极限点型 Sturm-Liouville 算子乘积的自伴性

Sturm-Liouville

(

210094)

󰀁l(y)=−(py󰀁)+qy,t∈[a,∞),lk(y)(k=1,2,3)

l(y)Li(i=1,2)L2L1

l(y)=−y󰀁󰀁

+qy,t∈[a,∞),

Li(i=1,2,3)L3L2L1

MR(2000)

34B20,47E05

1

Sturm-Liouville

=−¯2l(ψ)h

d2ψdx

+x22m2ψψ–m–

h¯

–L2[0,∞)

Sturm-Liouville

[1,2,3,4]

Sturm-Liouville

P=

󰀓NakDk

,

Dk

=dk

k=0

dt

k,

t∈I=[a,∞),ak

[a,∞)

P

+

=

󰀓N(−1)kDkak.

k=0

a

a

0,02

2×2

(aij)1≤i,j≤n,AT,A∗

A

rankA

[·,·]N(t)

(1.1)

Lagrange

󰀔∞

󰀔

∞

P(y)zdt−

yP+(z)dt=[y,z]a

a

N(∞)−[y,z]N(a).

2003-03-20,2005-06-28.

Hamilton

(1.1)

n

A

A

3

Sturm-Liouville

[5,6]

369

QN(t)

[y,z]N(t)=(σz(t))TNQN(t)(σy(t))N,

(σz(t))TN

=z(t),z

󰀁

(1)

(t),···,z

(N−1)

󰀂󰀁󰀂T

1(N−1)

(t),(σy(t))N=y(t),y(t),···,y(t),

(1.2)(1.3)

QN(t)=(fjk(t))0≤j,k≤N−1,

⎧N−k−1󰀃󰀄⎪⎨󰀓(−1)hhDh−ja

k+h+1(t),(0≤k+j≤N−1),

fjk(t)=j

⎪⎩h=j

0,(N−1[y,z]N(t),QN(t)

[5,6]

∀y,z∈D(T1(P)),[y,z]N(∞)=lim[y,z]N(t)

t→∞

1−1∗

(Q−N(t))=−QN(t).

;

Q∗N(t)=−QN(t),

1

P

[a,∞)

P

(

T1(P))

D(T1(P))

={g|g∈L2[a,∞),g(N−1)

[a,∞)

Pg∈L2[a,∞)},

T1(P)g=Pg,

g∈D(T1(P)).

2

P

3

P

D(T0(P))P[a,∞)

T1(P)

∞

C0(a,∞)

T0(P)

P

(

d(P))

d(P)=

4

P

[a,∞)P

D(T1(P))1dim.2D(T0(P))

N

d(P)=N,

1[5,6]P

d(P)=12N,

P

∀f∈D(T1(P))

g∈D(T1(P+)):lim[f,g]N(t)=0.

t→∞

2[5]y∈D(T0(P))

(1)y(a)=y(1)(a)=···=y(N−1)(a)=0;(2)∀z∈D(T1(P)),[y,z]N(∞)=0.

+1

3[7](Calkin)P(m,m)([N2]≤m≤N,m∈Z+),{vj(t)}(j=

D(T1(P))1,2,···,m)

m󰀒

(1)cjvj∈/D(T0(P));

(2)[vi,vj]N(∞)−[vi,vj]N(a)=0(i=1,2,···,m),D(T1(P))

j=1

[y,vj]N(∞)−[y,vj]N(a)=0(j=1,2,···,m)

(1.4)

370

P(y)P(y)

26

D

(1),(2),

D(T1(P))

{vj(t)}(j=1,2,···,m)

(1.5)

D={y∈D(T1(P))|[y,vj]N(∞)−[y,vj]N(a)=0,j=1,2,···,m}.

l=

n󰀓k=0

(−1)kDkakDk,

t∈I=[a,∞),

[a,∞)

(1.6)

(1.6)

M

ak∈Ck[a,∞).

d(l)n≤d(l)≤2n(3[8]4(1.6)l

lT(l)T(l)

[6]).

