A rigorous solution of Hill's equation

Chinese

Astronomy

E Astrophysics

5 (1981)

I’cr,camon

~~-67

Press. Printed in Crcat Brit:in 0146-6364/81/0301-0058-$07.50/O

Sin&a -21 119801 49-57

Act. As&on.

Received

1978 March 27

ABSTRACT Poincare for to

elucidate

analysis

the

in the

found,

exponent

solution

and the

generalized

for

of

it

is

a new type boundary

the

of

of

paper

By means of

between

are

the

The present

irregular

Hill

The

immediately

convergence

method

integrals

of

to

functions

series.

motion

onlv

lc;ding

attempts

correspondence

for

Taylor-Fourier

relation

expressions

equation

expression

mixed

solution. general

explicit

This

pseudo-periodic

the

pseudo-periodic

find

Hill’s

determinant.

an explicit

form of

studying

of

conclusion.

methods,

discussed

to

theory

an infinite

and the also

impossible

classical

Poincare’s

graph

I have

recovered.

of

essence

and tree

Floquet

that the

integral,

a numerical

principles is

concluded

an irregular

of

the

can be

non-Fuchsian

equations.

1.

INTRODUCTION

Hill’s

work

[l]

of

three

problem Poincare’

[Z]

Wintner

[S],

on the

for of

in

formal

[6],

solving

it

we can

truncate

off

and the

coefficients

parameters

of

the

equation.

namely,

is

impossible

now,

I have

equation

analytical

in

discarded and the

expression

that

regard

an explicit

the

formal of

Hill’s

for

approach

tree-graph

of

further

classical

the

[S]

restricted

developed

theory.

Later

developed

leads

to

of

infinite

theory

determinant the

solution

expression

method for

find

and the

determinant

the

In this to

analytical

a rigorous,

paper, the

it

a finite

study

by

Liapounoff

various

[4],

methods

of

equation.

infinite

can be proved

the

neccessarily

unknowns,

the

determinant,

of

Magnus and Winkler

Hill’s

When the

a new way in

infinite part

coefficients

number of

convergence,

[7],

effectively

solution.

of

has become

undetermined

infinite

Moon paved

method

Kosirev

Floquet’s

that

and his

bodies,

Brillouin

equations

so

the

[3],

The method

exponent

of

and von Kock

approximation

absolute

motion

satisfies

solution

is

for

a numerical be explicitly

Hill’s

for

in

certain

is

for

domain

However,

expressed

in

terms

the of

pronouncement, integral.

been

in

based

defined

obtained.

“correspondences”

succeeded

of

condition

evaluation.

has not

have

the

an irregular

function

and using induction,

determinant

made an important

expression

set

Poincarb’s

analytic

cannot Poincard

an infinite

finding

up to In this inherent

in

a rigorous,

function.

2 .“CORRESPONDENCE” RELATIONS 1

Volterra’s

Integral

Equations:

Consider

the

real

solution

of

the

equation

with

Hill’s

singly-periodic

‘2> cp+ i(th(r)

L($+y+ao,

known real

(a,, +

0, a. =

= 0,

;(t)-2~

number R> 0, and c _ 0, n > 1) and its basi:

For simplicity, including

we suppose

zero.

that

From Eqn (1)

g(# - 2’) is Green’s

decomposition

of

the

a”cos2r?f.

exponents

function.

solution

do not

differ

from

each

other

by an integer

For

c,(a

1, 2)

=

any two arbitrary

constants,

c

@(:I -

,&,(t)is 2.

not

satisfies

Jkr -

Volterra’s

t’, r’)cp,(z’)dr’

integral

equation

of

the

second

to

qo(Z)and

its

&&)is

the

zero,

of

The image

of

(2)

ea2nia,

-

l/L(s) kernal

=

G(J) as the

,&)has

a non-zero

Function

theorem

solution.

p,(i)

of

integral

transform,

the

relation

of

I&,(#)

between

a function

is

> sa =

Green’s

the

H(S)

To(S)

inRec

the

G(s)

the

fundamental

image

regular

The image

Define

indentically

kind,

- &(t),

0

The Imagecp,(J)of

According

a linear

c,@,(t).

h(l - t’, I’) - g(’ - t’)j(r’). As

equation

0

Independently

Qb(t) -

degenerate

exsists,

D

q.(r)

the

is

we have

cp(t)= 2 cdP&), Hence,

convergent,

a. absolutely solution

JIg(i- 2’)j(r’)&‘)dt’,

qJ(t)= $42) + where

59

co efficients:-

Lt

with

Equation

E >

function l/(22

h(z c’ n shift

is

0

the

index

of

growth

g(t) is f

/4’).

