AN EIGENVALUE PROBLEM FOR LINEAR HAMILTONIAN DYNAMIC SYSTEMS

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1 AN EIGENVALUE PROBLEM FOR LINEAR HAMILTONIAN DYNAMIC SYSTEMS Mrin Bohner Deprmen of Mhemics nd Sisics, Universiy of Missouri-Roll 115 Roll Building, Roll, MO , USA E-mil: Romn Hilscher Deprmen of Mhemics, Michign Se Universiy Es Lnsing, MI , USA E-mil: Asrc. In his pper we consider eigenvlue prolems on ime scles involving liner Hmilonin dynmic sysems. We give condiions h ensure h he eigenvlues of he prolem re isoled nd ounded elow. The presened resuls re pplicle lso o Surm Liouville dynmic equions of higher order, nd furher specil cses of our sysems re liner Hmilonin differenil sysems s well s liner Hmilonin difference sysems. De: Ferury 6, 2003 Running hed: Liner Hmilonin eigenvlue prolems 1991 Mhemics Sujec Clssificion. 34C10, 93B60, 39A12 Key words nd phrses. Time scle, Liner Hmilonin sysem, Eigenvlue, Eigenfuncion, Qudric funcionl, Focl poin, Normliy. Corresponding uhor. Reserch suppored y he Czech Grn Agency under grn 201/01/0079.

2 Liner Hmilonin eigenvlue prolems 1 1. Inroducion A ime scle is ny nonempy closed suse of R. For n inroducion o he ime scles clculus we refer he reder o [6, 7], see lso [8, 12]. If f is funcion on T, we revie f : f nd f : f, where is he forwrd jump operor. The ime scle derivive f reduces o he usul derivive f if T R nd o he forwrd difference f f +1 f if T Z. The grininess funcion of T is µ :. The se of rd-coninuous funcions is denoed y C rd nd he se of rd-coninuously differenile funcions y Crd 1. Le T : [, ] e ime scle inervl, <. The se T wihou is possile isoled i.e., lef-scered mimum will e denoed y T κ ; hus T κ T if is lef-dense. Consider he liner Hmilonin dynmic sysem A + B u, u C A T u, T κ, H where A, B, C : T κ R n n re rel rd-coninuous mrices, B, C symmeric, nd I µ A nonsingulr. Moived y [5], we consider eigenvlue prolems wih formlly self-djoin oundry condiions involving he sysem H, where he mrices A, B, nd C lso depend on n eigenvlue prmeer λ R. We give condiions, mong hem he noion of sric conrolliliy for sysem H, h imply h he eigenvlues of H re isoled nd ounded elow, i.e., hey my e rrnged s < λ 1 λ 2 λ 3..., couning mulipliciies. An eigenvlue prolem on ime scles of he Surm Liouville ype hs een recenly sudied in [1]. The seup of his pper is s follows. In he following Secion 2 we recll some preliminries on Hmilonin sysems h re needed ler. Then, in Secion 3, we inroduce our eigenvlue prolem in deil nd presen sic fcs ou his prolem, e.g., how i is possile o chrcerize he eigenvlues. In his secion we lso presen our min resul on isoledness nd lower oundedness of eigenvlues, which we prove y using some uiliry resuls h re given in deil in he ls Secion Preliminries: Hmilonin sysems By soluion of H we men pir, u wih, u Crd 1 T sisfying he sysem H on Tκ. When referring o soluions of H we use usul greemen h he vecor-vlued soluions of H re denoed y smll leers nd he n n-mri-vlued soluions y cpil ones. By rnk M, Ker M, Im M, def M, ind M, M T, M T 1, M, M 0, nd M > 0 we denoe he rnk, kernel, imge, defec dimension of he kernel, inde numer of negive eigenvlues, rnspose, inverse of he rnspose, Moore Penrose generlized inverse see [2, Chper 1], posiive semidefinieness, nd posiive definieness, respecively, of he mri M. By seing C A T H : A B, J : he liner Hmilonin sysem H hs he form 0 I, z :, z : I 0 u u, 1 L[z] J z + H z 0, T κ. H A soluion X, U of H is clled conjoined sis if rnkx T U T n some nd hence ny T, nd X T U U T X 0 on T. The Wronski mri W X T Ũ U T X is consn on T for ny wo soluions X, U, X, Ũ of H. These wo soluions re normlized if W I. The unique soluion X, U, resp. X, Ũ, of H sisfying he iniil condiions X 0, U I,

