Irih Mh. Soc. Bullein 63 (2009), 11 31 11 Applicion of Prüfer Trnformion in he Theory of Ordinry Differenil Equion GEORGE CHAILOS Abrc. Thi ricle i review ricle on he ue of Prüfer Trnformion echnique in proving clicl heorem from he heory of Ordinry Differenil Equion. We conider elf-djoin econd order liner differenil equion of he form Lx = (p()x ()) + g()x() = 0, (, b). We ue Prüfer rnformion echnique (which re generlizion of Poincré phe-plne nlyi) o obin ome of he min heorem of he clicl heory of liner differenil equion. Fir we prove heorem from he Ocillion Theory (Surm Comprion heorem nd Diconjugcy heorem). Furhermore we udy he ympoic behvior of he equion ( ) when nd we obin necery nd ufficien condiion in order o hve bounded oluion for ( ). Finlly, we conider cerin ype of regulr Surm Liouville eigenvlue problem wih boundry condiion nd we udy heir pecrum vi Prüfer rnformion. ( ) 1. Inroducion In hi review ricle we will preen min heorem reled o he udy of he oluion of elf-djoin econd order liner Differenil Equion of he form Lx = (p()x ()) + g()x() = 0, (, b), (1.1) where p() > 0, p() i boluely coninuou nd g() L 1 (, b) where, b re elemen in he exended rel line. For hi, we will develop nd ue he o clled Prüfer rnformion which re (roughly peking) generlizion of he Poincré phe plne 2000 Mhemic Subjec Clificion. 65L05, 34B24, 34L15. Key word nd phre. Ordinry Differenil Equion, Prüfer Trnformion, Surm Liouville Problem.
12 George Chilo nlyi. The Prüfer rnformion re generl polr coordine repreenion of he oluion of (1.1). The mo common Prüfer rnformion i { x() = r() in Θ() x () = r() p() co Θ(). (1.2) Subiuing (1.2) o (1.1) we obin he Prüfer yem { r () = ( 1 p() g())r in Θ co Θ Θ () = 1 p() co2 Θ + g() in 2 Θ. (1.3) In he econd ecion (Ocillion Theory) we will ue he rnformed yem (1.3) in order o prove min heorem from he Ocillion heory, like he Surm Comprion Theorem, he Ocillion heorem, nd Diconjugcy heorem. In he hird ecion (Bound of Soluion nd Aympoic Behvior) we will ue he Prüfer rnformion nd he modified Prüfer rnformion in order o udy he ympoic behvior of he equion (1.1) when (wihou conidering h g() L 1 (m)). Moreover we will prove necery nd ufficien condiion in order o hve bounded oluion for (1.1). Finlly, in he l ecion (Specrl Theory) we will conider he regulr Surm Liouville eigenvlue problem wih boundry condiion, (p()x ()) + (λr() q())x() = 0, [, b], λ 0 Ax() Bx () = 0 (1.4) Γx(b) x (b) = 0. We will ue he Prüfer rnformion o prove h here i n infinie number of eigenvlue of (1.4) forming monoone increing equence wih λ n, nd h he eigenfuncion Φ n correponding o he eigenvlue λ n hve excly n zero in (,b). Moreover, we will ue he Prüfer rnformed yem o derive upper nd lower bound for he pecrum of (1.4). A he end we preen li of reference which were ued in hi ricle. The reder my refer o hem for proof h re no included in hi pper. I would like o noe h I m priculrly in deb o Profeor D. Hinon for hi conn willingne o dicu ech ep of hi pper.
Applicion of Prüfer Trnformion 13 2. Ocillion Theory In hi ecion we will pply he Prüfer rnformion on regulr Surm Liouville problem in order o prove he Surm Comprion heorem, he Ocillion heorem nd Diconjugcy heorem. Conider he equion of he form Lx = (p()x ) + g()x = 0, (, b). (2.1) (Noe h he equion x + f()x + h()x = 0 cn be rnformed in he form of (2.1) by muliplying i wih e 0 f()d.) We ume h p() > 0 wih p boluely coninuou nd g L 1 (m). In (2.1) we conider he ubiuion y = p()x. From (2.1), x = y p(), y = g()x. (2.2) If we ue polr coordine, x = r() in θ(), y = r() co θ() on (2.2), nd olve for r, θ, hen we obin he Prüfer yem ( ) 1 r () = p() g() r in θ co θ (2.3) θ () = 1 p() co2 θ + g() in 2 θ. (2.4) In he equel we ue he bove rnformed yem o prove he following heorem reled o he oluion of (2.1). The fir wo reul re from [6]. Theorem 2.5 (Ocillion Theorem). Suppoe p i, g i re piecewie coninuou funcion in [, b], nd L i x = (p i x ) + g i x = 0, i = 1, 2. Le 0 < p 2 () p 1 (), g 2 () g 1 (). If L 1 φ 1 = 0, L 2 φ 2 = 0, where φ i re oluion of L i x, nd ω 2 () ω 1 (), where ω i re oluion of (2.4), hen ω 2 () ω 1 () (, b) (1). Moreover, if g 2 () > g 1 (), (, b), hen ω 2 () > ω 1 (), (, b] (2). Proof. From (2.4), ω i = 1 p i co 2 ω i + g i in 2 ω i, i = 1, 2. We hve (ω 2 ω 1 ) = (g 1 1 p 1 )(in 2 ω 2 in 2 ω 1 ) + h (3), where ( 1 h = 1 ) co 2 ω 2 + (g 2 g 1 ) in 2 ω 2. p 1 p 2 Noe h h 0.
