Decay of solutions of ordinary differential equations

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1 . Mth. Anl. Appl. 276 (2002) Decy of solutions of ordinry differentil equtions Suznne Collier Melescue Deprtment of Computer Science nd Mthemtics, Arknss Stte University, PO Bo 70, Stte University, AR 72467, USA Received 9 September 200 Submitted by C.W. Groetsch Abstrct We investigte the rte of decy of eigenfunctions of Schrödinger equtions using perturbtion method which consists of mking perturbtion B of the opertor L of the form B[y] =L[y] (g Lg)[y], whereg is n ppropritely chosen function. In our theory we llow B to be either reltively compct or stisfy certin boundedness condition. We give some emples which pply the results of our min theorems coupled with recent work on the reltive boundedness nd compctness of differentil opertors Elsevier Science (USA). All rights reserved.. Introduction We consider differentil epressions of the form n L[y]=w p i ()y (i), () i= < <, nd investigte the decy of L 2 w (, )-solutions of the eqution L[y]=λy. Conditions on the coefficients p i () re given in Section 2. Asymptotic solutions of differentil equtions hve long been studied using the WKB method nd vrints of it, e.g., see the book by Esthm [3]. WKB methods E-mil ddress: melescue@csm.stte.edu X/02/$ see front mtter 2002 Elsevier Science (USA). All rights reserved. PII: S X(02)

2 S.C. Melescue /. Mth. Anl. Appl. 276 (2002) require certin smoothness of the coefficients nd more importntly require regulrity of growth, which rules out mny oscilltory functions. A number of methods hve been developed for the decy of eigenfunctions of Schrödinger equtions. The reder is referred to Chpter 3 of [6] nd the references contined therein for results on Schrödinger equtions. Our opertors re one-dimensionl but my be of rbitrry order. The bsic tool is perturbtion method used by Kuffmn [8,9] which consists of mking perturbtion B of the opertor L of the form B[y]=L[y] (g Lg)[y], whereg is n ppropritely chosen function. In the Kuffmn theory B is required to be reltively compct with respect to L. In our theory we llow B to be either reltively compct or stisfy certin boundedness condition. (See condition (2) of Lemm.) This dditionl condition llows for stronger decy results. A number of emples re discussed in Section 4, but we illustrte now this stronger decy with the simple emple L[y] = y + q()y on the intervl < where q() is bounded, rel-vlued mesurble function. If we tke g() = e σ for σ>0 sufficiently smll, then the perturbtion B, B[y]=L[y] ( g Lg ) [y]= 2g g y + g g y, stisfies our hypothesis which yields tht solution of L[y]=λy, y L 2 (, ), where λ C is not in the essentil spectrum, stisfies e 2σ y() 2 d <. By using inequlities for derivtives of sum form, we conclude tht e σ y(), e σ y (), nde σ y () pproch zero s. Now, the perturbtion B is not reltively compct, but we produce reltively compct perturbtion if we tke g() = e σβ, β<. For this compct perturbtion, our theory gives tht y stisfies e 2σβ y() 2 d <, which is weker decy of y(). Note tht the order of decy, simple eponentil, is miml s is seen by the emple y = λy. However, the constnt σ produced by our theory is not shrp. We mention now some of the min fetures of our theory. The coefficients of the opertor nd the prmeter λ my be comple. Thus, the results re not just for self-djoint problems. The order of the differentil eqution cn be rbitrry. The decy of the solutions is of the sme order s tht given by WKB solutions, e.g., solutions of y + 2 y = 0 which re L 2 (0, ) re shown to decy like e σ2 for some σ>0. Recll the WKB gives decy /2 e 2 /2 for the L 2 (0, )-solutions. Our theory does not require the smoothness nd regulrity of growth demnded by WKB theory. The decy properties hold for ll solutions in the domin of the miml opertor. The theory pplies to weighted spces.

