- Email: [email protected]
n-Th Order Differential Equations
250
VI. n-th order differential equations
an Lp-function space and a finite-dimensional space of constants. On the function space, the resolvent is an integral operator whose kernel is the Green's function; on the space of constants, it is a multiplication operator (Theorem 6.4.1). The adjoint operator function of a boundary eigenvalue operator function defines the adjoint boundary eigenvalue problem (Theorem 6.5.1). For the adjoint problem in this operator theoretical sense no additional assumptions on the original boundary eigenvalue problem are needed. The adjoint operator function maps the direct sum of an Lp-function space and a finite-dimensional space of constants into a space of distributions. The realization of the original boundary eigenvalue problem within a n Lpfunction space is achieved in the following way: Take the original boundary eigenvalue problem with homogeneous boundary conditions and associate to it a family of closed linear operators whose domains consist of W~-functions which fulfil the boundary conditions. These closed linear operators are not necessarily densely defined and their domains may depend on the eigenvalue parameter. The adjoints of these closed linear operators are in general not operators but closed linear relations. Under additional assumptions these adjoints form a family of operators, in which case they yield the adjoint boundary eigenvalue problem in the parametrized form. The relationships between the adjoint boundary eigenvalue problem in operator theoretical sens~ and the corresponding problem in parametrized form is thoroughly discussed (Theorems 6.6.4 and 6.6.5). Finally, the special case of two-point boundary eigenvalue problems is considered. We state that the classical adjoint boundary eigenvalue problem coincides with the adjoint problem in the parametrized form. Root functions (eigenvectors and associated vectors) are defined for the above mentioned families of closed linear operators by taking root functions (eigenvectors and associated vectors) of the corresponding holomorphic boundary eigenvalue operator function. It is shown that the principal parts of the GREEN'S function can be represented in terms of eigenfunctions and associated functions of the family of closed linear operators and the family of the adjoints of these operators (Theorem 6.7.8).
6.1.
Differential equations and systems
In this chapter let ~ be a nonempty open subset of C, -oo < a < b < oo, 1 _
2. By ej w e denote the j-th unit vector in C". We consider the scalar n-th order differential equation n-1
(6.1.1)
o (") +
pi(.,z)o i=0
- o
(,T c Wf,(a,b),Z c a),
251
6.1. Differential equations and systems
where Pi E H ( ~ , L p ( a , b ) ) ( i - 0,... , n - 1). Together with this differential equation we consider the differential operator (6.1.2)
n-I L D ( ,~ ) ,I.I . _ ~ ( n ) + E P i ( "' ~' ) "O( i )
(rl CWp(a,b),2, C a ) .
i=0
LEMMA 6.1.1. L D E H ( n , L ( W ~ ( a , b ) , L p ( a , b ) ) ) .
Proof. From Proposition 2.3.3 we infer that Pi C H(~2,L(Wp-i(a,b),Lp(a,b))) for i - 0 , . . . , n - 1. D We associate a first order system to the n-th order differential equation. This system is defined by the operator (6.1.3)
TD(X)y "- y - A ( . , Z ) y
(y C (Wlp (a,b))n,x C C),
where 0
1 0
(6.1.4)
A "- (~i,j-1-
~i,nPj-1)inj=l
-
-Po
0 9 9
1 --Pn-1
PROPOSITION 6.1.2. Let rI E Wp(a,b), 2, C ~, and set
711 y
o~
o
9
rl(/,-ll Then y C (W~ (a, b) )n and 0 TD(2,)y -o
L~ Proof. The assertions y E (Wp1(a, b))n and (6.1.5)
e/TA(.,X)- e/T+,
( i - 1,...,n-- 1)
are obvious. For i - 1,... , n - 1, (6.1.3) and (6.1.5) yield e~TD(2,)y -- e~y' -- e~A(.,Z)y - 71(i) - e~+ly - O.
VI. n-th order differential equations
252 Finally we obtain
n-1
ernTO(Z)Y -- erny' + Z Pi("Z)e~+l y i=0 n-I
= 1"1(n) -I- E Pi (''z)rl(i) -- LD(~')I]" i=0
PROPOSITION 6.1.3. Let y C (W~ (a,b)) n, Z C a, and assume that e~TD(Z)y - 0
for i-- 1,... , n - 1. Then t 1 "- erlY C Wfl(a,b), 11 71' (6.1.6)
y0(n -1)
and LD(Z)rl -- ernTD(Z)y.