14 George Chilo If ω 2 ω 1 = u, hen by (3), u = fu + h, where ) f = (g 1 1p1 (in ω 1 + in ω 2 ) in ω 2 in ω 1. ω 2 ω 1 Hence, f i piecewie coninuou nd uniformly bounded funcion. Since h 0, u fu 0. Se F () = f() d. Then ef u + F e F u 0, nd by inegring hi, e F () u() e F () u() 0 (4). Now i i ey o ee h he bove prove (1). Now uppoe h (2) fil o hold. We how h here exi ome c > uch h ω 2 () = ω 1 () ( c) (5). Suppoe no, hen by (1) here exi equence { j } n j=1 uch h i limi poin of i wih ω 2 ( j ) > ω 1 ( j ), j = 1... n. Now uing (4) wih replced by j, i follow h for > j, ω 2 () > ω 1 (), j = 1... n. Wih j rbirrily cloe o we hve h (2) implie (5). Thi led o conrdicion. Uing (5), (3) i rue wih g 2 > g 1 only if ω 2 = ω 1 = 0 (mod π), nd if p 1 = p 2 in (,c). However, ince ω i, i = 1... n, re oluion of (2.4), he ce ω 1 = ω 2 = 0 (mod π) in (,c) i impoible. Thi prove (2) if g 1 > g 2, nd conclude he proof of he heorem. Theorem 2.6 (Surm Comprion). Suppoe φ i rel oluion of (px ) + g 1 x = 0 nd ψ i rel oluion of (px ) + g 2 x = 0, where x (, b). Le g 1 () > g 2 () in (,b). If 1, 2 re ucceive zero of φ in (,b), hen ψ mu vnih in ome poin in ( 1, 2 ). Proof. Clim 2.7. φ() cn vnih only where ω() = kπ, k Z (where ω() i oluion of (2.4).) Proof. For oluion φ of (2.1) here i oluion r = ϱ(), θ ( = ω() of (2.3), (2.4) repecively, where ϱ 2 = (pφ ) 2 +φ 2, ω = rcn φ pφ ). Since φ nd φ do no vnih imulneouly, i follow h ϱ 2 () > 0, nd wihou lo of generliy we cn ume h ϱ() > 0. A conequence of hi i h φ() = ϱ() in ω() cn vnih only where ω() = kπ, k Z. Now ince co 2 θ, in 2 θ re uniformly bounded, (2.4) h oluion over ny inervl on which p > 0 nd p, g re piecewie coninuou funcion (Picrd Theorem). Since he righ hnd ide of (2.4) i
Applicion of Prüfer Trnformion 15 differenible in θ, i follow h he oluion i unique in he uul ene. Now he proof of he heorem follow direcly from he clim, he monooniciy of ω(), nd Theorem 2.5. The following heorem i from [3]. Theorem 2.8 (Diconjugcy). Conider he problem Lx = (p()x ()) + g()x() = 0, where [, ), x() = 0, nd wihou lo of generliy x () > 0. If p(), g() re coninuou in [, ), nd ( ) 1 p() + g() d π wih p() > 0, [, ), hen no nonrivil oluion of Lx = 0 h wo zero in [, ). Proof. Recll h θ () = 1 p() co2 θ + g() in 2 θ, θ() = 0, nd inegre i o obin θ() = ( 1 p() co2 θ() + g() in 2 θ() ) d ( 1 p() + g() ) d. Hence, ( ) 1 θ() < p() + g() d π. (2.9) Since θ () > 0, θ() < ( ) 1 p() + g() d π, [, ). Since he zero of Lx = 0 occur when θ() = kπ, k Z, he bove inequliy prove he heorem. Theorem 2.10. Conider he problem Lx = (p()x ()) +g()x() = 0, x() = 0 (x () > 0), where g() < 0, [, ). Then he nonrivil oluion of Lx = 0 h mo one zero in [, ). Proof. We know h θ() = 0 nd θ () > 0. Now he monooniciy of θ() implie h for ome b (, ), θ(b) = π 2. Since θ (b) = 1 p() co2 θ(b) + g() in 2 θ(b) = g(b), we ge h 0 < θ() < π 2 in (, ). Therefore in (, ) here re no zero of ny nonrivil oluion of Lx = 0; ince if hey were ny, hey would occur θ() = kπ. Thi conclude he proof. In he following heorem, where i proof i ken from [9], we will mke ue of modified Prüfer rnformion in order o give n imporn reul bou he dince beween wo ucceive zero of fixed nonrivil oluion of he equion x () + p 1 ()x () + p 2 ()x() = 0, (2.11)