3 2 Mrin Bohner nd Romn Hilscher resp. X I, Ũ 0, is clled he principl, resp. ssocied, soluion of H. Togeher hey re clled he specil normlized conjoined ses of H. Lemm 1. For ny s T nd ny conjoined sis X, U of H here eiss noher conjoined sis X, Ũ such h hey re normlized, nd X s is inverile. Proof. See [13, Corollry 3.3.9], [9, Remrk 5]. A conjoined sis X, U of H is sid o hve no focl poins in he inervl, ], provided X is inverile ll dense poins T \ }, nd Ker X Ker X nd D : XX ÃB 0 on T κ. Recll h poin T is dense if i is righ-dense or lef-dense. Sysem H is clled disconjuge on T if he principl soluion of H hs no focl poins in, ]. A pir, u is clled dmissile if is piecewise rd-coninuously differenile, denoed y CpT, 1 u is piecewise rd-coninuous, denoed y u C p T, nd, u sisfies A + Bu on T κ poins T, where is no coninuous, his is o e red s he corresponding righ/lef-sided limi. Le R, S R 2n 2n wih S symmeric. The qudric funcionl F, u T C + u T Bu } T + S. is clled posiive definie F > 0, if F, u > 0 for ll dmissile pirs, u wih Im R T, 0. Following [11], sysem H is clled dense-norml on [, s] whenever s, ] is dense poin nd he only soluion of he sysem u A T u, B u 0, [, s] κ, 2 is he zero soluion u 0 on [, s]. The hypohesis of dense-normliy will e denoed y Sysem H is dense-norml on ny inervl of he form [, s] T. Moreover, we sy h H is norml on T if whenever 0 on T, hen u 0 on T, i.e., sysem H is norml on T if whenever u solves 2 wih s no necessrily dense, hen u 0 on T. The differeniion wih respec o λ will e denoed y d z ż. We require hroughou h d dλ X λ} U dλ D d X λ} 3 dλ U for every conjoined sis X, U of H. This ssumpion is rher resricive, u i cerinly holds for ny ime scle which hs consn grininess, in priculr for T R nd T Z. 3. The eigenvlue prolem Le e given consn mrices R, R # R 2n 2n such h rnkr # R 2n nd R # R T is symmeric. In his pper, he superscrip # does no men generlized inverse, u i is jus n ordinry upper inde. For λ R, we consider he liner Hmilonin sysem A λ + B λu, u C λ A T λu, T κ, H λ sujec o he formlly self-djoin oundry condiions u R # + R 0. 4 u

4 Liner Hmilonin eigenvlue prolems 3 We employ he following generl ssumpion For ll λ R, Aλ, Bλ, Cλ C rd T, R n n, Bλ, Cλ re symmeric, nd I µaλ is nonsingulr on T κ. For ll T κ, A, B, nd C re coninuously differenile wih respec o λ. We denoe Ãλ : [I µ A λ] 1. Firs we derive he Lgrnge ideniy for H on ny ime scle T. Lemm 2 Lgrnge ideniy. For ny z, w Crd 1 T, R2n, where z u nd w y v, nd wih noion 1, we hve w T L[z] L T [w] z } wt J z. Proof. Le z u, w y v, nd z u, w y v. For reviy, we omi he rgumen in he following compuion. The inegrion y prs in he hird equliy sign nd he symmery of H yield y w T T L[z] J + w H z} v u T y T u v T + w T H z } y T u vt + y T u + v T + w T H z } T y J v u y v T J + u w T J z + J w + H w } T z w T J z + L T [w] z. Therefore, he required ideniy follows. y T } v J T + w u T H z The oundry condiions 4 re clled formlly self-djoin if w T J z 0 for ll z, w Crd 1 T, R2n sisfying he given oundry condiions, i.e., z u nd w y v sisfy 4 nd y v R # + R 0, y v respecively. Le us now remrk h, in view of he ne resul, he symmery of R # R T is nurl ssumpion when considering formlly self-djoin eigenvlue prolems wih he sysem H. Lemm 3 Formlly self-djoin oundry condiions. Le R # nd R e rel 2n 2n-mrices such h rnkr # R 2n. Then he oundry condiions 4 re formlly self-djoin iff R # R T is symmeric. Proof. The proof is he sme s he proof of [13, Proposiion 2.1.1]. Remrk 1. By [13, Remrk 2.2.1], here eis mrices S, S # R 2n 2n, such h S is symmeric, rnks # R 2n, ImS # T Ker R, nd R # RS + S #.