3 534 S.C. Melescue /. Mth. Anl. Appl. 276 (2002) The theory of Kuffmn hs lso been used by Mergler nd Schultze [0,] nd Schultze [2,3] to study the invrince of deficiency inde nd essentil spectrum under reltively compct perturbtions. The clss of opertors tht they consider hve polynomil coefficients, but the perturbtions re only required to hve certin polynomil bounds. The cse of invrince of essentil spectrum hs immedite ppliction to our work since we require λ not to be in the essentil spectrum of L to obtin the decy of solution of Ly = λy. Thus, ny knowledge of the loction of the essentil spectrum is useful in pplying our theory. The criteri tht Mergler nd Schultze develop for reltively compct perturbtions of differentil opertors with polynomil coefficients is quite strong. It is pplicble to the theory in this pper in the cse tht our originl opertor L hs polynomil coefficients nd when the perturbtions tht we consider, which re of the form, L[y] (g Lg)[y], re reltively compct with respect to L. In Section 2 we stte the min hypotheses nd define the bsic concepts. The generl theorems re proved in Section 3. Section 4 contins emples which pply the results of Section 3 when coupled with recent work on the reltive boundedness nd compctness of differentil opertors. 2. Preliminries We use the following definitions s given by Goldberg [5] nd Weidmnn [4]. Let X nd Y be Bnch spces nd let B nd L be liner opertors, ech hving domin in X nd rnge in Y. By definition the grph norm of L on D(L), denoted L,isgivenby y L = y + Ly. We sy tht L is closed opertor if its grph is closed, i.e., if for ny sequence y n D(L) such tht y n y nd Ly n z, y D(L) nd Ly = z. We sy tht B is reltively bounded with respect to L (or L-bounded) if D(L) D(B) nd B is bounded on D(L) with respect to L, i.e., there eist constnts c,d > 0 such tht By c y +d Ly for ll y D(L). A sequence {y n } n= is L-bounded if there eists constnt C>0 such tht y n L <C for ech n. We sy tht B is reltively compct with respect to L (or L-compct) if D(L) D(B) nd B is compct on D(L) with respect to L, i.e., if {y n } n= is L-bounded sequence, then {By n } n= contins convergent subsequence. The function spce setting is the weighted, seprble Hilbert spce L 2 w ( ), n intervl, which is the spce of (equivlence clsses of) comple-vlued Lebesgue mesurble functions y such tht w y 2 <, where the weight w is positive Lebesgue mesurble function defined on.ifw, we denote this spce by L 2 ( ). We denote the inner product in L 2 w ( ) by f,g w. For the other Lebesgue spces we use the nottion L p ( ), p.

4 S.C. Melescue /. Mth. Anl. Appl. 276 (2002) We consider the differentil epression L[y]=w n p i ()y (i), i=0 where ech p i is comple-vlued function on =[, ) such tht p i L loc ( ), 0 i n. Themiml opertor L is the differentil opertor defined by L with the lrgest possible domin in L 2 w ( ) which is mpped into L2 w ( ), i.e., D(L ) = { y L 2 w ( ): y(n ) AC loc ( ), Ly L 2 w ( )}, where AC loc ( ) is the set of functions which re bsolutely continuous on compct subsets of.wedefinetheminiml unclosed opertor L 0 to be the restriction of L to the functions with compct support in the interior of.the miniml opertor L 0 is defined to be the closure of L 0. We cn integrte by prts to obtin n eqution of the form Ly, z w = [y, z]() + y,l + z w,where[y, z]() is clled the Lgrnge biliner form nd L + is clled the forml djoint of L. We sy tht L is formlly self djoint if L = L +. Note tht L 0, L, L + 0 nd L+ re closed opertors nd stisfy L 0 = L+ nd L = L+ 0 [4, p. 39]. Recll tht for functions y nd z with compct support, L 0 y,z w = y,l 0 z w. When L = L + the differentil epression bove cn be written in the form Γy(t)= W (t) { [n/2] ( ) j( P j (t)y (j) (t) ) (j) j=0 [(n )/2] +i j=0 ( ) j[( Q j (t)y (j) (t) ) (j+) + ( Q j (t)y (j+) (t) ) (j) ]}, (2) where [α] denotes the lrgest integer less thn or equl to α, the comple-vlued function y is defined on (, b), <b, the coefficients W,P j, nd Q j re rel-vlued functions on (, b) nd W(t) is positive. Ech coefficient is ssumed to be loclly Lebesgue integrble on (, b). The nturl number n is the order of the differentil epression Γ. Associted with the formlly self-djoint differentil epressions Γ re miml nd miniml opertors (Γ nd Γ 0, respectively) on the Hilbert spce L 2 W (, b), which re defined in similr mnner s the miml nd miniml opertors ssocited with the opertor L. For emple, the miml opertor Γ defined by Γ hs domin