(6.1.7)
Proof Let i C { 1,..., n - 1}. By assumption and from (6.1.5) we obtain (6.1.8)
e~y' - ery ' -- e~ T D (~)y -- e~a (., ~)y - eri+,y.
This proves 7/E Wr
and e 7 y - 71(i-l) f o r / -
i - 1. Assume that r/ E Wr (6.1.8) yields
and e r y -
1,... ,n. Indeed, this is true for
7/(i-1) holds for some i < n. Then
77 (i) -- eVy ' - e~+ly E W1 (a,b)
which proves r/ E Wr see Corollary 2.1.4. Thus 7/ E W~(a,b), and the equation (6.1.6) holds. Because of (6.1.6), the equation (6.1.7) immediately follows from Proposition 6.1.2. DEFINITION 6.1.4. Let ~0 C g2 and/71,... , Tin E Wp(a,b). Then {r/i,..., r/n} is
called a fundamental system of LD(~)rl -- 0 if for each 7/C N(LD(~o)) there are c j C C ( j - 1,..., n) such that n
77 -- 2 CjTIJ" j=l
A function (r/l,..., On)" ~ ~ Ml,n(W~(a,b)) is called a fundamental system function of LDy -- 0 if {r/1 ( ~ ) , . . . , 0n(X)} is a fundamental system of Lo(X)y - 0 for each X C g2.
6.1. Differential equations and systems
253
LEMMA 6.1.5. Let 2 0 E ~2 and Yo C Mn(W~ (a,b)) be a fundamental matrix of TD(~,o)y -- O. Then {erl Yoel,. . . ,erl Yoen} is a fundamental system of LD(~o)rl -- O, and (6.1.9)
(ellYoej)(i-1) - eli Yoe j
holds for i - 1,... ,n and j - 1,... ,n. Proof For each j E {1,... ,n}, Yoej fulfils the assumptions of Proposition 6.1.3.
Thus erlYoel,...,eTYoe n E Wp(a,b), and (6.1.9)holds. Now let 7/ C N(LD(2,o)) and set y " - (r/, r/',..., r/(n-1)) x. Then y E N(TD(~o)) by Proposition 6.1.2. Definition 2.5.2 yields a vector c - (Cl,...,Cn) T C C n such that y - Yoc. It follows that n
rI - eTlY- eTlt'oC-
cjeTlroej.
n
j=l
LEMMA 6.1.6. Let ~'o E D and rll,... , tin C W~(a,b) such that {rll,..., 7"/,,} is n a fundamental system of L D ( ~ ) - O. Then (Tl~i-1))i,j=l E M n ( W l ( a , b ) ) i s a fundamental matrix of rV(
)y - O.
Proof Let y C N(TD(~o)). Proposition 6.1.3 yields that r I " - ely y - (7/, rl',..., rl(n-1)) T and LD(;~0)rl -- e~TD(~.o)y- 0. Hence there is a vector c - (Cl,... , cn)r C C n such that tl
7 / - ~ cjrlj. j=l
This proves
Y --
11 rl I .
---
"
---(~tii-1))inj:lC. ["7
.I
PROPOSITION 6.1.7. Let ~o E D and rll,..., tin C Wp(a,b). Then the following conditions are equivalent." i) Tll,. . . , tin are linearly independent, LD(Xo)rlj --O for each j C { 1,... ,n}, and for each 0 E N(L~ there are cj C C (j - 1,... ,n) such that n -- E Cjl~j ; j=l
ii) (r/l,... , r/n} is a fundamental system of LD(~,o)rl -- O; iii) (rl!i-l))inj=l~ , is a fundamental matrix of T D ( ~ ) y -- O.
254
VI. n-th order differential equations
Proof i) ~ ii) is clear by definition of a fundamental systemand ii) ~ iii) follows 0
from Lemma 6.1.6. Assume that iii) holds. For j -
1,... ,n we set yj "-(71j , rlj,... , r/in-l)) r.
