HELP The

Photocopying permitted bylicenseonly the Gordon and BreachScience Publishersimprint.

Printed inMalaysia.

The HELP Inequality for Hamiltonian

Systems

B.M. BROWN

a,

W.D. EVANS^bandM.MARLETTA^c’*

aDepartmentofComputerScience, Universityof Wales,Cardiff,

PO Box916,CardiffCF23XF,UK;^bSchool of Mathematics,University ofWales, Cardiff,Senghennydd Road, Cardiff, CF2 4YH, UK; DepartmentofMathematics andComputerScience,University of Leicester, University Road,

LeicesterLE17RHUK

(Received30 July 1998; Revised 15September1998)

Wc extend the Hardy-Evcritt-Littlcwood-Polya inequality, hitherto established for 2nth orderformally sclfadjoint ordinary differential equations,toa wideclass oflinear Hamiltonian systems. The method follows Dias (Ph.D. ^{thesis, Cardiff:} University of Wales,1994)butwithouttheHilbcrtspace settingwhichheuses.

Keywords: Hardy; Everitt; Littlewood; Polya; Hamiltonian system; Inequality AMS(MOS) SubjectClassifications: ^26D10(26D1,34B, 34L,47F05)

1

INTRODUCTION

In

the seminalpaper

10]

Hardyand Littlewoodintroducedthe inequality

(/0 ^If’l

^2dx

^<

⁴

/0 ^Ifl

^2dx

/0 ^If"l

^2dx

⁽¹⁾

which is requiredtoholdfor all

functionsfsuch

^{that theright-handside}

of

(1)

is finite. Equalityisattainedwhen,for anyp

>

0 and for 0

<

x

<

o,

f(x)

^A

exp(-px/2)

sin(px/2

7r/3).

* Corresponding author.

57

No lessthan threeproofs of

(1)

may be found inthe book ofHardy

etal.

[11].

Everitt in

[9]

introducedthe inequality

2

{p(x)lf’(x)[

²

+ (q(x) TW(X))lf(x)[ }

^dx

fa 12fa

< _K(r) w(x)lf(x)

^dx

w(x)lA/l[f](x rf(x)[

^2dx

(2)

where isthe second-order Sturm-Liouville operator

kd[f]

^:=-1

(-(pf)’ +

^q

f)

w

with b

>

^a

>

cx,

w(x) >

0, p(x)

>

0, q(x)E with 1/p, q locally integrableandrareal parameter.In

[9]

itisassumed thatJ4^isregular at a and singular ^{at b} and satisfies the so-called strong-limit-point condition.Underthese conditions Everitt showed that theexistence or otherwiseof

K(r)

in

(2)

dependsonthespectral propertiesof

A[f]

^and the existence criteria for

(2)

^{can be}determined intermsof the behaviour of theTitchmarsh-Weylm-function associated with

.M.

Everitt’sproof in

[9]

ismodelledonthecalculusof variationsproofgiven in 11 for

(1).

Howeveranoperator theoreticproof^isgiven by Evansand Zettl in

[8]

and bothproofsarereproducedin the article

[6]

^{ofEvansandEverittin} which aneatcharacterisationofthe criteria foravalidinequalityisgiven in terms of theTitchmarsh-Weylm-function: it is thislatterapproach that will serveasamodel for the workreportedonin thispaper.The inequality

(2)

has been extended

[7]

^tothecasewhen bis aregular pointor k4^isinthe limit-circlecase atb.Thetheoryassociated with the existence ofananalogous inequality,inwhichA//isnowtheformally symmetric 2Nth order expression

N

:=

Wj=0

isproved byDiasin

[5] (see

also

[2]).

Inthislatterinequalitythe criteria for the existenceofavalid inequalitycan again be determinedby ^the spectral properties of the self-adjoint operator generated by

A

in

L2w(a,b)

^{but this time it is formulated in terms of} the associated Titchmarsh-Weyl Mmatrix.

In examplesofthe inequality,theproblem ofprovingthe existence and also ofdeterminingthe bestconstant is oftenahardanalyticproblem, depending asitdoeson knowledge of the closed form expression for themfunctionorMmatrix.

A

complete catalogueof all knownexamples tothe present time istobefoundin

[3]. In

view ofthe analytic difficulties inobtaininginformationabout the best constant in particularexamples, anumerical approachtotheproblemhas beenundertakenby Brown

etal. This isreported in,forexample,

[4].

This paper reports onthe development ofa

HELP-type

inequality associated with thelinearHamiltonian system

(AA + (3)

where Yis a 2n vector and

Jt

and

B

are 2n x 2n real matrices with

.A* Jt >

0 and/3*

B.

