Hill equation

An ordinary second-order differential equation

$$ w ^ {\prime\prime} ( z) + p ( z) w ( z) = 0 $$

with a periodic function $ p ( z) $; all quantities can be complex. The equation is named after G. Hill [1], who in studying the motion of the moon obtained the equation

$$ w ^ {\prime\prime} ( z) + \left ( \theta _ {0} + 2 \sum _ {r = 1 } ^ \infty \theta _ {2r} \cos 2rz \right ) w ( z) = 0 $$

with real numbers $ \theta _ {0} , \theta _ {2} \dots $ where the series $ \sum _ {r = 1 } ^ \infty | \theta _ {2r} | $ converges.

Hill gave a method for solving this equation with the use of determinants of infinite order. This was a source for the creation of the theory of such determinants, and later for the creation by E. Fredholm of the theory of integral equations (cf. Fredholm theorems). Most important for Hill equations are the problems of the stability of solutions and the presence or absence of periodic solutions. If in the real case one introduces in the Hill equation a parameter:

$$ x ^ {\prime\prime} + \lambda p ( t) x = 0, $$

then, as was established by A.M. Lyapunov in [2], there exists an infinite sequence

$$ {} \dots < \lambda _ {-} 1 \leq \ \lambda _ {0} = 0 < \lambda _ {1} \leq \ \lambda _ {2} < \dots < \ \lambda _ {2n - 1 } \leq $$

$$ \leq \ \lambda _ {2n} < \ \lambda _ {2n + 1 } \leq \dots , $$

such that for $ \lambda \in ( \lambda _ {2n} , \lambda _ {2n + 1 } ) $ the Hill equation is stable and for $ \lambda \in [ \lambda _ {2n - 1 } , \lambda _ {2n } ] $ it is unstable. Here $ \lambda _ {4n} $ and $ \lambda _ {4n + 3 } $ are the eigen values of the periodic, and $ \lambda _ {4n + 1 } $ and $ \lambda _ {4n + 2 } $ the eigen values of the semi-periodic boundary value problem. The Hill equation is well studied (see [3]).

References

[1]	G. Hill, Acta. Math. , 8 (1886) pp. 1
[2]	A.M. Lyapunov, , Collected works , 2 , Moscow (1966) pp. 407–409 (In Russian)
[3]	V.A. Yakubovich, V.M. Starzhinskii, "Linear differential equations with periodic coefficients" , Wiley (1975) (Translated from Russian)

Comments

An operator of the form $ Q = D ^ {2} + q $, where $ q $ is a periodic function, is called a Hill operator. Let $ q $ be of period $ 1 $. The periodic spectrum and the anti-periodic spectrum (or semi-periodic spectrum) of $ Q $ are obtained by solving $ Q f = \lambda f $ with $ f ( x + 1 ) = \pm f ( x) $. Together these spectra constitute a simple periodic ground state $ \lambda _ {0} $ followed by alternatively anti-periodic and periodic pairs of simple or double eigen values

$$ \lambda _ {0} < \lambda _ {1} \leq \lambda _ {2} < \lambda _ {3} \leq \ \lambda _ {4} < \dots . $$

The intervals $ ( - \infty , \lambda _ {0} ) $ and $ [ \lambda _ {2i-} 1 , \lambda _ {2i} ] $ are called lacunae or gaps. This term arises from the fact that the spectrum of $ Q $ viewed as acting on $ L _ {2} ( \mathbf R ) $ is in the closed complement of the union of these intervals.

The study of the Hill operator and its spectral data is important in the inverse scattering method for solving the (periodic) Korteweg–de Vries equation. A crucial result here is Borg's theorem, which says that the potential $ q $ can be recovered from the periodic and anti-periodic spectrum, the auxiliary spectrum (which is obtained by solving $ Q f = \mu f $ with $ f ( 0) = f ( 1) = 0 $), and certain norming constants obtained from the associated eigen functions. Cf. [a2] for more details. The $ \mu _ {i} $ are located in the gaps, $ \mu _ {i} \in [ \lambda _ {2i-} 1 , \lambda _ {2i} ] $, $ i = 1 , 2 ,\dots $.

In [a3] the term Borg's theorem refers to a different result which belongs to the context of "co-existence questions" , i.e. of the problem of determining when two linearly independent periodic solutions of period 1 and 2 can co-exist (cf. [a3], Sect. 2.6).

The result on the relative position of the periodic and anti-periodic spectrum together with the corresponding statements on the stability of the solutions of $ Q y = \lambda y $ in the various intervals is known as the oscillation theorem.

References

[a1]	M.S.P. Eastham, "The spectral theory of periodic differential equations" , Scottish Acad. Press (1973)
[a2]	H.P. McKean, "Integrable systems and algebraic curves" M. Grmela (ed.) J.E. Marsden (ed.) , Global analysis , Lect. notes in math. , 755 , Springer (1979) pp. 83–200
[a3]	W. Magnus, S. Winkler, "Hill's equation" , Dover, reprint (1979)