From Corollary 2.5.5 we infer TO(2o)yj --0 and hence, by Proposition 6.1.2, L~ - 0 for j - 1,... ,n. Since a fundamental matrix is invertible by Theorem 2.5.3 and Proposition 2.5.4, Yl,'" ,Yn are linearly independent. This implies that r/l,... , On are linearly independent. An application of Lemma 6.1.5 completes the proof. [--1 THEOREM 6.1.8. There is a fundamental system function (rll,... , On) of LDrl -- 0
such that rl~i-1)(a,~.) - Si,jfor Z C ~2and i , j - 1,... ,n. Furthermore, the fundamental system function is uniquely determined and depends holomorphically on Z E ~. Moreprecisely, we have rlj C H(~z, Wfl(a,b))for j - 1,...,n. Proof By Theorem 2.5.3 there is a fundamental matrix function Y of TOy = 0 such that Y(a,X) - I n for all ~ C I2. For j - 1 , . . . , n we set rlj "- eryej. By Lemma 6.1.5 we obtain that (r/l,... ,0n) is a fundamental system function. In addition, (6.1.9) and Y(a,Z) - I n yield r/Ji-1)(a,~.) - 6ij for ~. C f~ and i , j 1,...,n. Now let (r/l,... , On) be any fundamental system function of LDrl --0 with rlJi-l)(a,~) -- 6ij for 2 C ~2 and i , j - 1,... ,n. By Corollary 2.5.5 there is a unique fundamental matrix function Y of TDy = 0 with Y(a, &) = In for ~ E 12. Since (r/Ji-1))~,j=l is a fundamental matrix function with these properties by Lemma 6.1.6, we obtain that (r/l,... , 77,,) is uniquely determined. Since Y depends holomorphically on ~ by Theorem 2.5.3, it follows that r/J i-1) E H(~,W~(a,b)) for i , j - 1,...,n. For h E N, the indefinite integral defines a continuous linear map from Wph (a, b) to w h+l (a, b ) b y Proposition 2.1 8 9 , p
9
o
From rlJi-1)(a) -- 6ij and Proposition 2.1.5 i) we know that r/Ji-1)(x,Z) -- Sij +
/a x
rl!i)(t Z)dt
(x C (a,b))
for j - 1,..., n and i - 1,..., n - 1. Hence we obtain in view of Corollary 1.2.4 that
T~Jn-2) C H(a,W~(a,b)),...,
171 E H ( a , Wp(a,b)).
IS]
255
6.2. Boundary conditions 6.2.
Boundary conditions
Let L R C H(~,L(W;(a,b),C.n)). Suppose that p < co. We fix some ~'0 C ~ and l C { 1,... ,n}. By Theorem 2.2.5 there are uj C Lp,(a,b) (j -- 0,... ,n) such that n
eTLR(~o)-
~., (Uj)! j) and j=O
arab
e~Ln(~.o)rl -(rl,e~Ln(~,o))p,n - ~ j=O
(-1)Jrl(J)(x)uj(x) dx
for each 7/C Wp (a, b). Hence
e~LR(~'O)rl -for each r/ E
TR()t,) E
n
Wp(a,b).
This proves that for each ~. E ~2 there is an operator L((Wl(a,b))n,c n) such that 7/ 0'
LR(~,)rl -- TR(~,)
(6.2.1)
77(s holds for all 7/C W; (a, b). In applications, the boundary conditions are mostly given in a form such that it is easy to give a representation (6.2.1) with T R C H(~,L((Wfl(a,b))n,cn). For example, let
LR(,~,)TI "--
(O~ij(Z)Tl(J-l)(a) + ~ij(~,)rl(J-')(b))
(rl E Wp(a,b)), i=l
where the O~ij and ~ij are complex valued functions and where L R depends holomorphically on X. Choosing functions 7/ for which exactly one of the values r/(J-l)(a), rl(J-1)(b) ( j - 1,... ,n) is different from zero we see that the aij and ~ij are holomorphic functions. Then T R ()~), defined by n
n
TR(~,)y "-- (aij(X))i,j=lY(a) + (~ij(~,))i,j=lY(b)
n
(y e Wp (a,b)),
depends holomorphically on/~. Now we shall show that, if p < co, we can always choose T R(~) in such a way that it also depends holomorphically on X. This immediately follows from
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6.3. The boundary eigenvalue operator function
257
Hence F is (weakly) holomorphic in Wp.t+l [a,b]. The induction hypothesis yields that there are u 1,... ,u l E H(f~,Lp,(a,b)) such that
l-1 V(Z)- E Ui+I (Z)~ i) i=0
(Z C ~"~).
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UO(Z ) "-- b ~ a (1,v(Z))p,lZ(a,b) . Then u o C H(~2,Lp,(a,b)). From (6.2.2) we obtain
Y(Z) -- UO(Z ) --[-~(/~)t
l E l'li('~)~i) i=0
(Z C n).