Jisthe 2n x 2n matrix

0 ^-i)

^andIthenxn

identitymatrix.Itisshown in

[16]

that the 2nth orderformallysymmetric differential equation

A//[f]

^:=-

(pj(x) y

Wj=0

may bewritten as a Hamiltoniansystem, andin

[1]

it is shown that the matrix-vectorSturm-Liouvilleproblem

(P(x) Y’)’ + ^Q(x)

^{Y ,kY}

whereP and

Q

are nxn matrices and Yis an n vectormay also be embeddedinaHamiltonian system. Thiscurrentworkis notjustasimple extension of the work of Dias

[5],

since as

A

is semi-definite andnot definite,weareunabletousetheHilbertspace settingfor theproblem that hewasabletoexploit andareforcedtoworkwithout thisabstract structure. Howeverthegeneralapproachof the secondproofⁱⁿ

[6]

^can be made toworktoyielda

HELP-type

theoryfor linear Hamiltonian systems. Also examples can be found that are not covered by any previous work.

2

THE HAMILTONIAN SYSTEM AND THE DIRICHLET INTEGRAL Let C(x)

and

B(x)

be real symmetricn xnmatrices, and

A(x)

an nxn real matrix, such that the elements of

A,

BandCarelocally integrable over aninterval

[a, o).

^LetKbean nxndiagonalmatrixof the form

K

diag(kl

(x),..., km(x), O,

0,...,

0), (4)

wherern

<

nandwhere thediagonalelements

kj

^havelocallyintegrable reciprocals,togetherwiththeproperties

ess

inf(kj) > O,

j 1,...,m.

We

denote by Kthe 2n x 2n diagonal matrix whose first rndiagonal elementsarethe

kj

and whoseremaining2n rndiagonal elementsareall zero;wedenoteby

K and/

thepseudo-inverses

ofKand/respectively.

Thus,forexample,

K

diag(1/kl

(x),..., 1/km(x), ^O,

^0,...,

0). (6)

For absolutely continuous n-vector functions u and v we define the operators

L1

^and

L2

^by

[u, v] -v’ + C(x)u (x)v, Z:[u, v]

^:=

’- ^{(x)u (x)v.}

(7)

For

2n-vector functions y, partitionedin anobviousnotation as Y--

(UYvy )

wedefine

[, v])

z(y)

^:=

L:[y, Vy] (8)

Wedefinea bilinearform

(., .)

^by

If, g) rg ax u}u

^dx,

⁽⁹⁾

forany2n-vector

functionsfand

g for which theintegralconverges.

We

alsodefinetheset

of

^admissible

functions

^{4tobe theset}of absolutely continuous2n-vector

functionsfsuch

^that

I(f,f)l < +c, (10)

L2[uf, vf]

^OE]

n, ₍₁₁₎

(I-- KK)L1 [uf, f]

^{0 ]1}

n. ₍₁₂₎

We

make thefollowing crucialassumption.

ASSUMPTION

The bilinear form

(., .)

^ispositive definiteontheset

A

of admissible functions y which are solutions of

L(y) AKy

for any 6

C.

FollowingReid

[15]

^wedefine the

Dirichletform

associated withtwo n- vector

funetionsfand

^g:

O(f, g):-- fa [vnvg-+- ufug]dx.

Now

supposethat

f,

g 4. Wehave

(f,/tL(g)) ug(IZl ^{[Ug, llg])}

^dx

u)Ll[Ug, ^Vg]

^{dx (using}

⁽¹²⁾

^forg)

Uf (--

_Vg

+ ^Cug _AT ^lg)

^dx

[-u}g] + (Auf + Bf)*vf + u}Cug u}Ag

^dx

(a [z, z] ol [-}1 + ^e(f,g.

Thuswehave provedthe integration-by-parts formula

(f,/tL(g)) _{[-uvg] + D(f,} g) (13)

forf,

g

A.

3

THE HINTON-SHAW-TITCHMARSH-WEYL M-MATRIX

In this sectionwe review very briefly someresults from the paper of Hinton and Shaw

[12].

Thesewillbe very important in the^restof the paper.

Thepaperof Hinton andShaw dealswithsolutions of Hamiltonian systems of differential equations of the form

L(y) AK(x)y, A

E C.

(14)

In general,^notall solutions y of this equation will satisfy

(y, y) < + .

Thosewhichdowill lieintheset

.A

of admissible functions because the structureof ensures that

(11)

^and

(12)

^aresatisfied by_Uy,Vywhen y solves

(14).

Inaddition toAssumption 1,HintonandShaw require the following.

ASSUMPTION 2 (’Limit-point assumption’) Supposef,g E

Jr.

Then

(15)

Here denotes Hermitianconjugation.