16 George Chilo where p 1 (), p 2 () re piecewie coninuou, rel vlued funcion in cloed inervl. Theorem 2.12. Le nd b be conecuive zero of fixed nonrivil oluion of (2.11), nd le γ be differenible funcion defined on [, b]. Se Then M 1 up ( 2γ() p 1 () ), b M 2 up ( γ () p 2 () γ 2 () + p 1 ()γ() ). b b 2 0 d 1 + M 1 + M 2 2. Proof. Le x denoe he oluion referred o he emen of he heorem. Wihou lo of generliy ume h x() > 0 (, b), nd h x () > 0, x (b) < 0. Define he rel vlued funcion R nd Θ by he relion: R in Θ = x (1) R co Θ = x + γx (2), where R() > 0, Θ() [0, π], [, b]. We differenie (1), (2) nd ubiue ino (2.11). Then R co Θ RΘ in Θ = R(γ p 1 ) co Θ+R(γ p 2 γ 2 +p 1 γ) in Θ (3) R in Θ + RΘ co Θ = R co Θ Rγ in Θ (4). We elimine R from (3),(4), nd we hve Θ = co 2 Θ (2γ p 1 ) in Θ co Θ + (γ p 2 γ 2 + p 1 γ) in 2 Θ (5). From (1) we oberve h he zero of x occur when Θ() = kπ, nd from (5) we noe h Θ i increing kπ ince Θ (kπ) = 1. Hence we cn uppoe h Θ() = 0 nd Θ(b) = π. We ue he inermedie vlue heorem o obin (, b) uch h Θ() = π 2. Now le α denoe he le uch. For (, α), in Θ nd co Θ re boh poiive. Now we ue (5) o ge Θ co 2 Θ + M 1 in Θ co Θ + M 2 in 2 Θ,
Applicion of Prüfer Trnformion 17 nd o α π 2 o dθ co 2 Θ + M 1 in Θ co Θ + M 2 in 2 Θ = 0 d 1 + M 1 + M 2 2 (6). Similrly, if β denoe he lrge (, b) uch h Θ() = kπ, hen d b β 1 + M 1 + M 2 2 (7). By combining (6) nd (7) we hve b 2 0 0 d 1 + M 1 + M 2 2. By chooing pproprie vlue for γ nd impoing cerin condiion on p 1 () nd p 2 (), we cn derive ome very remrkble reul. Corollry 2.13. If p 1 0, hen up b p 2()d 2 b. Proof. Se γ Le M = up b b 2 p 2()d. Thu, M 1 = 2 up b M 2 = up 0 b p 2 ()d, 2 p 2 ()d. p 2()d. By Theorem 2.12 we hve d 1 + 2M + M 2 2 = 2 M. Corollry 2.14. If p 1 () i differenible hen up p 1 2 p 2 + p2 1 4 π 2 (b ) 2. b Proof. Chooe γ p 1 2. Then M 1 = 0 nd M 2 = up p 1 2 p 2 + p2 1 4. b
18 George Chilo From Theorem 2.12, b 2 o d 1 + M 2 2 = π 1. M2 Similrly we cn prove h if p 1 i differenible, hen up b p 1 () 2 p 1 () + (p 2 () p 1() 2 ) d 4 2 b. (To ee he bove, chooe γ p1() 2 + (p 2 p 1 Theorem 2.12.) 2 p2 1 4 )d nd pply 3. Bound of Soluion nd Aympoic Behvior In hi ecion we will ue he Prüfer rnformion in order o udy he ympoic behvior of oluion of he equion Lx = (p()x ) + g()x = 0 when, nd we will prove h every oluion of Lx = 0 i bounded if 1 p() g() d <. In he proof of he following heorem we will ue Gronwll Lemm. Lemm 3.1 (Gronwll Lemm). If u, v re rel vlued nonnegive funcion in L 1 (m) wih domin { : 0 }, nd if here exi conn M 0 uch h for every 0 hen u() M + 0 u()v() d, ( ) u() M exp v() d. 0 The following heorem i from [3]. Theorem 3.2. Every oluion x() of Lx = 0 ifie he inequliy [ 1 x() K exp 1 2 p() g() ], d where (, ) nd K = x 2 () + (p()x ()) 2. Moreover, if 1 p() g() d <,