['-]
We would like to note that the "canonical way" to associate T R to L R does not always yield a holomorphic operator function T R. For example, the identity aa 7/' (t) dt
7/(b) - 7"/(a)
yields that, for any complex valued function a on C,
(O/a,
)
defines a holomorphic operator function
c c H(C,L(W](a,b),C2)). But the operator function T R, given by
TR(Z)Y'-where y -
(yl(a) yl(b)+a(Z)(Yl(a)_y,(b)
)
) +Ot(z) fby2(t)dt
'
(Yl) Y2 E(W~(a,b))2 does notdependholomorphically onZ i f a d o e s
not depend holomorphically on X. We shall assume also in the case p - o o that L R C H(~2,L(W~(a,b),Cn)) is given in such a way that (6.2.1) holds for some T R C H(~2,L((W~(a,b))n,C~)). 6.3.
The boundary eigenvalue operator function
Let L D and L R be as defined in Sections 6.1 and 6.2. We call (6.3.1)
L - - (LD,L R) C H(~,L(Wp(a,b),Lp(a,b) x cn))
a boundary eigenvalue operator function.
258
VI. n-th order differential equations
Let {r/l,... , 0n} be the fundamental system function of Theorem 6.1.8 and set Y "- (o(i-1)] Define \ j !n i,j=l" (6.3.2)
ZL(~)C "- ( r / , ( . , X ) , . . . , r / n ( . , ~ ) ) c - e~r(.,2,)c
LDrl --0
given by
(c C cn,2, E a )
and (6.3.3)
(UL(Z)f)(x) "-- erl r ( x , Z ) f a X Y ( t , z ) - l e n f ( t ) dt
( f E Lp(a,b)).
From Lemma 6.1.6 we know that Y is a fundamental matrix function of TOy = O. Let 2~ C ~ and U(Z) be the fight inverse of T ~ given by (3.1.6). Then UL(Z ) -- erl U(Z)e n .
(6.3.4)
A characteristic matrix function of L is defined by
M(Z) - LR(Z)ZL()~).
(6.3.5)
Note that M is also a characteristic matrix function of the associated first order boundary eigenvalue operator function T - (T ~ T R) given by (6.1.3) and (6.2.1). THEOREM 6.3.1. L is an abstract boundary eigenvalue operator function in the sense of Section 1.11.
Proof We set E "- Wp(a,b), F 1 "-- Lp(a,b), G "- F2 "-- C m , T 1 "- L D, T2 "-- L g. We have to prove that (1.11.1) holds. For this let s C ~. For each f C Lp(a,b) we have T D ( z ) U ( Z ) e n f - enf. Thus we can apply Proposition 6.1.3 and obtain erlU(Z)en f C W~(a,b) and (6.3.6)
(erl U(X)enf) (j-1) _ er.U(Z)enf j
(j _ 1 ~~176176 n) 9
Then (6.1.7) and (6.3.4) yield
LD(2,)UL(Z)f - ernTD(Z)U(Z)e,f - f for each f E Lp(a,b), i. e., UL(Z ) is a right inverse of LD(z). For each ~ C I'~, ZL(2, ) is injective since r/l,... , On are linearly independent by Proposition 6.1.7. For the proof of (1.11.1) iii) let Z C ~2 and r/C N(LD(&)). Then, by Definition 6.1.4, there is a vector c E C n such that 7"/= (r/l(.,Z),... , On(',Z))c = ZL(Z)C, which proves 7/E R(ZL(Z)). Conversely, let 7"/E R(ZL(Z)). Then there is a vector c E C n such that 7"/= ZL(Z)c = (01 ( ' , ~ ' ) , ' " , rln(',Z))c" Proposition 6.1.7 proves
71 C N(LD(Z)). We shall show that UL is even a holomorphic right inverse. As in the proof of Proposition 2.1.6 we define
R "-- { (fj)kj:o " f i e Lp(a,b) (j - 0,... ,k), f j ' - fj+l (J - 0,... ,k - 1)}.