THEOREM 3.1 (Hinton ^and

Shaw) Suppose

that

A-0,

and that Assumptions and 2 both hold. Then the

differential

^equation

(14)

possessesnlinearly independent solutions

bl (x, A),..., bn(X, A)

such that

(bj,j) < +o,

j 1,...,n.

Moreover,

these solutionsmay be normalisedso

that/f(x, A)

^{isthe 2n xn}

matrixwhose columnsare

b](x, A),..., bn(X, A)

then

(a, A) (-M(A) )

^I

where

M(A)

^{is an} analytic n n matrix

function of ^{A (called}

^the

Titehmarsh-Weyl matrix). Thismatrixhasthepropertythat

M(A)

^{is a}

Nevanlinnafunetion,inthesensethat

is positive

definite for ^{A >}

^0,

M(A)

negative

definite for ^{< O. (16)}

Moreover,

4

CRITERIA FOR A VALID HELP

^INEQUALITY

In

this sectionwereproduce^theresultsofDias

[5]

onexistence ofa

HELP

inequality, but in themoregeneralcontextofaHamiltoniansystem

L(y) AK(x)y, (18)

orequivalently

L1 [u, v] AK(x)u, L2[u, v]

^0.

(19)

Weassumethat the followingstrong limitpointproperty holds.

ASSUMPTION

3 Forall

functions ^f,

^gE

^A,

lim

uf(x)*vg(X)

^{0 ]1}

n.

X----O

(20)

Notethat Assumption 3 impliesAssumption2.

We

definethespace

A0

toconsistofthose

functionsf ^A

^{such that}

uf(a)

⁰

n, _vf(a)

0 ^]t

n. (21)

ForA _#

+

^{iv,v 0,wedefinethedeficiencyspaces}

N+ ^(A)

^andN_

⁽⁾

by

N+(A) {f ^AlL(f) ARf), ^N_(A) {f

^E

^AlL(f) Rf}.

(22) As

the matrices

A,

BandC occurringinthedefinitions of

L1

^and

L2

^are

allreal-valued,andas isreal-valued,itiseasytoseethat y E

N+(A) = ^{37 N_(A).}

In

particular,therefore, dimN+(A)= dimN_(A). ^Recallthat the matrix

(x; A)

introducedinSection 3hascolumns whichformabasisof

N+(A).

Thus any function h in

N+(A)

^maybeexpressed^{as a}linear combination of the columns of

0(.; A),

h

0(.; A)a,

whereaisann-vectorofconstantsdepending onlyon

A.

We shallestablish

HELP

inequalitiesfor functions

fin

^theset

/x .=zx0,

N/(a),

^N_

(23)

The first resultwerequireisthat, despitetheappearanceof

A

^ontheright hand side of

(23), A

^doesnotdependon

A.

LEMMA4.1 A=.,4.

Proof

^{It is clearthat A C_}

^A

^{so wejust need to prove the opposite}

inclusion.

Following the notation ofSection 3, we form the 2n xn matrix whose columns are solutions of

L(y)= AKy,

subject to the initial conditions

(a;) (-M())I ⁽²⁴⁾

Itisnotdifficultto seethat the columns of

0(.; A)

span

N+(A)

and the columns of

0(.; A)

span

N_(A).

FromTheorem 3.1 wealso knowthat

M(A) M(A)

^{and hence}

Thus

(x;

((a;A),(a;X)) -()I

i 0

Hencethe 2n x 2n matrix

((a; A), (a; A))

^isoffull rank if andonlyif

3M(A)

^isnonsingular.

From

Hinton and Shaw

[12]

^weactually know that

M(A)

is positive definite for

A >

⁰ and negative definite for

A <

0,which certainlytherefore impliesthat

((a; A), (a; A))

^{isof full}

rank. Thusgivenanyfunction

fE ^A

^{wecanchoosea2n-vector}

e=(

^cl

)c2

suchthat

uf(a)) ^{((a; A) (a;/))c (a; A)Cl + (a; )c2.}

f(a)=

vf(a)

Defining

g(x) =f(x) (x;/)Cl I/(x; )C2,

itisclearthat gis anelementof.A suchthat

ug(a) vg(a)

^0E

Thus g E

A0.

^Also,

^{(.; )k)C} N+(,)k)

^and

(x;/)C2 N_().

Hence

f ^A,

^{whichcompletestheproof.}

Remark This result is essentially the

Von Neumann

decomposition formula see

[13,

Lemma

10.2.5].

^Wehave been forcedtoproveitin this verydirectway bythe fact that there doesnotappearto be anatural Hilbertspace setting for theseproblems,exceptinspecialcases.

Lemma4.1 establishes thedecomposition

e4

Ao

^@

N+ (/)

^@N_

(). (25)

Weshallusethisextensivelyinthesequel.