5 536 S.C. Melescue /. Mth. Anl. Appl. 276 (2002) D(Γ ) = { y L 2 W (, b): y{0},y {},...,y {n } AC loc (, b), Γy L 2 W (, b)}, where ech y {i},fori = 0,,...,n, is qusi-derivtive [4, pp. 26, 29]. We sy tht Γ or L is regulr t if > nd the ssumptions on the coefficients re stisfied on [,b) insted of (, b).we define regulr t b similrly. If Γ is regulr t nd regulr t b, then we sy tht Γ is regulr. Otherwise,Γ is singulr. The kernel inde of liner opertor A, denoted by α(a), isgivenbyα(a) = dimn(a),wheren(a) is the null spce of A. When the rnge R(A) is closed, we cn define the deficiency inde of A, denoted by β(a),byβ(a) = dimr(a). (See further eplntion of deficiency indices d ± in [4].) Note tht R(A) is closed iff 0 / σ ess (A),wheretheessentil spectrum of A is defined by σ ess (A) := {λ: R(λI A) is not closed} [4]. If not both α(a) nd β(a) re infinite, then A hs Fredholm inde defined s κ(a) := α(a) β(a). Note tht for ny rel number r we tke r = nd r =. A closed liner opertor with finite inde is clled Fredholm opertor. We reference the following result of Brown nd Hinton [2, Theorem 2.]. Theorem. Let =[, ), n > 0 nd j be integers such tht 0 j n, nd let N, W, nd P be positive mesurble functions such tht N,W, nd P L loc ( ). Suppose there eists n ε 0 > 0 nd positive continuous function f = f()on such tht nd S (ε) := sup S 2 (ε) := sup { { f 2(n j) [ εf f 2j [ εf +εf +εf P ][ εf W ][ εf +εf +εf ]} N < ]} N < for ll ε (0,ε 0 ). Then there eists constnt k>0 such tht for ll ε (0,ε 0 ) nd y D, N y (j) { 2 k ε 2j S 2 (ε) W y 2 + ε 2(n j) S (ε) P y (n) } 2 where D = { y: y AC loc ( ), W y 2 <, nd P y (n) } 2 <.

6 S.C. Melescue /. Mth. Anl. Appl. 276 (2002) Min results We use the following two lemms in the proof of Theorem 2. Lemm is tken in prt from [8, Lemm 2.9]. Lemm. Let F nd G be differentil epressions of the form () on the intervl (, b) where F is regulr t, the order of G is less thn the order of F,the coefficients of F nd G re sufficiently smooth so tht D(F 0 ) D(G 0 ), nd F 0 nd G 0 stisfy the following two conditions: () R(F 0 ) is closed. (2) Either G 0 is F 0 -compct or there eist constnts c,d > 0 such tht Gy c y +d Fy for every y D(F 0 ) nd c+dγ(f 0 )<γ(f 0 ),whereγ(f 0 ) = F 0. Then (i) D((F + G) 0 ) = D(F 0 ) nd (F + G) 0 = F 0 + G 0 D(F0 ). (ii) R((F + G) 0 ) is closed. (iii) α((f + + G + ) ) = α(f + ),wheref +,G + denote the forml djoints of F,G, respectively, nd (F + G) + = F + + G +. Proof. Note tht if R(F 0 ) is closed, then F 0 < vi the Closed Grph Theorem. (i) Let G = G 0 D(F0 ). Then Lemm V.3.5 or Corollry V.3.8 of [5, pp ] implies tht F 0 + G is closed. Since C0 D(F 0 + G) nd (F + G) 0 is the miniml closed etension of (F + G) 0,wehve(F + G) 0 F 0 + G. Hence, D ( ) (F + G) 0 D(F0 + G) = D(F 0 ). (3) Now, let y D(F 0 ). Then there eists sequence {y n } n= D(F 0 ) such tht y n y nd Fy n Fy s n. (4) The F 0 -boundedness of G 0 implies tht D(F 0 ) D(G 0 ). Hence, there eists constnt C>0 such tht Gy n Gy C( y n y + Fy n Fy ). Vi(4) nd the bove inequlity, we hve Gy n Gy s n. Therefore the sequence {(F + G)y n } n= converges to (F + G)y. By definition, y D((F + G) 0). Thus, D(F 0 ) D ( ) (F + G) 0. (5) Vi (3) nd (5) we hve tht (i) holds. (ii) Since F 0 is miniml opertor with regulr endpoint, we know tht α(f 0 ) = 0 ([4, Theorem 3.2] nd [5, Lemm VI.2.4]). Thus, F 0 hs n inde. Since R(F 0 ) is closed, Theorem V.3.6(i) or Corollry V.3.8 of [5, pp ] implies tht R((F + G) 0 ) is closed.