5 4 Mrin Bohner nd Romn Hilscher Definiion 1 Eigenvlue prolem. The eigenvlue prolem H λ, λ R, 4, E consiss of he liner Hmilonin dynmic sysem H λ nd he oundry condiions 4. A numer λ R is clled n eigenvlue of E if here eiss nonrivil soluion, u of H λ sisfying 4. Such soluion is hen clled n eigenfuncion corresponding o he eigenvlue λ. The se of ll eigenfuncions corresponding o λ ogeher wih he zero funcion is clled n eigenspce, nd is dimension is referred o s he mulipliciy of he eigenvlue λ. Theorem 1 Chrcerizion of eigenvlues. Le λ R nd le X, U, X, Ũ e ny normlized conjoined ses of H λ. Then λ is n eigenvlue of E iff he mri Λ R 2n 2n defined y X Λ : R X # U Ũ + R X X U Ũ is singulr, nd hen def Λ is he mulipliciy of he eigenvlue λ. Proof. Le, u e nonrivil soluion of H λ. We pu 1 X X Ũ T d : X T U Ũ u U T X T nd hus u X X U Ũ R # + R d on T. Hence, u u X R X # X X d + R 0, u U Ũ U Ũ d Λd. Thus,, u sisfies he oundry condiions 4 iff Λd 0, i.e., λ is n eigenvlue of E iff Λ is singulr. Corollry 1 Sepred oundry condiions. Assume h sepred oundry condiions re given, i.e., R 0 R # R, R 0 R # 0, 0 R # where he n n-mrices R, R, R #, R # sisfy rnkr# R rnkr # R n, R R # T R # R T, nd R R # T R # RT. Le X, U e he conjoined sis of H λ, λ R, wih X R T, U R # T. Then λ is n eigenvlue of E iff he mri Ω R n n given y is singulr. Ω : R # X + R U Proof. Le λ R. For X, U here eiss conjoined sis X, Ũ of H λ such h hey re normlized, y Lemm 1. Then, Theorem 1 implies h λ is n eigenvlue of E iff Λ is singulr. Since X Λ R X # U Ũ + R 0 I X Ω R # X, + R Ũ X we hve Λ c 1 c 2 0 iff c 2 0 nd Ωc 1 + R # singulr iff Ω is singulr. U Ũ X + R Ũ c 2 0, i.e., iff Ωc 1 0. Hence, Λ is Definiion 2 Sric dense-normliy. The se of sysems H R : H λ, λ R}, is clled sricly dense-norml on T if i H λ sisfies D for ll λ R.

6 Liner Hmilonin eigenvlue prolems 5 ii For ll λ R, for ny s T \ }, for ny soluion, u of H λ, if Ḣ λ 0 for ll [, s] u κ, hen u 0 on T. Remrk 2. We re priculrly ineresed in he cse when A λ A nd B λ B re independen of λ nd C depends on λ linerly, i.e., i is of he form C λ C. In his remrk we discuss some feures of his specil cse. i Firs, we noe h ii implies i in Definiion 2. To show his, le λ R nd ke ny soluion, u of H λ such h 0 on [, s], where s T is dense poin. We hve Ḣ λ u Ċ λ 0 Ċλ 0 0 u C 0 on [, s] κ, hence ii implies u 0 on T, so h H λ is dense-norml on [, s]. ii Ne, we show h eigenvecors corresponding o differen eigenvlues re orhogonl. More precisely, le R, S #, S, S R 2n 2n e such h S, S re symmeric, rnks # R 2n, ImS # T Ker R, nd pu Sλ : S λ S, R # λ : RSλ + S #. Consider he eigenvlue prolem A + B u, u C λ C A T u, T κ, u λ R, R # λ + R 0. u If, u nd y, v re eigenfuncions of Ẽ elonging o eigenvlues λ nd ν, respecively, λ ν, hen y wih respec o C nd S, i.e., T, y : T C y y + S 0. y To show his, we follow he proof of [13, Proposiion 2.2.2]. Since, u solves H λ nd y, v solves H ν, inegrion y prs implies T C λ Cy } + u T Bv u T y, 5 y T C ν C + v T Bu By susrcing 6 from 5 we oin } ν λ T Cy } Ẽ v T. 6 y T u T v. 7 Oserve h S # R T 0 nd R # λ + λr S RS + S #. Moreover, from [13, Proposiion 2.1.2] i follows h, u nd y, v sisfy he oundry condiions in Ẽ iff R T u c, R u # λ} T y c, R y T v d, R v # ν} T d,