LnMMA4.2 Thedirect sum appearingin

(25)

^{isanorthogonalsum with}

respecttothe sesquilinear

form

(f,g) (t (L(/) #f),t(L(g) #g)) + ^u2(f,g). (26) Proof ^Suppose thatf Ao

^{andg E}

^N+(,k),

^{so that}

^L(g)

^#Kg

+

^iuKg.

Then

(f, g), (/t (L(f) lzf), ivg) +

^u2

^{(f, g)}

iu

(L1 [uf, vf])*KtKKtKug -i#u u)KKtKKtKug

We

nowuse

(12)

^tosimplify the firstintegral,andtheresult

KK

K K tosimplify thesecond.Theseyield

fa fa

(f g)

_A ^it,,

L [uf, ^vf]*Ug

^iAu

uKug. ⁽²⁷⁾

Using theintegration-by-parts formula

(13)

^twice, togetherwith

(20)

and

(21)

^toeliminate theboundaryterms,weobtain

(28)

Now

L(g) AKg

implies that

Ll[Ug Vg]--,,Kug;

substitutingthisback into

(28)

gives

Ll [uf, vf]*Ug uKug. ⁽²⁹⁾

Substituting

(29)

^into

(27)

^yields

(f,g)

^=0,

asrequired.

Similarlyone mayprovethat

(f, g)=

0 for

fE ^A0

^{and gE}

^N_(A).

^It

remains toshow that

(f, g)

0

forf ^N+(A)

^{and g}

^N_(A).

This case is easy:wehave

L(f) Afand ^{L(g) A[fg,}

^sothat

(f,g), (iutf, iutg) + ^v2 (f,g)

u²

u}KId ^KKIO

^Ug

⁺

^u2

^{u}Kug O,}

since

KKtK=

K.Thiscompletestheproof.

LEMMA4.3

<I,, ff) 2u@(M(A)). (30)

Proof

Using the definition of

(., .)a

^wehave

(, ) (t(L() #),/t(L() #)) + v2(, ).

Nowwe usethe fact that

L(9) (# + iv)K9

^toobtain

(9, 9)

^u

(t/9)*KR(t/9) + u2(9, 9).

Fromtheidentity

RRtR R

thissimplifiestogive

(9, 9),\

^2v2

(9, 9). (31)

Butnow

1

9"L(9)

- {u,(a)v,(a) + D(9, 9)},

the last identityfollowingfrom theintegration-by-parts formula

(13).

Fromthe initial conditionson9,wehave

u(a) M(A)

^and

v(a) I,

yielding

1

{-M*(A) + D(9, 9)}

(,I,, ,I,)

Multiplying both sidesby

A

andtakingthe imaginary partsyields

v(9, 9) -(M*(A)) .(M(A)), (32)

thelast equalityonthe rightfollowingfromthewell known fact thatMis a symmetric matrix

[12].

^Combining

(31)

^and

(32)

gives the required result.

LEMMA 4.4 Let

f ^E.A

^{be a} real-valuedfunction, and suppose that

f fo +

^h

+

^h,where

fo Ao

^{and h}

^N+(A),

^{h EN_}

(A). ^Suppose

^that

h 9a,

(33)

wherea

(a

^l,

an)

isaconstantvector. Then

(f,f),x (J,J),x + 4va*(M(A))a. (34) Proof

^{From Lemma4.2we}know that the

sum ⁺

^h

⁺

^{hisorthogonal}

withrespectto

(., "/.

^Thuswehave

(f,f)a (fo,fo) + (h,h) + (h,h) (J,3)a + 2(h,h). (35) From (33),

^itisclearthat

a*( )aa.

(h,h)

Using

(30)

yields

(h, h), 2ua*(M(A))a. (36)

Combining

(35)

^and

(36)

^givesthe required result.

For

real-valuedfE ^A

^consider

^{(f, f).}

^From

⁽²⁶⁾

^wehave

(f,f), ([(tL(f), [(tL(f)) 2#(RtRf, RtL(f))

+ #2 ([(t[(f, [(tiff) + u (f,f)

(gL(f), gL(f)) 2/z(f, gL(f)) ₊ IAI

²

^(f,f),

wherewehave used the identity

*

^{severaltimes.Nowweapply}

theintegration-by-partsformula

(13)

^totheterm

(f, L(f))

^toobtain

(f, f) ([tL(f), ’tL(f)) 2#D(f, f) + I 12 ^(f,f 2#uf(a)*vf(a).