7 538 S.C. Melescue /. Mth. Anl. Appl. 276 (2002) (iii) Since R(F 0 ) = N(F0 ) [7, IV.5.6] nd F 0 = F +, R(F 0) = N(F + ). Hence, β(f 0 ) = dim R(F 0 ) = dim N ( F0 ) ( = dim N F + ) ( = α F + ), (6) nd similrly, β ( ) ( (F + G) 0 = α (F + G) + ). (7) Theorem V.3.6(ii) or Corollry V.3.8 of [5, pp ] implies tht κ(f 0 ) = κ((f + G) 0 ); nd since F 0 nd (F + G) 0 re miniml opertors with regulr endpoint, we conclude tht α(f 0 ) = α((f + G) 0 ) = 0 nd hence β(f 0 ) = β ( ) (F + G) 0. (8) Equtions (6), (7) nd (8) imply (iii) holds. Lemm 2. Let F be differentil epression of the form () on the intervl with smooth coefficients. Let G = g Fg F,whereg is positive, n times continuously differentible function on nd g = /g. Then the order of G is less thn the order of F nd (g Fg) + = gf + g. Proof. A long clcultion using the product rule for differentition shows tht the highest order derivtives of F nd g Fg cncel, thus the order of G is less thn the order of F. Since the djoint is found by successive integrtion by prts, we hve for y,z L 2 w ( ) of clss Cn with compct support in the interior of, g Fgy,z w = z()w() ( g Fgy ) () d ( = g z ) ()w()(fgy)() d ( = F + g z ) ()w()(gy)() d ( = gf + g z ) ()w()y() d = y,gf + g z w. Therefore, (g Fg) + = gf + g. Theorem 2. Let I be the identity opertor nd let F nd G be s in Lemm 2. Suppose (F λi) 0 nd G 0 stisfy conditions () nd (2) of Lemm.Ifg α on for some α>0, then the mp y g y is one-to-one from N((g(F + λi) g ) ) onto N((F + λi) ).