6.3. The boundary eigenvalue operator function
259
Because of the isomorphism proved in that proposition it is sufficient to show that for j = O,. .. ,n
(X,f) ~ (UL(~.)f)(J)
(~ C ~2, f E Lp(a,b))
defines a holomorphic map in L(Lp(a,b),Lp(a,b)). For j = 0,... , n - 1 this follows from (6.3.6)since U E H(~2,L((Lp(a,b))n,(W~(a,b))")). Since this also yields that
(X,f) ~-+ (UL(~,)f)(n-1)
(~ C ~, f E Lp(a,b))
defines a holomorphic map in L(Lp(a,b), Wfl (a,b)), we finally obtain that the assertion also holds for j - n. 0 As in Section 3.1 we apply Theorem 1.11.1 and obtain THEOREM 6.3.2. The boundary eigenvalue operator function L given by (6.3.1) is
holomorphically equivalent on ~2 to the Lp(a,b)-extension of M; more precisely, for ~, C ~ we have
idLp(a,b)
0 L(X)-idr
)(M~)~)
LR(Z)UL(~,)
O )(ZL(~I,),UL(~))_I
idLp(a,b)
'
and the operators (id0c,
idLp(a'b) ) C L(C n • Lp(a b) Lp(a,b) • C n) L R ( Z ) V , (Z)
'
,
and (ZL(~,),UL(~,)) E L(C n X Lp(a,b),Wfl(a,b))
are invertible and depend holomorphically on ~,. COROLLARY 6.3.3. The boundary eigenvalue operator function L is Fredholm
operator valued and p(L) = p(M) = p(r). Proof The first assertion and p(L) = p(M) immediately follow from Theorem 6.3.2 since M()~) is an operator in finite dimensional spaces. As M is also a characteristic matrix function of T, we have p(M) = p(T) by Theorem 3.1.2. IS] In the same way as Theorem 3.1.4 we obtain THEOREM 6.3.4. Let M be the characteristic matrix function given by (6.3.5). Suppose that p(M) # O. Let t2 E a(M) and r "- nulM(/2). Let {cl,... ,Cr} and {dl,..., dr } be biorthogonal CSRFs of M and M* at l.t. Define o j " - ZLcj,
vj . -
(-(LRUL)*dj) dj
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6.6. The adjoint problem in parametrized form
263
Since, for d E Cn, (rl,L*(Z)(u,d)) = (L(Z)rl, (u,d)) = (L~
u ) + (L~(Z)o,d)
:
we obtain the representation (6.5.2). For a function r/ E Wf(a,b) we set y " ( 0 , 0 ! ,...,r/ (n-l) )r and obtain (l~,tR* (Z)d)p,n - (tR(x)o,d)c
n
= (TR(Z)y,d)c, = (y, r R* (~,)d) p,l n = E(o(i-1),e~TR*(Z)d)p,1 i=1 n
= E(rl,(-1)i-l(eriTR*(Z)d)(i-1))p,n. i=1
This equation and (3.3.3) yield (6.5.3). Since LR(X)UL(X) - re(z)U(~,)en by (6.2.1), (6.3.4), and (6.3.6), we have
(LnUL) * (~,) -- ern(TRU) *(Z). With the aid of (3.3.4) we obtain the representation (6.5.4). 6.6.
VI
The adjoint boundary eigenvalue problem in parametrized form
In this section let p < oo. Let L be given by (6.3.1) and define L0(~, ) in Lp(a,b) by (6.6.1)
D(Lo(Z)) - {0 C Wp(a,b) " Lle(z)rl - 0 } C Lp(a,b)
and (6.6.2)
Lo(Z)O - LD(z)rl
(rI E D(Lo(~,)).
As for first order systems considered in Section 3.4, the domain D(Lo(~)) of L0(/~ ) may depend on Z and may be a non-dense subspace of Lp(a,b). Let P(Lo)"- {~, C I2" Lo(Z ) is bijective}, a ( L 0 ) " - ~ \P(Lo). THEOREM 6.6.1. Let L be the boundary eigenvalue operator function given by (6.3.1) and let L o be its restriction in Lp(a,b) with homogeneous boundary conditions as given by (6.6.1) and (6.6.2). i) P(Lo) -- p(L) and L o l ( Z ) f - L-1 (~,)(f,0)for Z E p(L) and f E Lp(a,b). If E P(Lo), then Lol(X) is continuous.
264
VI. n-th order differential equations
ii) Suppose that L R is of the form (6.3.8) and let G be the GREEN' S function given by (6.4.5). Then, for ~, C P(Lo) and f C Lp(a,b),
( L o l ( ~ ) f ) ( x ) --
/a
G(x,~,,~)f(~) d~.