Applicion of Prüfer Trnformion 19 hen every oluion of Lx = 0 i bounded. Proof. We will ue once more he rnformed yem [ ] 1 r () = p() g() r() in Θ() co Θ() (1) Θ () = 1 p() co2 Θ() + g() in 2 Θ() (2), where x() = r() in Θ(), o conclude h x() r() (3). From (1) we hve h [ ] 1 r() r() = p() g() r() 1 in 2Θ() d, (, ). 2 Hence, r() r() + 1 1 2 p() g() r() d. We pply Gronwll Lemm in he bove inequliy nd we ge ( 1 r() r() exp 1 2 p() g() ), d nd o from (3), x() r() exp ( 1 2 1 p() g() ). d Now oberve h if 1 p() g() d <, hen [ 1 ] 1 exp g() d R. 2 p() Thu if M = r() exp ( 1 2 1 p() g() d), hen M R nd x() M. Thi how h x() i bounded. Now we will udy he ympoic behvior of he oluion of he equion x () + (1 + g())x() = 0, (3.3) where if x 0 i fixed rel number h i ufficienly mll for lrge vlue of x, hen g() i rel coninuou funcion for x x 0. Now oberve h (3.3) i of he ndrd form Lx = (x p) +gx = 0, where p() 1, g() g()+1. In (3.3) we will ue modified Prüfer
20 George Chilo rnformion by ubiuing Θ() wih Θ() + 1. The rnformed equion re which yield o r () = g()r() in( + Θ()) co( + Θ()) ( + Θ()) = 1 + g() in 2 ( + Θ()), r () r() = 1 g() in 2( + Θ()), (3.4) 2 Θ () = 1 g()(1 co 2( + Θ()). (3.5) 2 Uing he bove rnformed yem we will prove he following heorem which er h he fundmenl yem of oluion x 1, x 2 of (3.3) when i { x 1 () = co() + o(1) x 2 () = in() + o(1) x 1() = in() + o(1) x (3.6) 2 = co() + o(1). The following heorem i from [8]. Theorem 3.7 (Aympoic Behvior). Le g() be rel coninuou funcion for 0, where 0 i fixed rel number, nd ume h he following inegrl exi: { g() d, g 1 () = g() co(2) d g 2 () = g() in(2) d, 0 g()g j () d, j = 1, 2. (3.8) Then he equion x + (1 + g()x) = 0 h fundmenl yem of oluion ifying (3.6). Noe h he bove umpion re cerinly ified if g L[ 0, ]. Proof. Sep 1: We will how h for ny nonrivil oluion of (3.3) he correponding Θ(), r(), given by (3.4) nd (3.5) repecively, end o finie limi. By uing g() co(2) co(2θ) = (g 1 co 2Θ) 2g 1 Θ in 2Θ nd g() in(2) in(2θ) = (g 2 in 2Θ) 2g 2 Θ co 2Θ, (3.5) cn be wrien : Θ () = 1 2 g+ 1 2 (g 1 co 2Θ) 1 2 (g 2 in 2Θ) +g 1 Θ in 2Θ+g 2 Θ co 2Θ. (3.9)
Applicion of Prüfer Trnformion 21 Since by (3.5) Θ g, i follow from he hypohei of he heorem h Θ i inegrble over [x 0, ). Now from (3.9) we hve Θ () = 1 2 g + 1 2 (g 1 co 2Θ) + 1 2 (g 2 in 2Θ) + g 1 g + g 2 g. From (3.8) nd (3.9) we conclude h Θ() end o finie limi. Similrly uing he relion we cn wrie (3.4) g() in 2 co 2 = (g 2 co 2Θ) 2g 2 Θ in 2θ g() co 2 in 2 = (g 1 in 2Θ) 2g 1 Θ co 2θ, r r = (1/2)(g 2 co 2Θ) + (1/2)(g 1 in 2Θ) + g 2 Θ in 2θ g 1 Θ co 2Θ. (3.10) Since (log r) = r r, from (3.4) we ge (log r) g, nd o (log r) i inegrble over [x 0, ]. Now uing (3.10) we obin (log r) (1/2) (g 2 co 2Θ) + (1/2) (g 1 in 2Θ) + g 1 g + g 2 g, nd hence by (3.8), log r end o finie limi. Therefore r h poiive (finie) limi. Thi conclude he proof of Sep 1. Sep 2: Now we will how h wo diinc oluion of (3.3) cnno end o he me limi. If we inegre (3.9), hen by (3.5) we ge Θ() = Θ( ) + 1/2 1/2 g() d + 1/2(g 1 co 2Θ g 2 in 2Θ) [g(g 1 in 2Θ) + g 2 co 2Θ)(1 co 2( + Θ)] d. (3.11) Now chooe 1 lrge enough uch h g 1 () 1/16 for every 1 nd 1 g j g d 1/16, where j = 1, 2. If ˆΘ() i oluion of (3.5) wih he me limi Θ nd Θ ˆΘ, hen if we ubrc from (3.11) he correponding relion wih Θ replced by ˆΘ, we ge h for 1, Θ() ˆΘ() 1/2 up 1 Θ() ˆΘ(). Hence up 1 Θ() ˆΘ() = 0, nd hu, Θ = ˆΘ. Thi i clerly conrdicion. Similrly we hve r() = ˆr(). Thi conclude he proof of ep 2.