7 6 Mrin Bohner nd Romn Hilscher for some c, d R 2n. Thus, from 7 we hve T ν λ, y ν λ T C y + ν λ S y T u T T v + ν λ S y y T T y u v + y u v y y T ν λ S d T RR # λ} T c + c T RR # ν} T d + ν λ c T R SR T d d T RRS + S # T c + c T RRS + S # T d 0. y y Hence, y nd he proof is complee. iii If he sysem is sricly dense-norml nd if S nd C re ll posiive semidefinie which is sisfied in he presen seing noe h hroughou his pper, wih he ecepion of his remrk, we ssume S 0 sujec o condiions V 1 nd V 2 given fer his remrk, hen ll eigenvlues re rel. To see his, le, u e n eigenfuncion corresponding o n eigenvlue λ. Then, ū is n eigenfuncion corresponding o he eigenvlue λ, nd we my use he clculion from he second pr of his remrk o oin 0 λ λ, λ λ } T C + T } S. Clerly,, 0, since oherwise he posiive semidefinieness of S nd C implies S 0 nd C 0 on [, ] κ nd hence u 0 y sric dense-normliy, which is impossile. Therefore, λ λ 0 nd our clim λ R follows. Le us coninue wih he invesigion of he generl eigenvlue prolem E. Given he eigenvlue prolem E, we define he qudric funcionl F, u; λ : T Cλ + u T Bλu } T + S, where he mri S is deermined y Remrk 1. We consider he following ssumpions: V 1 H R is sricly dense-norml on T. V 2 λ 1 λ 2 lwys implies H λ 1 H λ 2 for ll T κ. V 3 There eiss λ R such h F ; λ > 0 nd λ λ lwys imply for ll T κ Ker Bλ Ker Bλ nd Bλ B λ B λ } Bλ 0. V 4 H λ is norml on T for ll λ R. Now he min resul of his pper reds s follows. Theorem 2. Assume V 1 V 4. Then, if here eis eigenvlues of E, hey my e rrnged y < λ 1 λ 2 λ 3..., couning mulipliciies. More precisely,