(37) We

definetheform

J(f) <’tL(f), ilL(f)) 2#D(f, f) + IAI

²

^{(f,f). (38)}

Then

(37)

immediately yields

J(f) (f, f), + 2#uf(a)*vf(a). (39)

As f=fo +

^h

+

^{h with}

f0

^EA0,

^{Eq. (21)}

^yields

uf(a)= 2N(uh(a))

^and

vf(a) 2N(vh(a)).

Equations

(24)

^and

(33)

give

uh(a) -M(A)a, vh(a)

^a.

(40)

Substituting into

(39)

^weobtain

J(f) (f, f)A 8#((a))X(M(A)a). (41)

Combiningthiswith

(34)

yields

J(f) (J, f0) + 4ua*.(M())a 8#((a))(M()a). (42) We

nowseparate

M(A)

^{anda intoreal andimaginary}parts, denotedby

M(A) MR

^{q- iMi, a aR 4-}

iai.

Some simple algebrashows that

(42)

mayberearrangedintheform

4 4

J(f) (fo, fo), +- ^(uai +/ZaR)TMI(uaI + ^paR)+-- at.(-A:ZM(/))aR.

(43)

In order to make further progress we must introduce a further assumptionabout theset4of admissible functions.

ASSUMPTION4 Forall

non-zerofE ^A, (f,f) >

^O.

Withthisassumptionwecanrewritethe expression

(38)

^asaquadratic formin p :=

1)1.

^{Let peiso}

that/z

pcos4.Thensomesimplealgebra shows that

J(f) & ^(f)

^:=

^{(f, f)} (P ^{D(f, f)} (f,f)

^cos

^0’

²

+ (L(f) L(f)) ^{D(f, f)}

^cos2

^b

(f,f)

(44)

Similarly,whenever

A

peⁱ⁽⁺

r),

we obtain

D(f,f)

^cos

)

²

J(f) Jo+r(f)

^:=

(f,f)

P

+ _(f,f + (tL(f) tL(f)) ^D(f,f)cos

²

^b

(f,f)

(45)

Givenanynon-zero

fand

^any

b

^E

^[0, 7r/2)

^onemayalways choose

D(f,f)

^cos

(46)

Thiseliminatesthefirst term onthe right hand side ofoneof

(44, 45).

^The remainingtermwill

b

non-negative if andonlyif

D(f,f) <

^sec2

(f,f)(tL(f), tL(f)). (47)

Thisisclearlya

HELP

inequality,provided0

< b < 7r/2.

^{Weare nowin a}

positiontoprovethe following theorem.

THEOREM 4.5 Let Sbetheset

of

^{all values}

of 4

^E

^{(0, 7r/2]}

^{such thatthe}

followingtwoconditionshold:

(-A2M(A))>0 (A=pe i0)

}

^Vp>0.

⁽⁴⁸⁾

(A2M(A)) ^>_

⁰

(A

^pe

i(o+r))

Then

7r/2

^$.Let

00

inf(S). ^ThenaHELPinequality

D(f,f) <_ t(f,f)([ftL(f),tL(f)) (49)

holds

for

^all

f ^A if

^andonly

if Oo < 7r/2.

^{Moreover,the bestconstant in}

the inequality^isgivenbyt

sec20o.

Proof ^Suppose

^that

00 < 7r/2.

^Choose

b=00.

^Given

f,

^choose p accordingto

(46),

eliminating the firstterm onthe right handside of one of

(44, 45).

The conditions

(48)

^{thenimplythat}

(47)

holds, giving^a

HELP

inequalitywith

sec200.

Thus the condition

00 < r/2

is sufficient fora

HELP

inequality.

To

see that sec²

00

^isthe best constant, suppose^we look for a smallerconstant2 sec²

q

forsome

b

^E

^{(0, 00). By}

definition of

00

^there

will exist somep

>

0such thatatleast one of thefollowingconditions fails:

(-)2M(,)) ^>_

^{0, ,k--pe}

^i, (50)

(A2M(A)) ^_>

^0,

^A

^pe

^i(o+r). (51) Suppose

the first condition fails. Thenwe canfindarealvectoraRsuch that

With #=pcos0, v=psin0, we choose a real vector

a =-/z/PaR.

Nextwedefine h

(aR +

^iai)and

f

^{h/hsothat from}

⁽⁴³⁾

^wehave

J(f) <

0. From

(44)

^this shows that the HELP inequality fails with

=sec2b.

^{This covers the case where}

⁽⁵⁰⁾

^{does not hold.}

^A

^similar

argument dealswith thecase where

(51)

^doesnothold, so the choice

secZ0

is clearly best-possible. ^This also proves that the condition

00 < 7r/2

^isnecessary fora

HELP

inequalitytohold.