I’, r’> is

an operator:

a,e-2”iar. operator

with

respect

to’

s

,

.

When c 7.50

Hill’s

50

e*“ie;F(s) By the convolution

-

Equation

F(sfZni).

theorem and the shift

rule,

we obtain

CXX: I?)

i .i) Assuming the existence

The existence

of the inverse

of the inverse

operator

proved by the method of successive Expression

result.

the following

apart from set of poles possible

{fin -k Zni)(n

the Theorem of

As positive equal to zero.

= [1-

and the legitimacy solution

H(s)]-‘,

of its

or established

and negitive

5 0, f 1, zk2, - - *) T

n co-exist

expansion

by verifying up,

the2 entire

can he

the final

is the basis

of

it has no significant

on the imaginary axis.

to surround these

.‘iesidues then gives

we have

binomial

of the image function

P&S) is meromorphic on the s plane;

to choose a suitablecontour

at infiniiy,

T(s)

approximation,

[4) for the basic

discussion.

operator

It is

Iloles and Jordan% Lcmmnbeing vnKi structure

of the solution.

the sum C oi may be greater than, less than, or ‘i\’ the exponent and this gives rise to

WhenXni -pi 0 there is contraction

sequences higher-order

poles.

The form of contraction

becomes more and more complicated

For a general term (the term of order h in ur) let us use the notation as ii increases. * ** *** to express the different choices of n corresponding to non-re~lice~~completelyf , > reduced and semi-reduced

We proceed

to find

tcns.~~(S)

the functions

and the corresponding

can also be divided

corresponding

into

3 parts

to the 3 separate

in a unique manner:

portions

relation = T@(L)&‘,

%O) T,(t)

The Normal Portion

I.

Consider poles

== T,*(t) + 2?Y(r> + T,***(t1,

I

first

the sequence of first-order

From the

Theorem

of

Residues

poles{@,

-k 2ix~ni) ;tnd neglect

we have

tbr hi:!?-order

Hill’s

where

the

suffix

in

am_Ini

a, = a_,,

81 = ah

is

never

-h

=

=

ab-.

Equation

zero.

-ip,

Since,

L(B,f

= pL

ah

=

for

2inl) al,-,

61

=

the

Hill’s

L&T

equation

2iq),

= pT..,.

we have

(p:(r)

5

=i e+’

q:(t)

Linear

combinations h,*(t)

It

is

for

the

2

Hill’s

is

two sucessive

expansion;

series,

forming

successors,

cp:,

(alo -

Contracted is

and

A,*(:),

p$J =

h:(r):

1)

(7b)

and by convention

p +n,

Also

the

terms,

usual

leading

of

new series

in

and so on,

Sunrmation Formula and the

corresponding connected

there

For this

residue,

the

the

convergent.

is

rise use

the

then

leading

by Rcs(nl, for

to

the

combine the

na, -. *, nz).

example,

for

the

exists

method its

the

order The

of

of tree

successors leading

series

term,

the

which

graphs.

term of

the

relation

coefficients

series,

in

but

solution.

no iterative for

new types

limit,A

term of

any contracted

expression

to

with

for

form of there

no general

gives

pass

residue

general

X (X=1,2,3...)

and eventually

by an over-bar,

absolutely

contraction,

I shall

necessary,

Hereafter

finding

finding

in the

contraction

order if

in

that

of

form.

term

wil 1 be

so

process

are

&2p:,

Portion

no difficulty

come arbitrariness

the

the

+ p)t,

in principle

between

combine

and odd functions

C p$sin (2n + p) 1. -m is absolutely convergent,

There

a difficulty

in

even

(7a)

2dil + 4,ne-2mit)},

02.e

the

ulneVznir) , )

c-C 1

problem

there

be written

olne2nir +

Ia,/ IG(B + 2in)l < 1

Because

the

*( c1

p:,cos(2n .

2

The Wholly

is

1 +

1+

give =

The Series there

I

{

h:(tTn -, known that& a, II

c’ hence

e--fpg

in

cannot

I shall determinate

order

A+1 with

its

-t m. h will ni

a second-order

be denoted

participating contraction

by(n,,

ttl, . . ..nA)

in a contraction term

(n, =

-ma)

Hill’s

62

will

Here A,(@)

be called Series

The O-Series contractions

the second-order

Equation

leading

factor.

formed by the contraction

of the successor

terms according

of the leading

to increasing

n.of

term and the successive

the successive

orders,

The formulae

Series

“J’heH-Sesie_l;_

formed by contractions

repestedly

formed through the form

(d%lr:. 1.4,) of the increasing

n of successive

H-Series,

the 2-plot

(a-2)

series *

For example,

ObVioUSly, only the H-series Analytical

by Coursat’s

summation over Expression

gives

of the

obtained

to the conplex

the coefficients

= n! 2ni h,,

In the

arc the secondary

is

by replacillgnl(B)bynr(B)Oin

contracted

portion

in L

the above.