8 S.C. Melescue /. Mth. Anl. Appl. 276 (2002) Proof. Since g is bounded bove on, w() y() 2 d < implies tht w()g 2 () y() 2 d <. Moreover, if we let g(f + λi)g y = 0for y L 2 w ( ),then(f + λi)g y = 0. Therefore, g y N((F + λi) ). Hence, the mp y g y mps N((g(F + λi)g ) ) into N((F + λi) ) nd is oneto-one since g is positive. By Lemm 2 G + = ( g Fg F ) + = ( g Fg ) + F + = gf + g F +. Using this epression for G + nd (F λi) + = F + λi,wehvevilemm (iii) tht α ( (F + ) ( λi) = α (F λi) + ) (( = α (F λi) + + G +) ) = α ( (F + λi + G + ) (( ) = α gf + g λi ) ) = α (( g(f + λi)g ) ). Since the null spces hve the sme finite dimension nd the mp y g y is one-to-one, the mp is onto. Remrk. We conclude from Theorem 2 tht the inverse mp z gz is one-to-one from N((F + λi) ) onto N((g(F + λi)g ) ). Therefore, z N((F + λi) ) implies tht gz N((g(F + λi)g ) ), i.e., w()g2 () z() 2 d <. 4. Applictions Now, we emine severl emples to obtin upper bounds on the rte of eigenfunction decy. In ech emple the given differentil epression L is formlly self-djoint, i.e., L = L +.IfL = L + nd λ is nonrel, then λ/ σ ess (L 0 ). Therefore, for ll comple-vlued λ, λ / σ ess (L 0 ) is equivlent to λ/ σ ess (L 0 ). We use this fct implicitly in pplying Theorem 2. In ll three emples below, we tke the weight function w. Emple. Suppose p nd q re rel-vlued functions stisfying p AC loc ( ), =[, ), >0, q L ( ), p>0nd p () K p().e. on for some positive constnt K. LetL (L 0 ) be the miml (miniml) opertor ssocited with the differentil epression L[y] = (p()y ) + q()y, ndletb (B 0 ) be the miml (miniml) opertor ssocited with the differentil epression B[y]=L[y] (g Lg)[y] with g defined by [ g() = ep σ After some clcultions ] ds. p(s)

9 540 S.C. Melescue /. Mth. Anl. Appl. 276 (2002) B[y]=B [] [y]+b [0] [y], where B [] [y]=2σ ( ) p()y nd B [0] [y]= σ 2 + σp () 2 y. p() The proof of Theorem 2. of Anderson nd Hinton [] shows tht B [j] 0 is reltively bounded with respect to L 0 λi nd the constnts c j nd d j in B [j] y c j y +d j (L λi)y, y D(L 0 ), re bounded by constnt (independent of ε nd y) times the quntities ε 2j S 2 (ε), ε 2(2 j) S (ε), j = 0,. Hence, B 0 [y]= B [] 0 [y]+b[0] 0 [y] hs the bound By (c 0 + c ) y +(d 0 + d ) (L λi)y, y D(L 0 ). It is further shown in the proof of Theorem 2. of [] tht the ppliction of our Theorem with n = 2, W() =, P() = p() 2, f() = p(),ndn()= (σ 2 + σp ()/ p()) 2 for j = 0ndN()= (2σ p()) 2 for j = yields tht the quntities S (ε) nd S 2 (ε) re proportionl to sup ε p() +ε p() σ 2 + σp (s) 2 2 p(s) ds in the cse j = 0, nd in the cse j = sup ε p() +ε p() p(s) 2σ p(s) 2 ds. Notice tht the first epression is bounded bove by constnt multiple of σ 2 becuse of the condition on p (), nd the second epression is equivlent to 4σ 2. Hence, we cn choose σ>0 sufficiently smll so tht there eist constnts c,d > 0 such tht By c y +d (L λi)y for every y D(L 0 ) nd c + dγ(l 0 λi) < γ (L 0 λi). (9) Therefore, L nd B stisfy the hypotheses of Theorem 2 for λ / σ ess (L 0 ).We pply the remrk following Theorem 2 to conclude tht if y N((L λi) ),then [ ep 2σ p(s) ds] y() 2 d <. (0) Hence, every solution y D(L ) of Ly = λy, λ/ σ ess (L ), decys eponentilly s in (0). Noticethtifweletp() on, we obtin e 2σ y() 2 d <.Since y + q()y = λy nd q() L ( ), it follows tht e 2σ y () 2 d <.