Proof. Up to some changes in notations and references, the proof coincides with the proof of Theorem 3.4.1. ~ As in Section 3.4 we can prove PROPOSITION 6.6.2. For all ~, C D the operator Lo(~ ) " Lp(a,b) --+ Lp(a,b) is closed. In the same way as Proposition 3.4.7 we prove PROPOSITION 6.6.3. Let M be the characteristic matrix function given by (6.3.5) and suppose that p(M) 7~ O. Let l,t E t~(M) and r "- nulM(~). Let {Cl,... ,Cr} and {dl , . . . ,dr} be biorthogonal CSRF of M and M* at la. Define -
ZLcj,
-
(j -
1,...
,r),
where Z L and UL are given by (6.3.2) and (6.3.3), respectively. Let mj "- v(tlj ), the multiplicity of the root function cj. Then the operator function t"
L o l - Z (" -- ~t)-mj~j @ Uj
j=l is holomorphic at tt. The adjoint L~(~,) is a linear relation in Lp,(a,b) defined by its graph
G(L;(Z))-
(G(-Lo(Z))) -L,
i. e.,
u C D(L~(~,)) r 3w C Lp,(a,b)Vy C D(Lo(X)) (Lo(X)y,u) - (y,w) and
L~(~,)u - {w C Lp,(a,b) " Vy C D(Lo(A)) (Lo(A)y,u) - (y,w)}. Here ( , ) is the canonical bilinear form on Lp(a,b) x Lp,(a,b). In the same way as Theorem 3.4.3 we prove THEOREM 6.6.4. i) Let ~, C ~ and u C Lp,(a,b). Then u C D(L~(A)) if and only if there is a vector d C C n such that L*(~,)(u,d) C Lp,(IR). ii) Let 2~ C D and u C D(L~(~,)). Then
t~(~)u -- {(L*(~i,)(u,d))r " d C C n, L*(~)(u,d) E
Lp,(I~)}.
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6.6. The adjoint problem in parametrized form
269
Obviously, {e z x - XxeZX,xe zx} is a fundamental system of LD(Z)rl - O. The corresponding fundamental matrix is
xeZX Y (x, ~,) - ( ezx - ~xeZX \ -- ~ 2xe~.X ezx + ~xeZXJ and fulfils Y (0, ,~) = 12. The characteristic determinant is given by M ( ) ' ) - ( 0-1
;1)+(
_(eZ-l-,Z,e --
_2ee
ez--e'z,/3., 2'2'e z e z +ez~eZ) z
z
eZ-1 ) e z +2e
z
9
From detM(~) - e 2~" - e z - ~ e ;t - ~ 2 e ; ~ we infer p(M) r 0 and that detM has zero of order 2 at 0. The vector function given by M(~)(~)-(
a
ez-l-~ez_~2e z )
has a zero of order ~ at 0 .ence ( : ) is a root function of ~, at 0 of mu,tiplic~ty
2, and by Proposition 1.8.5 it is also a CSRF of M at 0. In the same way,
M,(~)(-2-8~')_ 2~
(e~'-l-~e~" ez - 1
-"~2e~" ) ( ) e z + X ez 2~ 8
_((eZ-l-ZeZ)(-2-8Z)
-2'q'3e;t )
2( 1 - e z + ~ e z + ,q.,ee z ) _ 8 ~ (e z _ 1) shows that (
~~ 8 ~ ) is a ~ ' ~
~2
2~,
of~'* at 0 of multiplicity ~ ~r~
M(~,) =
2~
--~ -- 89
-1
/q.,2hl (,~)
= 1 + ~ 2 h 2 ( ~, .) . where h 1 and h 2 are holomorphic functions on C, we see that the CSRFs are biorthogonal. It is easy to see that
Y- l (x, ,~ ) - (e-~X~ + ~xe-zx2xe_xx
--xe-XX e-ZX_ ~,xe-ZXj 9
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6.7. Two-point boundary eigenvalue problems in Lp(a,b)
275
DEFINITION 6.7.4. Let 7/E H (g~, W~ (a, b)) and g C 12. 7/is called a root function
of L 0 at U if and only if r/(g) -J: 0, (LDrl)(g) -- 0 and wa(l.t)(rl(i)(a, la)) n-1 i=0 -+Wb(la)(rl(i)(b, la))in-d -- O. The minimum of the orders of the zero of LOt/ and
wa(o(i)(a,
. ) ) ni-0 - 1 q.. W
b (rl(i)(b,.))i=o n-1 at g is called the multiplicity of 7/.