8 Liner Hmilonin eigenvlue prolems 7 i V 1 nd V 2 imply h he eigenvlues re isoled. ii V 2 V 4 imply h he eigenvlues re ounded elow y λ, provided H λ sisfies D for ll λ R. Proof. Pr i isoledness. Le Xλ, Uλ, Xλ, Ũλ e he specil normlized conjoined ses of H λ for ech λ R. Fi λ 0 R. Then y Lemm 6 here eiss ε > 0 such h X λ is inverile nd 1 I 0 0 I U λ Ũ λ X λ X λ is sricly decresing for ll λ Uλ 0, ε, where Uλ 0, ε : [λ 0 ε, λ 0 + ε] \ λ 0 } is he closed ε-inervl round λ 0 wihou λ 0. I follows from he Inde Theorem Proposiion 1 in he ne secion h he singulr poins of 0 I Λλ R # X λ X λ X λ R X # λ X λ X λ I 0 + R U λ Ũ λ U λ Ũ + R λ, U λ Ũ λ i.e., he eigenvlues of E y Theorem 1, re isoled. Furhermore, he mulipliciy of n eigenvlue λ 0 is def Λλ 0 ind Mλ + 0 ind Mλ 0, since X # 0 I X λ 0 X λ 0 is inverile he mri Mλ is defined in Proposiion 1. Hence, pr i is proved. Pr ii lower oundedness. Assume h H λ sisfies D for ll λ R, nd h V 2 V 4 hold. For λ R define 1 I 0 0 I Mλ : R S + Q # λ} RT, Q # λ :. U λ Ũ λ X λ X λ We pick λ 0 λ. Then F ; λ 0 > 0 y he Comprison Theorem Theorem 3 in he ne secion. Since H λ0 is norml on T, Proposiion 2 implies h X λ 0 nd hence X # 0 I X λ 0 X λ 0 re inverile, nd Mλ 0 > 0 on Im R. I follows h X # λ is inverile on some open inervl J round λ 0. Moreover, he mri Q # λ defined ove is sricly decresing on J, y Lemm 5, nd ind Mλ ind Mλ 0. Now, we my pply he Inde Theorem Proposiion 1 o oin def Λλ 0 ind Mλ + 0 ind Mλ 0 + def X # 0, i.e., Λλ 0 is inverile. This mens in view of Theorem 1 h λ 0 is no n eigenvlue of E. Therefore, if here eiss n eigenvlue ll, here is he smlles one λ 1 nd sisfies λ 1 > λ. The proof is complee. 4. Auiliry resuls In his secion we collec uiliry resuls needed in our work. Recll h Uλ 0, ε is he closed ε-inervl round λ 0 he cener is removed.

9 8 Mrin Bohner nd Romn Hilscher Proposiion 1 Inde Theorem [13, Theorem 3.4.1, Corollry 3.4.4]. Le m N nd le here e given mrices R, R #, X, U R m m wih rnkr # R rnk X U m nd RR# T R # R T, X T U U T X. Le Xλ, Uλ R m m e mrices such h X T λuλ re symmeric for ll λ Uλ 0, ε, for some ε > 0, Xλ X, Uλ U s λ λ 0, nd Xλ is inverile for λ Uλ 0, ε. Suppose h UλX 1 λ decreses sricly on [λ 0 ε, λ 0 nd on λ 0, λ 0 + ε], nd denoe Then Mλ : R # R T + RUλX 1 λr T, Λλ : R # Xλ + RUλ, Λ : R # X + RU. ind Mλ 0 : lim λ λ 0 ind Mλ, ind Mλ + 0 : lim ind Mλ λ λ + 0 oh eis, Λλ is inverile for ll λ Uλ 0, δ for some δ 0, ε, nd def Λ ind Mλ + 0 ind Mλ 0 + def X. Proposiion 2 Jcoi Condiion [10, 11]. Suppose D holds. Le X, U, X, Ũ e he specil normlized conjoined ses of H. Then F > 0 iff X, U hs no focl poins in, ] nd S + Q # > 0 on Im RT Im X #, where X # : 0 I X nd X Q# is cerin 2n 2nmri uil up from X, U, X, Ũ. Moreover, if H is norml on T, hen F > 0 implies X nd hence X # is inverile. Lemm 4. Suppose h Xλ, Uλ is conjoined sis of H λ for ll λ R wih Ẋλ 0 U, i.e., X nd U re independen of λ. Then X T λ U X λ U T T λẋλ τ λ X Ḣτ λ τ λ τ U τ λ U τ λ holds for ll T nd for ll λ R. Proof. In he compuion elow we skip he evluion T. Compre [5, Lemm 4]. We hve X T ν [Uλ Uν] U T ν[xλ Xν] } Xν T } Uλ Uν Xλ T Uν X λ X T ν Uλ Xν Uλ Uν Xλ X T ν X Hν Hλ} λ. Uν Uλ Now, dividing y λ ν nd leing ν λ oserve h 3 is used yields X T λ Uλ } X U T λẋλ T λ X λ Ḣλ. Uλ Uλ Inegring from o we ge X T τ λ X Ḣτ λ τ λ τ X U τ λ U τ λ τ T λ U τ λ Uτ T λẋτλ nd Ẋλ 0 U yields he resul.,