22 George Chilo In erm of (3.3), hi men h for ny nonrivil oluion x here exi conn A, α, (A > 0, 0 α < 2π) uch h for lim x() = A in( + α) + o(1), lim x () = A co( + α) + o(1). Moreover, if x 1, x 2 re linerly independen oluion of (3.3), hen he correponding phe hif α 1, α 2 cnno differ by n ineger muliple of π. Conequenly, by forming uible combinion of x 1, x 2, we cn obin oluion wih ympoic behvior i i decribed in (3.6). Exmple 3.12. (1). Given he equion y ± ky = 0, k > 0, from Theorem 2.6 wih p() = 1, g() = ±k, we ge y() y 2 () + (y ) 2 () exp (1/2 1 ± k ). (2). Conider he equion (p()y ) + k p() y = 0, where 1 p() L1 (m) nd k > 0. Then by Theorem 3.2, ince 1 k p() d 1 k d p() <, we conclude h he oluion y() re bounded. (3). In hi exmple we will illure n pplicion of Theorem 3.7. We will udy he ympoic behvior of he equion x () + ( 1 + in 2 ) x() = 0, 0, λ (Z \ {±2}). 2 in λ Conider he funcion g() =, λ ±2, λ Z. Oberve h g() / L 1 (m), bu he hypohei of he Theorem 3.7 re ified. Indeed, in λ d in 2 λ d (1 co 2λ) = d 2 1 co u = 1/2 du 2 u du = 1/2 u 1/2 co u du. (3.13) u 2
Applicion of Prüfer Trnformion 23 Moreover, du 2 u = nd co u 2 u. Thi how h g() / L 1 (m). Oberve h g() d = Thu, Addiionlly, in λ in λ d co λ λ du <, hence in λ d (1/ 1/ λ + 1/ + g 1 () = (ee [15], p.96 (15.34)) nd g 2 () = in λ in λ in λ d = d exi, ince if 0 < <, co λ 1 co λ d λ λ. co 2 d < in 2 d < ) 1 2 d = 2 λ. (ee [15], p.96 (15.38)). Moreover, ince g 2 L 1 (m), i i elemenry o how h gg j d < for > 0, j = 1, 2. Thi how h he hypohei of he heorem re ified, nd hu he equion x () + (1 + in 2 )x() = 0, 0, λ (Z \ {±2}) h fundmenl yem of oluion ifying (3.6). Remrk 3.14. If λ = ±2 we cn eily ee h where h in 2 λ d = 1 co 4 2 d = d 2 co 4 d, 2 d 2 = nd co 4 d < [0, ). Thi how 2 in 2 λ d = nd hu he inegrl g 2 () = doe no exi nd he Theorem 3.7 doe no pply. in λ in 2 d
24 George Chilo 4. Specrl Theory In hi ecion we will conider he regulr Surm Liouville eigenvlue problem wih boundry condiion. We ue once more Prüfer rnformion echnique o obin heorem concerning he pecrum of uch problem. Conider he yem (p()x ()) + (λr() q())x() = 0, [, b], Ax() Bx () = 0 Γx(b) x (b) = 0. There i no lo of generliy in uming h 0 A 1, 0 B/p() 1 nd A2 +B 2 p 2 () = 1. Thi men h here i unique conn α, 0 α π, uch h he expreion Ax() Bx () = 0 cn be wrien (co α)x() (in α)p()x () = 0. Similrly, here i unique conn β, 0 β π, uch h Γx(b) x (b) = 0 cn be wrien (co β)x(β) (in β)p(b)x (b) = 0. Hence he bove yem i equivlen o he following yem (p()x ()) + (λr() q())x() = 0, [, b], λ 0, (co α)x() (in α)p()x () = 0 (4.1) (co β)x(β) (in β)p(b)x (b) = 0, where λ i rel prmeer nd p, r, q re rel nd piecewie coninuou funcion in [, b] wih p > 0, r > 0 in [, b]. The vlue of λ for which he yem (4.1) h nonrivil oluion re clled eigenvlue nd he correponding (nonrivil) oluion, eigenfuncion. Nex, we preen he mo imporn heorem (which i ken from [4]) bou he eigenvlue nd he zero of eigenfuncion of (4.1). Theorem 4.2. There i n infinie number of eigenvlue λ 0, λ 1, λ 2,... forming monoone increing equence wih λ n n of (4.1). Moreover, he eigenfuncion φ n correponding o λ n hve excly n zero in (, b). Noe h by Theorem 2.6 (Surm Comprion) he zero of φ n epre hoe of φ n+1. Proof. Le φ(, λ) be he unique oluion of he fir equion of (4.1) which ifie φ(, λ) = in α, φ (, λ) = co α. Then φ ifie he econd equion of (4.1). Le r(, λ), ω(, λ) be he correponding Prüfer rnformion of φ(, λ). The iniil condiion re rnformed o r(, λ) = 1, ω(, λ) = α. Eigenvlue re hoe vlue of