From LRrl - wa(r~(i)(a,'))n-~ -k- wb(r~(i)(b,'))in_=7 w e obtain PROPOSITION 6.7.5. Let rI C H(~2,Wp(a,b)), l.t C ~2 and v C N. Then t 1 is a
root function of L 0 of multiplicity v at la if and only if 11 is a root function of L of multiplicity v at l.t. Canonical systems of root functions of L0 are defined in the same way as for L. Hence a system of root functions is a canonical system of root functions of L 0 at g if and only if it is a canonical system of root functions of L at ~. The situation is different for L~- - L~ and L*. PROPOSITION 6.7.6. Let (~,d) E n(~2,Lp,(a,b) x C n) be a root function of L*
of multiplicity v at It. We may assume that ~ is a polynomial of order <_ v - 1. Then ~ E H(~Z, Wpn (a,b)), ~ is a root function of L-~ of multiplicity >_ v at l.t, and d + Cu(a, .) + Du(b, .) has a zero of order >_ v at tt. Proof By assumption v-1
((z)-
(z-
(z c c),
i=0
where ~i C Lp,(a,b) ( i - 0,..., v W7, (a, b). For this, define
1). First we shall show that ~'i belongs to
(i- 0,...,v-
1).
Since (~', d) is a root function of L* of multiplicity v at/1, we have i
L*(,U)(~i,di) - - E ~.1 ( dzJ dJ L *'~)
(i- 0,...,v-
1).
j=l
Since the restriction of LR* to ~ (a,b) is zero, we obtain i
(6.7.9)
r
j=l
~
/
(~)~i--J/r"
For i - 0 , (6.7.9) yields (tD*(l~)~O)r- 0 and hence (0 E Wpn(a,b) by (6.5.2) and Proposition 2.6.1. Now, for i - 1, the right hand side of (6.7.9) belongs to Lp, (a, b), and Proposition 2.6.1 yields ~'1 C Wp~,(a, b). Repeating this procedure
VI. n-th order differential equations
276
we obtain ~i E Wpn (a, b) for i - 0, 1,..., v - 1. As in the proof of Corollary 6.6.6 we obtain that (6.7.10)
Hau(a,') - Ward
and
Hbu(b,. ) + w b r d
have a zero of order >_ v at/~. From the representations (6.7.2) and (6.7.3) we infer that L D+ ~ has a zero of order _ v at/2. Hence ~" is a root function of L~- of order _> v at/2. From (6.7.10) and the definition of H we obtain that
(W aT ~) (C.
(6.7.1 1)
/ 9 ) ( (a,.))u
.(b, )
-]-
(W aT ~) (d) \w
has a zero of order _> v at/2. Hence it follows in view of the invertibility of W
bT
that (~u (a,.) +/gu (b,.) + d has a zero of order _> v at g.
V]
PROPOSITION 6.7.7. Let ~ E H(~2,Wp(a,b)) be a root function of L-~ of multiplicity v at It. Set d "- -Cu(a, .) - Du(b,.). Then (~,d) is a root function of L* of multiplicity >_ v at la. Proof By assumption, Wau(a, .) + ~,'bu(b, .) has a zero of order >__v at g. Hence the matrix function (6.7.11) has a zero of order _> v at/2. Now the assertion is clear because of (6.7.2), (6.7.3), (6.7.5) and (6.5.2). [-] A canonical system of eigenfunctions and associated functions of the family of operators L0(&) is defined by taking a canonical system of eigenfunctions and associated functions of the holomorphic boundary eigenvalue operator function L. THEOREM 6.7.8. We consider the families of operators Lo(Z ) and L-~ (Z ) defined by (6.6.1), (6.6.2) and (6.7.6), (6.7.7), respectively. Assume that 12 E a(Lo) and let { ~i,h " 1 < i < r, 0 <_ h <_ m i -- 1} be a canonical system of eigenvectors and associated vectors of L o at It. Then there is a canonical system of eigenvectors and associated vectors { ~i,h " 1 < i <_ r, 0 < h < m i - 1} of L-~ at !a such that the principal part of the GREEN'S function G(x,~,.) at 12 has the form
r mi-1 ]E (.-
(6.7.12)
j ]E
i= 1 j = 0
k=0
If W a and W b do not depend on ~, then the biorthogonal relationships
m 1 fab (ff-~-fOih)(X,~2)~j,m_k(X) Ok dx-- l~ijl~mi_h,m
~_~ ~
(6.7.13)
k=0
(1 <_ h < mi;
0
mj-
1;i, j -- 1,..., r) hold, where m i-
1
Oih "-- LD E (" -- ~2)l-hoi,rn" /=0
6.7. Two-point boundary eigenvalue problems in Lp(a,b)
277
Proof We set mi--1 Oi(Z) "-- E (Z -- ~)hOi,h
(i-- 1,...,r).