5

GENERALISATIONS

Itwould be nicetoremoveAssumption 4,and indeedwe cando thisforat least part of the resultⁱⁿTheorem 4.5. If Assumption 4^isremoved then the condition

00 < 7r/2

^{ofTheorem4.5 stillimpliesa}

^HELP

^inequality

for

thosenonzero

functions f

^E

^.At

^suchthat

^{(f,f) >}

^O.

Whathappens

iffis

^anonzerofunction such that

(f,f)

^0?

^In

^thiscase

(38)

loses theterm

IAI

²

^(f,f)

^{ontherighthandside. Equation}

⁽⁴³⁾

^still

holds,andsothe condition

00 < 7r/2

^impliespositive-definitenessof

j(f) (t L(f), L(f)) 2/zD(f,f).

Wecan

take/z

^tobe arbitrarilylargeandofeithersign in

(38)

^{andso we} deducethat,when

00 < 7r/2,

(f,f)

⁰

= ^D(f,f)

^O.

Thuswestillhavea

HELP

inequalityinthiscase,albeitatrivialone.

6

EXAMPLES

We present three examples of

HELP

inequalities associated with Hamiltonian systems. Two of these may be ^derived from

HELP

inequalities for even-order differential operators; one, however, is completely^new.First,however,^werequireatechnicallemma.

LEMMA

6.1 (Compactness

Lemma) Suppose

that there exists

b

^E

(0, 7r/2)

such that the conditions

(-A2M(A)) >

^0,

A

pe

i, (52)

(A2M(A)) >

^0,

A

-peⁱ

(53)

hold

for

^{all 0 [,}

^7r/21, for

^allsufficientlylarge p and

for

all sufficiently smallp. Then, in thenotation

of

Theorem 4.5,

00 < 7r/2,

^andsoaHELP inequality holds.

Proof

The Nevanlinna property of

M(A)

ensures that

(52)

^holdsfor 0

7r/2. Hence

bycontinuity, for each p

>

0 there exists

Cp (0, 7r/2)

^such that

(52)

^{holds forany0}E

[bp, 7r/2].

^{Wecan choose}

Cp

^{as a}continuous functionof_p.Thus given any^set[Pmin,

Pmax],

^{we can}choose

b. ^{(0, 7r/2)}

such that

(- A2M(A)) >

0 for

A

pe

i0,

for 0

[b., 7r/2]

^and P^E[Pmin,

Pmax]. However,

by hypothesis,wecanchoose_Pminsufficiently small and Pmax sufficiently large to ensure that

(52)

holds for all 0 [b,

Tr/2],

^{for all} p(O, Pmin]l,.J[Pmax,X3). Hence

(52)

holds for all 0 [min(b,,

), 7r/2]

^{for all}p

>

0.

A

similarargument deals with

(53).

6.1 Example

I

We

considerthe Hamiltonian system withn 2for which, in our earlier notation,

(0 0) A=(0 1)

C= 0 0 0

(1 0) B__(2 0)

K= 0 0 0

(54)

Inordertodeterminethe square-integrablesolutionsofthis systemwe require theeigenvaluesand eigenvectors of the matrix

Theeigenvaluesarereadilyseentobe 4-#+,where

#+/-

(1 ^A

^4-

v/A - ^{A) 1/2. (56)}

Hereweadoptthe convention that the squareroothaspositivereal part.

Theeigenvectorassociated withaneigenvalue_#is givenby

V--

(]Z(lZ

^2-]-

1),

^{]Z2t_ _]_1,}

-/]2,/)T. (57)

Given aneigenvalue _#and eigenvectorvthe associated solution of the Hamiltonian systemis vexp(#x). It^iseasy tocheck that thepositive- definiteness condition inAssumption holds for any^linearcombination ofourfour solutions here.

For

A

in thefirstquadrantof thecomplex planewith

IA[

large,itiseasy to seethat the

eigenvalues

with negative realpartare _-/+ and#_.

The associatedsolutionsyl(x)andy2(x)of theHamiltoniansystemare givenby

#_

(#2_ + 1) exp(#_x)

(#2 + ^A + ^1)exp(#_x)

(Yl (x)y2(x)) --A#

²

^exp(#_x)

A#_

exp(#_x)

_#+(#2+ + 1)exp(-#+x)

(#+ + A +1)exp(-#+x)

_p,2+ ^exp(-#+x)

-A#2+ ^exp(-/z+x)

From

Theorem 3.1weknow that

v

(yl (0)y2 (0))

i

forsomenon-singularn n matrixV.Thisallowsus todeduce that

{ ^(u-

-(1

-I-

#_/z+)/

-(1 -+-/z_U+)/A

)

(U- -/z+)(A + 1)/A (58)

Putting

A

pexp(i0),0

<

⁰

< _7r/2,

p

>

^{0,it is easytoseethat}#+

1/

and#_

ix/x/ ^exp(i0/2)

^forlargep.This allowsustodeducethat

(-,,2ml ()) -x/p

^3/2

_{cos(30/2), (59)}

(-,2m22())) -xp

^5/2

_cos(50/2), (60)

(_2m12() ^p3/2 cos(30/2). (61)

Itisthereforeeasytocheck that for allsufficiently largep,thereexistsa sectorsurrounding 0

7r/2

^inwhich

(-A2M(A))

^ispositive definite.