10 Liner Hmilonin eigenvlue prolems 9 Lemm 5. Suppose h Xλ, Uλ, Xλ, Ũλ re normlized conjoined ses of H λ for ech λ R wih Ẋλ U λ 0 X λ Ũ λ. Le T, >. Assume h X λ is inverile for λ in some open inervl J. For λ J pu 1 I 0 0 I Q λ :. U λ Ũ λ X λ X λ Then V 2 implies h Q λ decreses on J. decreses sricly on J. Moreover, V 1 nd V 2 imply Q λ h Proof. The proof is similr o he proof of [5, Lemm 5], so we skech i only. Le T, >, nd λ J. We pply Lemm 4 o 0 I I 0 X # λ :, U # λ :. X λ X λ U λ Ũ λ Then for d R 2n 2n i follows h where d T Q λd τ u τ T τ Ḣτ 0, u τ Xλ Xλ : X u # 1 λd, Uλ Ũλ nd where we used V 2, i.e., Ḣλ 0. Suppose h V 1 nd V 2 hold wih d T Q λd 0. Then Ḣτλ τ u τ 0 for ll τ [, ] κ. Sric dense-normliy implies u 0 on T, i.e., d 0. Thus, Q < 0 follows. Lemm 6. Le Xλ, Uλ, Xλ, Ũλ e he specil normlized conjoined ses of H λ for ech λ R. Assumpions V 1 nd V 2 imply h for ll λ 0 R here eiss ε > 0 such h X λ is inverile nd Q λ defined y 1 I 0 0 I Q λ : 8 U λ Ũ λ X λ X λ is sricly decresing for ll λ Uλ 0, ε. Proof. Fi λ 0 R nd le ˆX, Û e he conjoined sis of H λ 0 such h Xλ 0, Uλ 0 nd ˆX, Û re normlized nd ˆX is inverile, see Lemm 1. Le ˆXλ, Ûλ e he conjoined sis of H λ wih ˆX λ ˆX, Ûλ Û, λ R. Due o coninuiy, ˆXλ is inverile on some open inervl round λ 0 nd on h inervl we hve, y Lemm 5 wih ˆXλ, Ûλ nd Xλ, Uλ, h he mri 1 I 0 0 I ˆX 1 1 Ûλ U λ ˆX λx λ ˆX λ T 1 λ X λ ˆX λ Û λ λ is sricly decresing. Consequenly, ˆX 1 λx λ is sricly decresing s well. I follows h X λ is inverile on Uλ 0, ε for some ε > 0. Applying Lemm 5 gin, he sric monooniciy of he mri Q λ in 8 follows. ˆX 1

11 10 Mrin Bohner nd Romn Hilscher Lemm 7. Le m N nd le e given rel m m-mrices A, A, B, B, C, C such h he Hmilonin mrices C A T C A T H :, H : A B A B re symmeric. Suppose h hold. Then H H, Ker B Ker B, BB B B 0 T C + u T Bu T C + u T Bu for ll, u,, u R m wih Bu Bu A A. Moreover, here eiss mri E R m m such h A A B BE nd E T B BE C C. Proof. The proof is similr o he discree cse [5, Lemm 7], compre lso he coninuous cse [13, Lemm ]. Remrk 3. Oserve h Ker B Ker B from he ove lemm is equivlen o B BB B BB B, see [3, Lemm A5, pg. 94] or [4, Remrk 2iii]. Theorem 3 Comprison Theorem. Suppose h V 2 nd V 3 hold. Then F ; λ > 0 for ll λ λ. Proof. Suppose F ; λ > 0 nd le λ λ. From V 2 nd V 3 we hve } B λ B λ, Ker B λ Ker B λ, B λ B λ B λ B λ 0. Le, u e dmissile for F ; λ, i.e., nd 0. For T κ we define u : B λb λu A λ + B λu, T κ, wih Im R T, } I B λb λ E, where E : T κ R n n is such h A λ A λ B λ B λ}e, y Lemm 7. Noe lso h B λb λb λ B λ y Remrk 3. Then ll funcions evlued so h Bλu Bλu Bλu BλB λbλu + Bλ BλB λbλ } E Bλ Bλ} E Aλ Aλ}, Aλ + Bλu Aλ + Bλu, i.e.,, u is dmissile for F ; λ. Applying Lemm 7 gin we ge 0 < F, u; λ Hence, F ; λ > 0 s well. T Cλ + u T Bλu } T + S T Cλ + u T Bλu } + T S F, u; λ.

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