TOPICS
Search

Hill Determinant

A determinant which arises in the solution of the second-order ordinary differential equation

x^2(d^2psi)/(dx^2)+x(dpsi)/(dx)+(1/4h^2x^2+1/2h^2-b+(h^2)/(4x^2))psi=0.
(1)

Writing the solution as a power series

psi=sum_(n=-infty)^inftya_nx^(s+2n)
(2)

gives a recurrence relation

h^2a_(n+1)+[2h^2-4b+16(n+1/2s)^2]a_n+h^2a_(n-1)=0.
(3)

The value of s can be computed using the Hill determinant

Delta(s)=|... | | | | ...; ... ((sigma+2)-alpha^2)/(4-alpha^2) (beta^2)/(4-alpha^2) 0 0 ...; ... 0 -(beta^2)/(alpha^2) -(sigma^2-alpha^2)/(alpha^2) -(beta^2)/(alpha^2) ...; ... 0 0 -(beta^2)/(1-alpha^2) ((sigma-1)^2-alpha^2)/(1-alpha^2) ...; ... | | | | ...|,
(4)

where

sigma=1/2s
(5)
alpha^2=1/4b-1/8h^2
(6)
beta=1/4h,
(7)

and sigma is the variable to solve for. The determinant can be given explicitly by the amazing formula

Delta(s)=Delta(0)-(sin^2(pis/2))/(sin^2(1/2pisqrt(b-1/2h^2))),
(8)

where

Delta(0)=|... | | | | ...; ... 1 (h^2)/(144+2h^2-4b) 0 0 ...; ... (h^2)/(64+2h^2-4b) 1 (h^2)/(64+2h^2-4b) 0 ...; ... 0 (h^2)/(16+2h^2-4b) 1 (h^2)/(16+2h^2-4b) ...; ... 0 0 (h^2)/(2h^2-4b) 1 ...; ... 0 0 0 (h^2)/(16+2h^2-4b) ...; ... | | | | ...|,
(9)

leading to the implicit equation for s,

sin^2(1/2pis)=Delta(0)sin^2(1/2pisqrt(b-1/2h^2)).
(10)

See also

Hill's Differential Equation

Explore with Wolfram|Alpha

References

Hill, G. W. "On the Part of the Motion of Lunar Perigee Which is a Function of the Mean Motions of the Sum and Moon." Acta Math. 8, 1-36, 1886.Magnus, W. and Winkler, S. Hill's Equation. New York: Dover, 1979.Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 555-562, 1953.

Referenced on Wolfram|Alpha

Hill Determinant

Cite this as:

Weisstein, Eric W. "Hill Determinant." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/HillDeterminant.html

Subject classifications