Nextwecheck thecasewhere issmall.Itiseasytocheck that when issmall,theeigenvaluesof thematrixSin

(55)

with negative real part are _#+and _# Also,

#+ ^,,-1 #_

2’

2

The M-matrix is obtainedbyreplacing#_ by #_ in

(58),

and hence

(-2 .2/2)

’k2M(A) _,k2/2-2,X

Putting pexp(i0)weget

_,k2M(A) ( _p2/2

^2psinO

sin 20

\

_p2/2

sin20

’

2psin 0

J

Againitis easytocheck that for all sufficientlysmall p, thereexists a sectorsurrounding 0

7r/2

in which this matrix ispositivedefinite.

Wemust nowcheck the case where

A

pexp(i(0

+ ^7r)),

⁰

^<

⁰

< 7r/2:

againwe mustconsiderseparatelythecasesoflargeandsmallp.Wehave

/z+

(1 +

^pei4-

V/p2e

^2io

+

^pe

^io) 1/2,

andit iseasytocheckthat,bothinthecaseof small p andinthecaseof largep, _-#+ and -#_ arethe eigenvalues ofSwithnegativereal part.

Thus, changing#_ to _{-#_} in

(58),

+ v-)/.

M(,) -(1 -v+v-)/,

-( +-/)

’

-[( + )/](+ + -) )

For largep^wehave

#+

x/v/-fiexp(i0),

whilefor small p

q- ^4-

21-V/-pe ^i0/2.

Henceitiseasily shown that forlargep,

sin(30/2)

9(2M()) x/’p3/2

sin(30/2)

sin(30/2)

)

-psin(50/2)

Itiseasily checked that for all sufficientlylargep thereexists a sector surrounding0

7r/2

^{in whichthis is}positive definite.

Forsmall pweobtain

M(,k)

2psin0

21

^{P cos 0}

1/2PCOS0)

and for all sufficiently smallpthereexists a sectorsurrounding 0

7r/2

ⁱⁿ

which this is positive definite.

By

the compactness lemma

(Lemma 6.1)

^wetherefore havea

HELP

inequality for thisproblem,valid forall

functionsffor

^which

(f,f) >

^0.

It is interesting to observe that for this Hamiltonian system, Assumption 4 fails. The function

0

f(x) _(1/2)e_X/C e-X/

(1/ x/’

^{e Xv5}

is anadmissible function with

(f,f)

^0.

6.2 Example 2

For

oursecondexampleweconsider the Hamiltonian system associated withthesecondordermatrix-vectorSturm-Liouville equation

_y,,+(Xo -xO) ^{Y=AY’ x(Oc)., (62)}

Here

Yis a2-vector.Thesystemcanbecastin the Hamiltonian form

-V’= AU+

0

U’=

V.

ItiseasytocheckthatourDirichlet form

D(f,f)

^{isthesame astheusual} Dirichlet integral for suchaproblem:

D(f,f) Y’ll

²

^+Y*

^x_{0 -x}^{0 Y dx, where}

^f= y,

Itisalso easytocheck that Assumptions 1-4 all hold.

For

fixed

A

with

(A) :

^{0 letyl(x)denoteanontrivialL2solutionofthe}

scalar equation

y" +

^xy

^Ay

^{and let}Y2denoteanontrivialL²solution of the scalar equation

y"

xy

Ay.

Then

(y,)

⁰

^Y= (0)

^y

areL²solutions of

(62). We

^formthe matrices

U(x)= (yl(x)

⁰

^y2(x)

⁰

) ^{V(x)--(yx) y(x)}

⁰

)

and observe that

(W)

^{specifies an} admissible matrix solution of the Hamiltonian system. TheM-matrixistherefore givenby

M(,X) v(o) r(o) -yl(O)/y[(O)

⁰

0

-y2(O)/yi(O

We

nowrecognise that

M(A)

diag(ml

(A), m2(A))

where _ml and _m2 are, respectively, the scalar m-functions associated withtheSturm-Liouvilleequations

-y" +

^xy

^{Ay, -y"-xy Ay,}

The conditions for a

HELP

inequality reduce to the corresponding conditionsfor each of thesetwoscalar equations.