I

2n

(

d(P)=n)

n×2n

D={y∈D(T1(l))|M(σy(a))2n=0},

(1)rankM=n,

1∗

(2)MQ−2n(a)M=0,

1

Q−2n(a)

(1.2)

(1.3)

2

Sturm-Liouville

Sturm-Liouville

l(y)=−(py(1))(1)+qy,t∈I=[a,∞),

(2.1)

p,q∈C3(I)

p>0,

l2(y)=l(l(y))=p2y(4)+4pp(1)y(3)+(3pp(2)+2(p(1))2−2pq)y(2)

+(pp(3)+p(1)p(2)−2p(1)q−2pq(1))y(1)+(q2−pq(2)−p(1)q(1))y.

d(l)l

(d(l)=2)A.lk(k=1,2,···)

(2.2)

1977KauffmanR.M.l

[3]

F,F(l).EvansW.D.

[2].lk(k=1,2)

Zettl

Li(i=1,2)(2.1)

L=L2L1

l(y)

Q2(t),Q4(t)

D(Li)

l(y),l2(y)

Lagrange

󰀅Q=

0−11

0

󰀆,

Q−1=

󰀅

01

󰀆.

(2.3)

−10

3

Sturm-Liouville

(1.2)

(1.3)

󰀅

󰀆

Q−21(t)=

1−Q−02

H0(t)

41p(t)

Q1

,(t)=

1

p4

(t)

−H0∗(t)

(p(t)p

(2)

(t)−2p(t)q(t))Q−1

,󰀅

H0

−p2(t)

󰀆

0(t)=

p2(t)−2p(t)p(1)(t)

.

Li(i=1,2):

󰀇

Li(y)=l(y),

y∈D(Li),

D(Li)={y∈D(T1(l))|Ai(σy(a))2=0},

Ai

2×2

rankAi=1.

(σl(y)(a))=󰀁l(y)(a)l(y)(a)󰀂T2(1)

=(H1H2)(σy(a))4,

󰀅q(a)

−p(1)(a)󰀆󰀅

0

󰀆H1=

q

(1)

(a)q(a)−p

(2)

(a)

,H2=−p(a)−2p

(1)

(a)−p(a)

.

L=L2L1,D(L)

L

L

󰀇

L(y)=l2(y),

y∈D(L),

D(L)={y∈D(T1(l2))|M(σy(a))4=0},

󰀅󰀆M=

A102

A.

2H1

A2H2

p=0

rankH2=2,(2.7)

rankM=2.

1L=L2L1

A1Q−1A∗2=0,

A1,A2,Q−1

(2.6)

(2.7)(2.3)L=L2L1

(2.10),(2.11)

l2(y)

4

L

MQ−41(a)M∗

=0,

371

(2.4)

(2.5)

(2.6)

(2.7)

(2.8)

(2.9)

(2.10)

(2.11)

4

372

26

M,Q−41

(a)

(2.11)

(2.4)

󰀅

MQ−1∗

1

02

󰀆

󰀅

󰀆

4(a)M=

AAQ−A∗1H1∗A∗

2

2H41(a)

󰀅1A2H202H2∗A∗

=02A1Q−1A∗󰀆

2

2p(1a)A−1A∗.

2Q1

02−MQ41(a)M∗

=0

−1A1QA∗2=0,

3

Sturm-Liouville

l(y)=−y󰀄󰀄+qy,t∈I=[a,∞),

q∈C4(I),

lk(k=1,2,3)

l3(y)=l2(l(y))=−y(6)+3qy(4)+6q(1)y(3)+(7q(2)−3q2)y(2)

+(4q(3)−6qq(1))y(1)+(q3−3qq(2)−2(q(1))2+q(4))y,

Lagrange

Q6(t),⎛

02

02

Q−1

⎞

Q−61(t)=⎜⎜⎝

02Q−1

H)⎟

3(t⎟⎠,Q−1

−H3∗(t)

H4(t)

Q−1

(2.3)

󰀅

󰀆H3(t)=

03q(t)

(2)−3q(t)−3q

(1)

(t)

,

H4(t)=(4q(t)+6q2(t))Q−1.

Li(i=1,2,3):

󰀇

Li(y)=−y󰀄󰀄+q(t)y,

y∈D(Li),

D(Li)={y∈D(T1(l))|Ai(σy(a))2=0},

Ai

2×2

rankAi=1.