Applicion of Prüfer Trnformion 25 λ for which φ(, λ) ifie he hird equion of (4.1). Th i, re hoe vlue of λ for which ω(b, λ) = β +nπ, n Z. By Theorem 2.5 (Ocillion) for ny fixed [, b], ω(, λ) i monoone nd increing in λ. Noe h ω(, λ) = 0 (mod π) if nd only if φ(, λ) = 0. From θ = 1 p co2 θ + (λr q) in 2 θ i i cler h θ = 1 p > 0 zero of φ, nd hence ω(, λ) i ricly increing in neighborhood of zero. Clim 4.3. For ny fixed = c, c [, b], lim λ ω(c, λ) =. Proof. Since α 0 nd ince ω > 0 for ω = 0 (mod π), ω(, λ) 0. Thu i uffice o how h for ome 0, α < 0 < c, lim [ω(c, λ) ω( 0, λ)] =. λ Le 0 = +b 2, nd P, Q, R be conn uch h over ( 0, c), p() P, r() R > 0 nd q() Q. Then he equion P x + (λr Q) = 0 (4.4) wih oluion ˆφ ifying ˆφ( 0, λ) = φ( 0, λ), ˆφ ( 0, λ) = φ ( 0, λ), h ˆω( 0, λ) = ω( 0, λ), nd hence by Theorem 2.5 ω(c, λ) ω( 0, λ) ˆω(c, λ) ˆω( 0, λ). (4.5) (4.4) implie h he ucceive zero of ˆφ P hve pcing π λr Q, P nd hence lim λ π λr Q = 0. Then for ny ineger j > 1, ˆφ will hve j zero beween 0 nd c for λ lrge enough. Thu, ˆω(c, λ) ˆω( 0, λ) jπ. Since j i rbirry, by (4.5), lim λ [ω(c, λ) ω( 0, λ)] =. Thi prove he clim. Clim 4.6. For fixed = c, c (, b], we hve lim ω(c, λ) = 0. λ Proof. We will ue he equion θ = 1 p co2 θ+(λr q) in 2 θ. Chooe δ > 0 ufficienly mll uch h α < π δ. If δ ω π δ, λ < 0, nd if 0 < P p, 0 < R r, Q[ q, hen ω ] < 1/p λ R in 2 δ + Q α δ c α < 0 whenever λ < α δ α c Q 1/p R in 2 δ < 0. Hence ω(c, λ) δ for λ ufficienly lrge. Since δ i rbirry, lim ω(c, λ) λ = 0, nd hi prove he clim. Now for c = b, lim λ ω(b, λ) = 0. Since β > 0, nd ince ω(b, λ) i monoone nd increing in λ, i follow h here i vlue λ = λ o for which ω(b, λ o ) = β. Since 0 α < π nd β π,
26 George Chilo 0 < ω(, λ o ) < π in (, b). Now from hi we immediely obin h he oluion φ(, λ o ) ifie he hird equion of (4.1) nd in ddiion i doe no vnih. Now le λ incree beyond λ o. Then here i unique λ 1 for which ω(b, λ 1 ) = β + π. Clerly, φ(, λ 1 ) ifie he hird equion of (4.1) nd h excly one zero in (, b). If we coninue in hi mnner, he n h eigenvlue i deermined by ω(b, λ n ) = β + nπ nd he n h correponding eigenfuncion h excly n zero in (, b). Thi conclude he proof of he heorem. A Prüfer rnformion, in combinion wih one dimenionl Sobolev inequliy, cn be ued o derive upper nd lower bound for he pecrum (e of eigenvlue) of regulr elf djoin econd order eigenvlue problem. For he nex heorem conider he following eigenvlue problem. Le q be rel funcion in L (, b), 1, nd le λ o < λ 1 < λ 2 <... nd φ o, φ 1, φ 2,... denoe he eigenvlue nd rel orhonorml eigenfuncion (ee Theorem 4.2) of y + q(x)y = λy y() = y(b) = 0. (4.7) We inroduce he noion f + (x) mx (f(x), 0) nd f (x) f + (x) f(x) for rel funcion f. The following heorem i ken from [5]. Theorem 4.8. Le λ < λ n. Then he eigenvlue of (4.7) ify he following inequliy, ( π(n + 1) [ (n + 1) 2 λ n λ + 2(b ) + π 2 b 4(b ) 2 + (λ q(x)) dx] 1/2 ) 2 (1), b which implie h λ n λ + (n + 1)2 π 2 b (b ) 2 + 2 (λ q(x)) dx b (2). Proof. In he equion φ n + qφ n = λ n φ n we pply he modified Prüfer rnformion Thi yield o φ n = r in θ, φ n = λ n λr co θ, r(x) > 0. θ = λ n λ co 2 θ (q λ n) in 2 θ λn λ = λ n λ (q λ) in2 θ λn λ (3).