10 S.C. Melescue /. Mth. Anl. Appl. 276 (2002) A stndrd interpoltion inequlity (see Lemm 2. of [2]) gives tht there is constnt k so tht if [c,c + ], then for [c,c + ] f (j) () ( c+ c+ k f() d + f () d ), j = 0,, () c c for ll f such tht f,f AC loc ( ). Using f = y in () nd pplying the Cuchy Schwrz inequlity gives for [c,c + ], y (j) () {( c+ ) /2 } k e 2σ d c {( c+ c e 2σ y() 2 d ) /2 + ( c+ c e 2σ y () 2 d ) /2 } from which it follows tht e σ y(), e σ y () 0s ; hence e σ y () 0s since q L ( ). This gives the first result mentioned in the Introduction. Emple 2. Suppose q AC loc ( ), =[, ), >0, is rel-vlued function stisfying q K nd q () K 2 q() 3/2 for nd some positive constnts K nd K 2.LetL (L 0 ) be the miml (miniml) opertor ssocited with the differentil epression L[y] = y + q()y,ndletb (B 0 ) be the miml (miniml) opertor ssocited with the differentil epression B[y] =L[y] (g Lg)[y] with g defined by [ ] g() = ep σ q(s)ds. After some clcultions B[y]=B [] [y]+b [0] [y],where B [] [y]=2σ ( ) q()y nd B [0] [y]= σ 2 q()+ σq () 2 y. q() The proof of Theorem 3. of Anderson nd Hinton [] shows tht B [j] y d j Ly, y D(L 0 ), where the constnt d j (independent of ε nd y) is liner combintion the quntities ε 2(2 j) S (ε) nd ε 2j S 2 (ε), j = 0,. Hence, B 0 [y]= B [] 0 [y]+b[0] 0 [y] hs the bound By (d 0 + d ) λ y +(d 0 + d ) (L λi)y, y D(L0 ). It is further shown in the proof of Theorem 3. of [] tht the ppliction of our Theorem with n = 2,

11 542 S.C. Melescue /. Mth. Anl. Appl. 276 (2002) W()= q() 2, P()=, f()= q(), ( ) N()= σ 2 q()+ σq () 2 2 for j = 0 nd q() N()= ( 2σ q() ) 2 for j = nd yields tht the quntities S (ε) nd S 2 (ε) re proportionl to q() sup ε +ε/ q() in the cse j = 0, nd in the cse j = q() sup ε +ε/ q() q(s) 2 σ 2 q(s)+ σq (s) 2 2 q(s) ds q(s) 2σ q(s) 2 ds. Notice tht the first epression is bounded bove by constnt multiple of σ 2 becuse of the condition on q (), nd the second epression is equivlent to 4σ 2. As in Emple, we cn choose σ>0sufficiently smll so tht there eist constnts c, d>0such tht By c y +d (L λi)y for every y D(L 0 ) nd inequlity (9) holds. Therefore, L nd B stisfy thehypothesesof Theorem2 for λ / σ ess (L 0 ). We pply the remrk following Theorem 2 to conclude tht if y N((L λi) ),then [ ] ep 2σ q(s)ds y() 2 d <. (2) Hence, every solution y D(L ) of Ly = λy, λ / σ ess (L ), decys eponentilly s in (2). Notice tht if we let q()= 2 on, we obtin the second result mentioned in the Introduction. Emple 3. Suppose q L loc ( ), =[, ), >0, is rel-vlued function stisfying q() K γ 2 for, γ 2 nd some sufficiently smll positive constnt K. LetL (L 0 ) be the miml (miniml) opertor ssocited with the differentil epression L[y] = ( γ y ) + q()y, ndletb (B 0 )bethe miml (miniml) opertor ssocited with the differentil epression B[y] = L[y] (g Lg)[y] with g defined by g() = σ σ for sufficiently smll σ>0. After some clcultions B[y] =B [] [y] +B [0] [y], whereb [] [y] =2σ γ y nd B [0] [y]=σ(σ + γ ) γ 2 y. The proof of Theorem 4. of Anderson nd Hinton [] shows tht B [j] y d j Ly, y D(L 0 ), where the constnt d j