L=L3L2L1,D(L)

L

L

󰀇

L(y)=l3(y),

y∈D(L),

D(L)={y∈D(T1(l3))|M(σy(a))6=0},

⎛

A1

02

02⎞

󰀅

M=⎜⎜⎝A2K1

−A0󰀆20q(a2⎟

⎟)⎠,K1=A3K2

Aq

(1)

3K3

A(a)

q(a)

,

3

.2)

.3)

.4)

.5)

.6)

.7)

(3.1)

(3(3(3(3(3(33

Sturm-Liouville

󰀆,

373

0

󰀆.

(3.8)

󰀅K2=

q2(a)−q(2)(a)−2q(1)(a)

󰀅K3=

−2q(a)

2q(a)q(1)(a)−q(3)(a)q2(a)−3q(2)(a)

−4q(1)(a)−2q(a)

⎛A1

0202⎞⎛

⎞

M=⎜⎜⎝

02

A2⎟E1

0202⎟⎜20⎜K⎟1−E202⎟02

0⎠⎝⎠

2A3K2K3E3

(3.5)

rankM=3.

⎛

02

02

A1Q

−1

A∗⎞3

MQ−61(a)M∗=⎜⎜⎝

02

A2Q−1A∗2

02⎟

⎟.A3Q−1A∗1

00⎠

2

2

2

L3L2L1

A1Q−1A∗3=A2Q

−1A∗

2=0,A1,A2,A3,Q−1(3.4)(3.5)(2.3)

1L1,L2,L3(3.4)(3.5)L(3.6)

L=L3L2L1L2L,L1,L3

Hamilton

l(y):=−¯h

2d2ymdx

+x222y,0≤x<∞,

y–

Li(i=1,2,3)

󰀇

L1y=l(y),x∈[0,∞),y(0)=y󰀄(0);

󰀇L2y=l(y),x∈[0,∞),y(0)=−y󰀄(0);󰀇

L3y=l(y),x∈[0,∞),y(0)=0.

[2]

Levinson-Evans-Zettl

Li(i=1,2,3)L1,L2,L3

L3L2L1

(3.7)L1=L3.

(3.8)

374

26

[1]CaoZJ,SunJandEdmundsDE.Onself-adjointnessoftheproductoftwosecond-orderdifferential

operators.ActaMath.Sinica(EnglishSeries),1999,15(3):375–386.

[2]EvansWDandZettlA.Levinson’slimit-pointcriterionandpowers.J.Math.Anal.andAppl.,

1978,62:629–639.

[3]KauffmanRM,ReadTandZettlA.TheDeficiencyIndexProblemofPowersofOrdinaryDiffer-entialExpressions.LectureNotesinMath.621,Berlin/NewYork:Springer-Verlag,1977.

[4]RaceDandZettlA.Onthecommutativityofcertainquasi-differentialexpression,I.J.London

Math.Soc.,1990,42(2):489–504.

1987.[5]

[6]NaimarkMA.LinearDifferentialOperators,II.NewYork:Ungar,1968.

Calkin(),1988,19(4):573–587.[7]

[8]SunJ.Ontheself-adjointextensionsofsymmetricdifferentialoperatorswithmiddledeficiency

indices.ActaMath.Sinica(EnglishSeries),1986,2(2):152–167.

1983,12(3):161–178.[9]

[10]CoddingtonEA.Thespectralrepresentationofordinaryself-adjointdifferentialoperators.Ann.

Math.,1954,60(1):192–211.

[11]EdmundsDEandEvansWD.SpectralTheoryandDifferentialOperators.Oxford:Oxford

UniversityPress,1987.

SELF-ADJOINTNESSOFPRODUCTSOFTHELIMIT-POINT

STURM-LIOUVILLEOPERATORS

YangChuanfu

(DepartmentofAppliedMathematics,NanjingUniversityofScienceandTechnology,Nanjing210094)

AbstractForthedifferentialexpressionl(y)=−(py󰀄)󰀄+qy,t∈[a,∞),undertheassumptionthatlk(k=1,2,3)arelimit-pointed,theauthorstudiestheself-adjointnessoftheproductoperatorL2L1,whichLi(i=1,2)aregeneratedbyl(y),andobtainsanecessaryandsufficientconditionforself-adjointnessofL2L1.Also,anecessaryandsufficientconditionfortheself-adjointnessofL3L2L1,whichLi(i=1,2,3)areassociatedwithl(y)=−y󰀄󰀄+qy,t∈[a,∞),isobtained.

KeywordsProductsofdifferentialoperators,limit-pointeddifferentialexpression,self-adjointboundaryconditions.