atkinson's theorem and Atkinson–Wilcox theorem are nam ...

and A. B. Mingarelli, concerns eigenvalues of certain Sturm–Liouville

differential operator In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a function and retur ...

s. In the simplest of formulations let ''p'', ''q'', ''w'' be real-valued

piecewise continuous In mathematics, a piecewise function (also called a piecewise-defined function, a hybrid function, or a function defined by cases) is a function whose domain is partitioned into several intervals ("subdomains") on which the function may be ...

functions defined on a closed bounded real interval, . The function ''w''(''x''), which is sometimes denoted by ''r''(''x''), is called the "weight" or "density" function. Consider the Sturm–Liouville differential equation 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 () at all except a finite number of points in (''a'',''b''). The unknown function ''y'' is typically required to satisfy some

boundary condition In the study of differential equations, a boundary-value problem is a differential equation subjected to constraints called boundary conditions. A solution to a boundary value problem is a solution to the differential equation which also satis ...

s at ''a'' and ''b''. The boundary conditions under consideration here are usually called separated boundary conditions and they are of the form: where the

\

, are real numbers. We define

The theorem

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

(w/p)_(x) = \max \

, (resp.

(w/p)_(x) = \max \)

are not identically zero functions over I. Then the eigenvalue problem (), ()–() has an infinite number of real positive eigenvalues

^

0 < ^ < ^ < ^ < \cdots < ^ < \cdots \to \infty;

and an infinite number of negative eigenvalues

^

0 > ^ > ^ > ^ > \cdots > ^ > \cdots \to - \infty;

whose spectral asymptotics are given by their solution of Jörgens' Conjecture

^ \sim \frac,\quad n \to \infty,

and

^ \sim \frac,\quad n \to \infty.

For more information on the general theory behind () see the article on

Sturm–Liouville theory In mathematics and its applications, a Sturm–Liouville problem is a second-order linear ordinary differential equation of the form \frac \left (x) \frac\right+ q(x)y = -\lambda w(x) y for given functions p(x), q(x) and w(x), together with some ...

. The stated theorem is actually valid more generally for coefficient functions

1/p,\, q,\, w

that are

Lebesgue integrable In mathematics, the integral of a non-negative Function (mathematics), function of a single variable can be regarded, in the simplest case, as the area between the Graph of a function, graph of that function and the axis. The Lebesgue integral, ...

over .

References

# F. V. Atkinson, A. B. Mingarelli, ''Multiparameter Eigenvalue Problems – Sturm–Liouville Theory'', CRC Press, Taylor and Francis, 2010. # 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 als
free download of the original paper
# K. Jörgens, ''Spectral theory of second-order ordinary differential operators'', Lectures delivered at Aarhus Universitet, 1962/63. {{DEFAULTSORT:Atkinson-Mingarelli theorem Ordinary differential equations Theorems in mathematical analysis