To find a series

contour of integration

J fm#

&)‘+g’

can be expressed

of order

of the series.

Since

r, @ -

the others

solution,

h and 6 .

The wholly

domain, and choose a suitable

j’“‘(&)

S- plct

a power expression

Theorem we can prove the convergcnces

pole 8 = & and no other,

term of known contraction.

is the main series,

the high order factor

such as HO-series

Composite Series we must consider

order for a leading

contraction

as an integral

P is

For this, r,

we extend

which contains

6 the

Hill’s

A,

For determinate

Let

It

A*(,?,,),

is

is

finite

and

the maximum moduli

L’I*(@~)(~~C r,> be respectively,

obvious

Resolving

A(8)

that

into

G

real

hence

1hckj”k -+ 0,

k-r

and imaginary

63

Equation

parts

as

C);I,rk

1‘ -+ co

according

to

even

is

of

A@),

a Taylor

A(p)

Then

orTO

series:

and odd powers

of

L’(,I~,)

=

_+rip

we have

p,(t)

q?*(t)

=

Mt)

-I- iqp(t))e+‘,

p:*(t)

=

{p&)

-

= poo + po*rZ+ po,t’ +

44L))e”‘.

- . - , 40(r) = 401’+ qsp + - * -, Haf&

s Z’a’,,&( 1 + Cc&b, + . . *) , qol = Xa2,,al( 1 + PO0 For = ~a~,c&nz(

1 -I- Xai,b,

aa = nlmlM1/2p’,

aI =

b, =

M,[(rn,

b, =

(m,

i- ma +

-t- ma +

a3=i M1M1/4,u2, b, A M,(rnl

Mi = rapidly

-1/‘4ni(ni decrease

By linear

as

O(ni/n‘)

combination,

* . *), 403=

Ca:,a:,af.,a3(

- - s),

I -i- Xa:,b3

+

-

- -), l.3a)

MJ2p,

2/p)’

-

rnj -

rn: -

4/pz]/2n,ml,

2/pL)M,,'2p, a3 =

-i- ma f

+ p),

i-

+

MIMIM3/8p3,

3//r)&,

mi -

l/(ni

. we get

6, =

-

/x),

Similarly the

[(8mi

MO = we obtain

even

i- 8;~)’

-

li4n(n

-I- %m; +

-I- p),

the higher

64,‘$]M,!32p,

and the

order

coefficients

terms.

and odd functions

(8b)

3.

Semi-Contracted -.

The S-Series shifted

back

(shifted

one position

Portion series)

- series

in successive

formed

orders.

by the

contraction

of

the

leading

term

Hill’s

64

Form111 ae :

ResS(nl,n2,..*,nl)

=

A,@)[ 1 -

A,(p)s Composite series

of convergence

repeated.

The calculation leading

.ZCZ~,~-~~“~~~G(~-I- 2in)]-‘.

Formulae2

Discussion

the 3’d -order

is the same as in the wholly contracted in the semi-contracted

synthesized

before,we

and imaginary parts

the real

q:**(t)

= 2 n

&**(t)

Or, using linear

portion

portion,

and need not be

is more complicated,

for example

term may have 3 possiblities

They should be separately obtain

Equation

=

x

n

combinations

according

to the tree graph. and resolutions

Using the same method as

bv even and other powers

{Pn(t> -#- iqn(t)}ef(Zn--a)r, {p,(t)

-

iqn(c)}e-i(2n-c)t,

we get the even and the odd functions

.

65

Hill's Equations

aI =.-

’

MOMI,

2P

(9b)

6,=~*M,M,M,M,+~*M~M~fM,M,(M,+

b, -~MoM,M2MsM,

16,~'

;+Zrni

MI),

+-&(M,+

M,). MOMI~MIM3MI.

......

4. HILL FUNCTIONS 1. Hill Functions of the First Kind Continuing the above results (7) - (91, we find that between the basic solutions of Hill's equation and of its degenerate equation, there exist the correspondences

H,(t) - T*eQ - C{P,(r) + iQ"(t)}ei(2"--r)r, H,(t) -

(101

T/r' - E{P.(t) - iQn(t)}e-'(2n-~)'e

The corresponding function is a Taylor-Fourier series and this series and its second-order derivative are absolutely convergent.

