Atkinson–Mingarelli theorem

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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

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

 

 

 

 

(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:

[math]\displaystyle{ \alpha_{1}y(a)+\alpha_{2}y'(a)=0\qquad(\alpha_1^2+\alpha_2^2\gt 0), }[/math]

 

 

 

 

(2)

[math]\displaystyle{ \beta_{1}y(b)+\beta_{2}y'(b)=0\qquad(\beta_1^2+\beta_2^2\gt 0), }[/math]

 

 

 

 

(3)

where the [math]\displaystyle{ \{\alpha_i, \beta_i\} }[/math], i = 1, 2 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 [math]\displaystyle{ (w/p)_{+}(x) = \max \{w(x)/p(x), 0\} }[/math], (resp. [math]\displaystyle{ (w/p)_{-}(x) = \max \{ -w(x)/p(x), 0\}) }[/math] are not identically zero functions over I. Then the eigenvalue problem (1), (2)–(3) has an infinite number of real positive eigenvalues [math]\displaystyle{ {\lambda_i}^{+} }[/math], [math]\displaystyle{ 0 \lt {\lambda_1}^{+} \lt {\lambda_2}^{+} \lt {\lambda_3}^{+} \lt \cdots \lt {\lambda_n}^{+} \lt \cdots \to \infty; }[/math] and an infinite number of negative eigenvalues [math]\displaystyle{ {\lambda_i}^{-} }[/math], [math]\displaystyle{ 0 \gt {\lambda_1}^{-} \gt {\lambda_2}^{-} \gt {\lambda_3}^{-} \gt \cdots \gt {\lambda_n}^{-} \gt \cdots \to - \infty; }[/math] whose spectral asymptotics are given by their solution [2] of Jörgens' Conjecture [3]: [math]\displaystyle{ {\lambda_n}^{+} \sim \frac{n^2 \pi^2}{\left(\int_a^b \sqrt{(w/p)_{+}(x)}\, dx\right)^2},\quad n \to \infty, }[/math] and [math]\displaystyle{ {\lambda_n}^{-} \sim \frac{- n^2 \pi^2}{\left(\int_a^b \sqrt{(w/p)_{-}(x)}\, dx\right)^2},\quad n \to \infty. }[/math]

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 [math]\displaystyle{ 1/p,\, q,\, w }[/math] that are Lebesgue integrable over I.

References

  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.