h=0
{rll,..., Or} is a CSRF of L at g by Propositions 1.6.2 and 6.7.5. By Theorem 1.5.4 there are polynomials (~i,di) "C ~ Lp,(a,b) x C n of degree < m i such that { ( ~ l , d l ) , . . . , ( ~ r , d r ) } is a C S R F o f L* at/t, r
L-1 - E ( " - ld)-miFli (~ (~i'di)
(6.7.14)
i=1
is holomorphic at g and the biorthogonal relationships (6.7.15)
1 dm ~
m---~,dX m (Oih' (~j'dj))(l't) -- (~ij(~rni-h'm
hold for 1 <_ h <_ mi, 0 <_ m <_ m j -
1, i, j - 1,..., r, where we use the notation Oih "-- ( " - g)-hLrli. By Proposition 6.7.6, ~ 1 , " ' , ~r are root functions of L~- at
g and di(g ) - - C ( g ) u i ( a , g ) -D(la)ui(b,g ). Hence ~ ' l ( # ) , " ' , r are linearly independent as (~'1,d l ) ( g ) , " - , (r dr)(g) are linearly independent. Since the multiplicities of a CSRF of L+ at g cannot exceed the multiplicities of a CSRF of L* at g by Proposition 6.7.7, { ~1,..., ~'r} is a CSRF of L + at g. We set
mi--1 ~i (z) --" E (Z -- ~t)h~i,h
( i - 1,...,r)
h=0
and infer that { ~i,h" 1 <_i <_r, 0 <_h <_m i -- 1} is a canonical system of eigenvectors and associated vectors of L~- at g. By Theorem 6.6.1 i) and (6.7.14) the principal part of Lol at g is equal to the principal part of r
E ( " -- ]A)-mioi(~ ~i i=1
at #, and Theorem 6.6.1 ii) yields that the principal part of G(x, ~, .) at g is
r mi-1 9 1 dj E E ( " - ld)J-m'-(Oi( x,')~i(~
j!
i=1 j = 0
d~J
"))(~) '
r mi-1 j = E E (" --~)j--mi E Oi,k(X)~i,j-k (~)" i= I j=O
k=O
LRFli is a polynomial of degree _< m i -- 1 and has a zero of order _> m i at g. Hence LRrli -- 0 for i -- 1,..., r. Thus Oih -- (Oih, 0), If
W a and W b are constant,
and (6.7.15) leads to (6.7.13).
then
V1
278 6.8.
VI. n-th order differential equations Notes
Mostly, the operators associated with boundary eigenvalue problems are considered as operators from Lp(a,b) to Lp(a,b). In that case, it it sufficient to consider somewhat weaker conditions on the coefficients, see [NA2, Chapter V]. However, in general it is much more advantageous to have bounded operators, and we are therefore going to use the operator L'Wp(a,b) --+ Lp(a,b) x C n in subsequent chapters. An important advantage of this approach is the fact that the adjoint operator L*(~) is defined on the whole space Lp,(a,b) x C" and is a bounded operator with values in some space of distributions. As a consequence, the associated adjoint boundary eigenvalue problem is defined without any restrictions. This implies that the eigenvectors and associated vectors of the adjoint problem are always defined. The associated vectors may not belong to the domain of the classical adjoint problem. This explains why in the classical approach problems containing associated vectors are mostly disallowed. Adjoint boundary conditions for two-point boundary eigenvalue problems were introduced by G. D. Birkhoff in [BI2]. G. FROBENIUS has shown in [FRO] that the adjoint differential expression A~ of a differential expression of the form n j A ~j=oAjD is uniquely determined by the identity vA(u) - uA'(v) - DA(u, v), -
where A(u, v) - 2 nj,k(- 1)k (DJu) D k (Aj+k+ 1v).