For the first equation there is only point spectrum. ^{The first few} eigenvalues, subjecttotheNeumann boundarycondition

y’(O)=

0,are given approximatelyby

A0

1.01879297,

A1

10.5507875,

A2

23.2333564.

(These

^were obtained using the code SL02F

[14].)

The equation

y" +

^xy

^Ay

^hasanassociated ’shifted’

HELP

inequalityoftheform

(2)

^for7- Ak,^k 0,1,2, Forthe second equation, the whole real line consists of continuous spectrum, and there ^{is a}

HELP

inequality [6, Example

5]

of theform

(2)

for any value of^7-.Thusfor the equation

(62)

thereis no

HELP

inequality, but thereare

HELP

inequalities associated witheach of the ’shifted’ equations

x(O, oo),

^k=O,1,2,...

As

anasideweobserve thatfor

(62)

^thereiscontinuous spectrumonthe whole real line with discontinuities of the spectral function at the eigenvalues of the equation

-y"+

xy

Ay

with boundary condition

y’(0)

0.

6.3 Example3

Weconsiderafourth-ordermatrix-vectorSturm-Liouvilleproblem

y(iv)+ (ql(x)

⁰

^q.(x)

⁰

)

^Y=

^AY,

^x

^{(0, o). (63)}

We

convertthisinto aHamiltonian systemby defining

(,) (-,,,,)

U

y

^V

y.

sothat

A-q(x)

0 0 0 0 0 0 0

-V’--

⁰

^A-qv.(x)

^{0 0}

U+

^{0 0 0 0}

0 0 0 0 0 0 0

V,

0 0 0 0 0 0 0

0 0 0 0 0 0 0

U’

^{0 0 0}

U+

^{0 0 0 0} V.

0 0 0 0 0 0 0

0 0 0 0 0 0 0

Once againit iseasytocheckAssumptions 1-4andit isalsoeasyto see that

D(f,f)

is equivalenttothe usual Dirichlet form associated with

(63).

Clearlytheoriginalfourth order equation

(63)

canbedecoupledinto twoscalarfourth order equations. Under suitable limit-pointhypoth- eses, letyland_Zldenote linearlyindependent L²solutions of

y(iV) +

^ql

^{(x)y Ay,}

and lety2and_z2denote linearlyindependent L²solutionsof

y(iV) + ^{q2(x)y Ay.}

Wecanthen form four admissible solutions forourHamiltonian system:

f (x)a" (y (x). O. y (x). O. y’ (x). O. y (x). 0).

f2(x)

^T

(Zl (x),

^0,zl

(x),

^0,

z"(x), O,zt(x), 0),

f3 (x)T _(0,

Y2

(x),

^0,

Yl ^(x),

^0,

^y’(x),

^0,

y _(x)),

f4(x)

^T

(0, ^{z2(x) O, zi(x O,-}

^zm[x

,,

^O,

_zo.x...

The 4 x 4 M-matrix is thus givenby

M()

ylO) ^zl(O)

₀

_y2(O)

⁰

_z2(O)

⁰

|y(O) z ⁽⁰⁾

^{0 0}

\o

⁰

^{y(O) (0)}

-y]" (0) -z]" (0)

⁰

0 0

-y’(0)

y’(o) z’(o) o

0 0

y(O)

-1

-’(0)

0

(o)

Thisgives

M(A) P[

₀

()

\

where 0 denotes the 2 x 2zeromatrix,

M(A)

^(j

^1,2)

denote the 2 x 2 Titchmarsh-Weylmatrices

zZ l l)(

andPisthe 4 x 4permutationmatrix

0 0 0

0 0 0

0 0 0

0 0 0

The matrices

M1

^and

M2

^areTitchmarsh-Weylmatricesfor the fourth orderproblems

y(iv) +

^ql

^{(x)y Ay, y(iV)} + ^{q2(x)y Ay.} (64)

Thus, as in the case of Example 2, the conditions for a

HELP

inequality in Theorem 4.5 become equivalenttothe conditionsfor both ofthe scalar equations in

(64)

tohave associatedHELP inequalities.

The coefficientmatrix

Q(x) (

^ql0

^{(x) q2(x)}

⁰

)

in

(63)

^canbereplaced by anymatrixof the form

RQ(x)R

^TwhereRis orthogonalandconstant,andthe same result will hold: the conditions for a

HELP

inequality for the matrix-vector system reducetothe conditions for

HELP

inequalitiesfor the associated scalarproblems.

7

CONCLUDING REMARKS

In this paper we have generalised the work of Dias to Hamiltonian systems.

In

so doing, we have been forced to abandon the natural Hilbert-space setting which Diasusesfor the 2nth order scalarselfadjoint case; in particular, therefore, ^our formula for the so-called

Von

Neumann decomposition has^hadtobeprovedinaverydirectway.

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