Applicion of Prüfer Trnformion 27 From (3) we hve, θ λ n λ (λ q) in 2 θ λn λ λ n λ (λ q) λn λ (4). Since φ n h excly n zero in (,b) nd vnihe, b, (ee Theorem 4.2) we my ke θ() = 0 which implie h θ(b) = (n + 1)π. Now we inegre (4) o conclude h (n + 1)π (b ) b λ n λ (λ q(x)) dx (5). λn λ Thi i equivlen o A B A + C, where b A (λ n λ), B (b ) 1 (n+1)π, C (b ) 1 (λ q(x)) dx. Hence A [B+ B 2 + 4C]/2, which i equivlen o (1). Moreover B2 + 4C = B 1 + 4C/B 2 B(1 + 2C/B 2 ) = B + 2C/B, ince 1 + x 1 + x/2 x 0. Thu, A [B 2 + 2B(B + 2C/B) + B 2 + 4C]/4 B 2 +2C, which i excly (2). Thi conclude he proof. Remrk 4.9. The proof of Theorem 4.8 give necery nd ufficien condiion for conrucing q which will mke (1) equliy. Equliy hold in (1) if nd only if equliy hold in (5). Conequenly, b (λ q(x)) in 2 θ(x) dx = b (λ q(x)) dx, which implie h (λ q(x)) co 2 θ(x) = 0,.e[m]. We inegre (3) nd we ge (λ q(x)) + in 2 θ(x) = 0,.e[m]. Since θ (x) > 0, if x = kπ, (k Z), hen (λ q(x)) + = 0,.e[m]. Thu, (λ q(x)) = (λ q(x)) nd (λ q(x)) = 0.e on E, where E = {x : in 2 θ(x) 1}. For exmple ke n = 0 nd λ = 0. Then q(x) 0, nd by (3) we hve h { λo {x : θ(x) = π/2} q(x) = 0 elewhere. In he following heorem we re rying o find, under cerin generl condiion on he coefficien q, be poible (opiml) upper bound on he rel prmeer λ in order for he differenil equion y (x)+ (λ q(x))y(x) = 0, x [, ), o hve nonrivil oluion in L 2 (, ).
28 George Chilo We conider he equion of he form y + (f 2 + fg + fk)y = 0, x [, ), (4.10) where ll quniie re rel, ubjec o he following: (i) f(x) i poiive, loclly boluely coninuou in [, ), nd ifie lim f (x)f 2 (x) = 0. (4.11) x (ii) g(x) i loclly L 1 [, ) nd ifie We define nd we ume h lim g(x)f 1 (x) = 0, (4.12) x k(x) L 1 (, ). (4.13) ψ 1 (x) = up f ()/f 2 (), (4.14) x ψ 2 (x) = up g()/f(), (4.15) x ψ 2 1f, ψ 2 2f re boh in L 1 (, ). (4.16) Then we hve he following heorem (ee [2]). Theorem 4.17. Le he bove (4.11) o (4.16) condiion hold, nd le y be nonrivil oluion for (4.10). Define R(x) by Then for ome conn A we hve, log R(x) A+1/π R 2 = fy 2 + f 1 (y ) 2 R > 0. (4.18) f()(ψ 1 ()+ψ 2 ()) d x [, ). (4.19) Proof. Conider he modified Prüfer rnformion y = Rf 1/2 co θ, y = Rf 1/2 in θ. (4.20) Then we obin he following differenil equion for R, θ : θ = f (1/2)f f 1 in 2θ + g co 2 θ + k co 2 θ (4.21) R R 1 = (1/2)f f 1 co 2θ + (1/2)g in 2θ + (1/2)k in 2θ. (4.22) We inegre (4.22) over (, x) in order o obin he bound in (4.19). The l erm of (4.22), due o (4.13), yield bounded inegrl.
Applicion of Prüfer Trnformion 29 Thu we only hve o conider he oher wo erm. The fir erm in (4.22), uing (4.14), yield o f f 1 co 2θ d ψ 1 f co 2θ d. Now we ue (4.21) o obin, f f 1 co 2θ d ψ 1 θ co 2θ d = + + + ψ 1 (1/2ψ 1 f + g + k ) d ψ 1 θ (2/π) d Subiuing θ from (4.21), we hve, f f 1 co 2θ d ψ 1 f(2/π) d + + (1 + 2/π) ψ 1 θ ( co 2θ 2/π) d ψ 1 (1/2ψ 1 f + g + k ) d. ψ 1 θ ( co 2θ 2/π) d [(1/2)ψ 2 1f + ψ 1 g + ψ 1 k ] d. (4.23) The fir inegrl on he righ yield o he erm ψ 1 (ee (4.14)). Now we will prove h he oher wo erm in (4.23) re bounded. Indeed, for he ce of he econd inegrl in (4.23), we oberve h ψ 1 i non-negive, non-increing nd h θ ( co 2θ 2/π) d = ( co 2θ 2/π) d (4.24) i uniformly bounded for x. From he men vlue heorem for inegrl, nd ince (θ co 2θ 2/π) d i uniformly bounded, we obin ξ (, x) uch h ψ 1 θ ( co 2θ 2/π) d = ξ (θ co 2θ 2/π) d C, where C i conn. Hence he econd erm in (4.23) i bounded. Now in he l inegrl in (4.23) ll hree ummnd of he inegrnd re in L 1 (, ). For he fir erm hi w umed in (4.16). For