12 S.C. Melescue /. Mth. Anl. Appl. 276 (2002) (independent of ε nd y) is liner combintion the quntities ε 2(2 j) S (ε) nd ε 2j S 2 (ε), j = 0,. Hence, B 0 [y]=b [] 0 [y]+b[0] 0 [y] hs the bound By (d 0 + d ) λ y +(d 0 + d ) (L λi)y, y D(L0 ). It is further shown in the proof of Theorem 4. of [] tht the ppliction of our Theorem with n = 2, W() = 2γ 4, P() = 2γ, f() =, nd N()= (σ (σ + γ ) γ 2 ) 2 for j = 0ndN()= (2σ γ ) 2 for j = yields tht the quntities S (ε) nd S 2 (ε) re proportionl to sup ε +ε s 2(2 γ) σ(σ + γ )s γ 2 2 ds in the cse j = 0, nd in the cse j = sup ε +ε s 2( γ) 2σs γ 2 ds. Notice tht ech epression is constnt multiple of σ 2. As in Emple, we cn choose σ>0sufficiently smll so tht there eist constnts c,d > 0such tht By c y +d (L λi)y for every y D(L 0 ) nd inequlity (9) holds. Therefore, L nd B stisfy the hypotheses of Theorem 2 for λ / σ ess (L 0 ).We pply the remrk following Theorem 2 to conclude tht if y N((L λi) ),then 2σ y() 2 d <. (3) Hence, every solution y D(L ) of Ly = λy, λ / σ ess (L ), decys s in (3). Note tht we only get polynomil decy in Emple 3. This is to epected for if γ = 2ndq() = 0, then the essentil spectrum is [/4, ); ndfor λ</4 thel 2 ( )-solutions of ( 2 y ) = λy re multiples of the function r, r = ( ( 4λ) /2 )/2. We remrk tht Lemms nd 2 nd Theorem 2 pply to the more generl forml differentil epressions of the form (2) in which the coefficients W,P j, nd Q j re (m m)-mtri vlued functions on (, b) nd W(t) is positive definite. Acknowledgments The uthor would like to thnk Professor Don B. Hinton for his suggestions nd thnk the referee for bringing her ttention to the work of Mergler nd Schultze.

13 544 S.C. Melescue /. Mth. Anl. Appl. 276 (2002) References [] T.G. Anderson, D.B. Hinton, Reltive boundedness nd compctness theory for second-order differentil opertors,. Inequl. Appl. (997) [2] R.C. Brown, D.B. Hinton, Sufficient conditions for weighted inequlities of sum form,. Mth. Anl. Appl. 2 (985) [3] M.S.P. Esthm, The Asymptotic Solution of Liner Differentil Systems, Clrendon, Oford, 989. [4] H. Frentzen, Equivlence, djoints, nd symmetry of qusi-differentil epressions with mtrivlued coefficients nd polynomils in them, Proc. Roy. Soc. Edinburgh Sect. A 92 (982) [5] S. Goldberg, Unbounded Liner Opertors: Theory nd Applictions, Dover, New York, 985. [6] P.D. Hislop, I.M. Sigl, Introduction to Spectrl Theory with Applictions to Schrödinger Opertors, Springer-Verlg, New York, 996. [7] T. Kto, Perturbtion Theory for Liner Opertors, Springer-Verlg, New York, 980. [8] R.M. Kuffmn, On the limit-n clssifiction of ordinry differentil opertors with positive coefficients, Proc. London Mth. Soc. (3) 35 (977) [9] R.M. Kuffmn, On the limit-n clssifiction of ordinry differentil opertors with positive coefficients II, Proc. London Mth. Soc. (3) 4 (980) [0] B. Mergler, B. Schultze, A perturbtion method nd the limit-point cse of even order symmetric differentil epressions, Proc. Roy. Soc. Edinburgh Sect. A 94 (983) [] B. Mergler, B. Schultze, On the stbility of the limit-point property of Kuffmn epressions, Proc. Roy. Soc. Edinburgh Sect. A 03 (986) [2] B. Schultze, Odd-order differentil epressions with positive supporting coefficients, Proc. Roy. Soc. Edinburgh Sect. A 05 (987) [3] B. Schultze, Spectrl properties of not necessrily self-djoint liner differentil opertors, Adv. Mth. 83 (990) [4]. Weidmnn, Spectrl Theory of Ordinry Differentil Opertors, in: Lecture Notes in Mth., Vol. 258, Springer-Verlg, New York, 987.