P,(t),

Q”(t)

are respectively series of ascending

even and odd powers, whose coefficients series in the constant term of the parameters of the equation: P&t) = Zpo,2J2, P.(t) -

pzcl+ p.0 + ~pnd;

t+',Q.(t)= Q..zr+ltt+l. (fJa 1) @l(t)- ~qo,,r+d .. . Verlflcatlon.

Substitute CP:(t)intothe equation, the whole of *(XB*.*X)portion

and the remaining terms are the wholly and semi-contractedportions in T' Substitute qz*,qz** %

disappears

*(HH***P)

into the equation, compare the coefficients of

of various orders they exactly cancel out the remaining terms. 2. Hill Functions of the Second Kind

Linear combinations of H,(t),H,(t) give the even

and odd functions H,(t).H,(t):

(11)

X(--r)

=H,(t), HA-t) 3. -H,(t). 3. Floquet's Solution Because %(t)(o= 1, 2) is a basic set of solutions and %(r + X) is also one, we have %(t + z) = c

r

G,(P&),

(a, r = 1, 2).

66

Equation

Hill's

Here aOr are constants. From pm(r)we form the general solution

Adding the condition of solution with a multiplier we obtain Floquet's solution -_

i.e. F(c + R) = U(t),

1 - e'"Y*

Hence we obtain Fuchs basic equation laij

ksii\

-

-- 0,

0.

241 + 1 I-

or a2 -

where 9 is a constant independent of the choice of the basic set 4(8,, 6,

* - *) * (ait + au)/2 = Erpdn) -t ipJ!=Il/Z.

Hence we have gtv

1=3f&+f*

E

and x {c2m --PP.(O) + p:(o)),! Let the solution (11) be divided, respectively, by cPn(a> I 0 then the Hill Functions of the second kind are normalized in the following sense: r-I:(O) = 0,

E&CO) - ‘1,

H:(o) = 1,

H,(O) -0, in this case we have

To find the boundary of pseudo-periodicmotion, we have a(&,

01,

* * *) *

$-

-

2

Pm -

mw

+

H;(n))

#wosda8%

t

##(n)

f

g&f)]

%I), “a /c

t9& The nature

if

* ‘P.(O),

Q*(=)P,(=).

of motion is determined by VR > O(~V -UR

UR - 0,

If yK(o

-

H*(=)

Yl

*

0;

;

divergent motion if

YR>

$- hf).

We have pseudo-periodic motion

0; ; and periodic motion if pa = IQ== 0;

then we have decaying motion. See the following table.

a VIZ

vi

motion

a-c-

I

h(akv5TT) 0

divergent

-1cago

o&a<1

0

-&s-y -

0

3)

1ca -

h(afv?Fi)

fecda

pseudo-periodic pseudo-periodic

Thus the condition for pseudo-periodic motion is

--I ~*c%,~,cct***>
0

divergent

Hill’s

If

9>1or4<-

Equation

67

1 then the motion is divergent.

5. DISCUSSION Unlike the theory of infinite

1.

give an explicit

expression

The essence

2. explicitly

an irregular

the usually

to Hill

of Poincark’s integral”

inexhaustible,

hence this

is that the primitive

contains

a definite

type,

that is,

type of series

algorithm

in this

the higher

may be called

is impossible

form of an irregular

but a kind of new function,

The method proposed here is to synthesize

3.

the method proposed

(1887) pronouncement that “it

assumed Floquet solution,

expansion are of a proliferate

determinant,

paper can

function. to write

integral

the coefficients

order correction

out

is not

of its

terms are

tree series. such functions

and can be made accurate

by tree graph method, it

to any order in a,

according

to the

equations,

thus

tree graph. Therefore,

4. extending

the present

the range of application

Because of limitation orders

of space,

and the expressions

published

method can be used to solve of the analytic

certain

for their

general

discussions,

coefficients

non-Fuchsian

theory. the tree graphs of the various

have been omitted.

elsewhere.

REFERENCES LI ]

Eill,

H. 0..

dm.

J.

Xufh.,

l(1878).

i 2 1 f%wnrQ, II., Null. Sot. Jfiatlt. France, l-t(lSSti); 1:: j \-on Koch, W., .4cfa Ytd., %!18%), 16(1892), 14 j Linponmoff, A., C. R., 73(1896~, 78ClPP9). i 9 ]

Wintrier,

A., Math.

Z., 24(19”G).

1 6 1 Kosi~v, N. A., Bstm. Xaclrr., 239(193(l). 1 7 1 Erillouin, L., Quart. :lppl. Math., 6(1948). j h 1 Sl;~gnus,IV., Winklct, S.. Hill’s Equation

(1966).

.-tcfn Jfrrlh., ~(1901).

IOilPTJi)

These details

will

be