30 George Chilo he econd erm noe h g ψ 2 f, nd gin he reul follow from (4.16). For he hird erm noe h ψ 1 i bounded, nd hen ue (4.13). Therefore f f 1 co 2θ d i bounded. Similrly we cn prove h he econd erm in (4.22) yield o bounded inegrl. (Ju replce ψ 1 by ψ 2 nd co 2θ by in 2θ.) Thi conclude he proof of he heorem. The following heorem i from [2]. Theorem 4.25. Le r(x) be loclly L 1 (, ) nd le lim x r(x) = 0. Se p(x) = up x r(x) nd ume h p L 2 (, ). Moreover, le y be nonrivil oluion o y +(λ r(x))y = 0, where x [, ), nd define R > 0 by R 2 = λ 1/2 y 2 + λ 1/2 (y ) 2. Then for fixed λ > 0 nd for ome fixed conn A > 0, we hve h log R(x) A + λ 1/2 π p(x) d x [, ). (4.26) Proof. Apply Theorem 4.17 wih f = λ 1/2, g = rλ 1/2, k = 0. The reul now i immedie. Uing he bove heorem i i poible o prove h he conn λ 1/2 π in (4.26) i he be poible, in he wy h for ny oher conn c < λ 1/2 π he inequliy in (4.26) doe no hold. The Prüfer rnformion re exremely ueful in obining upper bound for rio of eigenvlue of cerin Differenil Operor. Before we cloe hi ecion we will give heorem decribing he opiml bound for rio of eigenvlue of one dimenionl Schrödinger Operor wih Dirichle boundry condiion nd poiive poenil. Theorem 4.27. Le H = d2 + V (x) be Schrödinger Operor dx2 cing on L 2 (I), where I R i finie cloed inervl nd where Dirichle boundry condiion re impoed boh endpoin of I. Aume h V L 1 (I) nd V (x) 0.e on I. Then he rio λn λ 1, of he n h eigenvlue of H o he fir eigenvlue of H, ifie he bound λn λ 1 n 2. Thi bound i opiml, nd for V L 2 (I) nd n > 1 equliy i obined if nd only if V 0.e. on I. The proof of he heorem i given in [1]. The Prüfer rnformion needed for he proof i y(x) = r(x) in( λθ(x)) y (x) = λr(x) co( λθ(x)).
Applicion of Prüfer Trnformion 31 Reference [1] M.S. Ahbugh nd R. Benguri, Opiml Bound for rio of eigenvlue of one dimenionl Schrödinger operor wih Dirichle boundry condiion nd poiive poenil, Comm. of Mh. Phyic 124 (1989), 403 415. [2] F.V. Akinon nd W. Everi, Bound for he poin pecrum for Surm Liouville equion, Proc. of Royl Soc. Edinburgh 80 A (1978), 57-66. [3] J.H. Brre nd J.S Brdley, Ordinry Differenil Equion (1989), 328 348. [4] G. Birkhoff nd G-C. Ro, Ordinry Differenil Equion, Blidell Publihing Compny (1969), 288 308. [5] R.C. Brown, D.B Hinon nd Š. Schwbik, Applicion of one-dimenionl Sobolev inequliy o eigenvlue problem, Differenil nd Inegrl Equion 9 (1996), 481 498. [6] E. Coddingon nd N. Levinon, Theory of Ordinry Differenil Equion, Mc-Grw Hill Book Compny (1995), 208 213. [7] J.H.Cohn, On n ocillion crierion de l Vlle Pouin, Qur. Journl of Mh. Oxford 39 (1988), 173 174. [8] W.A. Coppel, Sbiliy nd ympoic Behvior of Differenil Equion, D.C. Heh nd Compny (1965), 123 128. [9] W.B. Hrri, On n Ocillion Crierion Of Cohn, Qur. Journl of Mh. Oxford 42 (1991), 309 313. [10] W.B. Hrri nd D. Luz, Aympoic Inegrion of dibic ocillor, J.Mh. Anl. Appl. 51 (1975), 76 93. [11] P. Hrmn On he zero of oluion of econd order liner differenil equion, J. London Mh. Soc. 27 (1952), 492 496. [12] P. Hrmn, On cl of perurbion of he hrmonic ocillor, Proc. AMS 19 (1968), 533 540. [13] J. Pryce, Numericl oluion of Surm Liouville problem, Clrendon Pre, Oxford-New York-Tokyo, 1993. [14] W.T. Reid, Surmin heory of ordinry differenil equion, Mc-Grw Hill Compny (1988), 53 62. [15] M.R. Spiegel, Mhemicl Hndbook of formul nd ble, McGrw Hill Compny, 1968. George Chilo, Deprmen of Compuer Science, Univeriy of Nicoi, Nicoi 1700, Cypru chilo.g@iunic.c.cy Received on 20 Augu 2008 nd in revied verion on 28 July 2009.