Periodic and antiperiodic eigenvalues for half-linear version of Hill s equation

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1 INERNAIONAL JOURNAL OF MAHEMAICAL MODELS AND MEHODS IN APPLIED SCIENCES INERNAIONAL JOURNAL OF MAHEMAICAL MODELS AND MEHODS IN APPLIED SCIENCES Periodic antieriodic eigenvalues for half-linear version of Hill s equation Gabriella Bognár University of Miskolc Deartment of Analysis Miskolc-Egyetemváros Hungary matvbg@uni-miskolc.hu Abstract he nonlinear eigenvalue roblem of the differential equation x 2 x + + ct x 2 x = > with resect to the eriodic boundary conditions: x = x x = x or to the antieriodic boundary conditions: x = x x = x are considered. Various results on the set of eigenvalues concerning both roblems are resented. Some estimates are given for the eriodic antieriodic eigenvalues. Key Words: Hill s equation eriodic solution eigenvalues I. INRODUCION A Hill s equation is a differential equation of the tye x + qt x = where qt is an integrable real function of eriod. his tye of equation was first investigated in connection with the theory of lunar motion by G. W. Hill [9]. It is also well-known in the quantum theory of metals semi-conductiors see e.g. [4] [7] [8] or in otics when ultrashort otical ulses are examined see e.g. [2] [5] [6]. he value of the eriod of the solution lays an imortant role in the discussion of eriodic solutions. A secific question is the case of solutions of eriod 2 see [5] [7] [3]. We consider the half-linear version of Hill s differential equation x x 2 + qt x x 2 = >. It is called half-linear differential equation by I. Bihari [2]. Its solution set reserves the half of the roerties of the linear differential equation since it is homogeneous but not additive. In [8] Á. Elbert established the existence uniqueness of solutions to the initial value roblem for differential equation equation of tye. he aim of this aer is to examine the eriodic solutions of equations with eriodic or antieriodic boundary conditions or x = x x = x 2 x = x x = x 3 resectively when qt = + ct R the otential c t is eriodic. he value is called an eigenvalue x an eigenfunction if the air x satisfies -2 or -3. We investigate the asymtotic behavior of large eigenvalues. II. PRELIMINARIES In this section we recall some known results techniques. A. Generalized sine function For the secial case qt the solution of equation x x 2 + x x 2 = 4 with the initial conditions x = x = called the generalized sine function x = S t t + 5 was introduced by Á. Elbert in [8]. For t [ π/2] where function S satisfies π/2 = π / sin π t = S dx x. 6 Formula 6 defines uniquely function S on [ π/2] with S π/2 =. We extend S to all R still denote this extension by S as a 2 π eriodic function: S t = S π t for t [ π/2 π] S t = S t for t [ π ] 7 S t = S t + 2 π for t R. herefore function S has the following roerties: i S t + ˆπ = S t for all t S t is an odd function having zeros at t = jˆπ j = Z ii S t has zeros only at t = ˆπ + jˆπ j = Z. 2 iii From 4 by integration we have the generalized Pythagorean relation S t + S t = for all t R. 8 For = 2 we have that S 2t = sin t ˆπ = π equation 8 is reduced to the usual Pythagorean relation sin 2 t + cos 2 t =. B. Generalized Prüfer transformation It is convenient to introduce the generalized Prüfer transformation for the examination of the solutions of the quasilinear differential equation using the above defined generalized trigonometric function. For xt of the generalized olar functions ϕt ρt are defined by Issue Volume Manuscrit received Oct. 9 27; Revised received Jan

2 INERNAIONAL JOURNAL OF MAHEMAICAL MODELS AND MEHODS IN APPLIED SCIENCES 2 INERNAIONAL JOURNAL OF MAHEMAICAL MODELS AND MEHODS IN APPLIED SCIENCES where xt = ρt S ϕt x t = ρt S ϕt ρt = [ xt + x t ] / moreover we have that x W x W x W x / W x W x W moreover ϕt ρt are continuously differentiable functions of t. hen the air ϕ; ρ = ϕt ; ρt is a solution of the system of differential equations ϕ = S ϕ + qt Sϕ ρ = ρ qt S ϕ S ϕ 2 S ϕ. III. PERIODIC AND ANIPERIODIC EIGENVALUE PROBLEMS We consider differential equation for qt = + ct : x 2 x + + ct x 2 x = 9 in where > > is real number ct is a ositive continuous eriodic function on. he boundary conditions are x = x x = x called eriodic boundaryconditions or x = x x = x P AP called antieriodic conditions. Let x = xt be a solution of 9 with P. We extend x as a eriodic function on R such as xt + = xt for any t R then x is a -eriodic solution of 9 on the whole of R. Let x = xt t [ ] be a solution of 9 with AP extend x as follows: then xt = xt for t 2 xt = xt + 2 for any t R. herefore x is a 2 -eriodic solution of 9 on the whole R. For the functional settings we define W as a function sace of all continuous functions y = yt t [ ] such that y = [ y + ct y ] dt / < y satisfies P. Let us define W as a function sace of all continuous functions y = yt t [ ] such that y < y satisfies AP. Both W W are Banach saces Sobolev saces of - eriodic -antieriodic functions. defines a norm in both saces. We can summarize the following roerties: W W = { x : x = x = x = x = }. he formulation of the eigenvalue roblem of 9-P in a weak sense is the following: Definition Function x W is called the weak solution of 9-P if for all y W x 2 x y dt + ct x 2 xydt = is satisfied. Analogously we have Definition 2 he weak solution of 9-AP is a function x if x 2 x y dt + ct x 2 xydt = W holds for all y W. he regularity of the weak solution can be considered by stard regularity argument given by M. Otani [4]. If x is a weak solution of 9-P then x C [ ]. We have the same roerty for x. Moreover if x is a weak solution of 9-P then x C 2 with excetion of the oints t where x t = for > 2. he same holds for x. We note that the initial value roblem of 9 under initial conditions xt = x x t = x admits unique solution x C R x C 2 R with excetion of the oints where x = for > 2 see [6]. For the variational characterization of the eigenvalues we set for y W W S := y = y dt / we use the notation { } y W : y =. For a closed symmetric set A S we define the Krasnoselski genus of A as follows: Let us define γa := inf {m N : continuous odd maing A into R m \ {}} γa := if such m does not exist. F k := {A S : A = A γa = k} k N. Issue Volume

3 INERNAIONAL JOURNAL OF MAHEMAICAL MODELS AND MEHODS IN APPLIED SCIENCES 3 INERNAIONAL JOURNAL OF MAHEMAICAL MODELS AND MEHODS IN APPLIED SCIENCES We denote by S à F k the same sets if W is relaced by W. he eigenvalues of 9-P or 9-AP are those values of R for which there exists non-zero solution of 9-P or 9-AP. Definition 3. Let us denote by k k the eigenvalues of 9-P 9-AP. hen we have k := min max A F k+ x A x for k {} N k := min max x for k N. à F k+ x à We consider the eigenvalues of 9 with resect to the eriodic P or antieriodic AP boundary conditions. When no otentials are resent ct the eriodic antieriodic eigenvalues of 9 are known because 9 is integrable. hus the eigenvalue roblem IV. ASYMPOIC RESULS Henceforth we consider differential equation 9 for sufficiently large value of such that + ct >. Without loss of generality we assume that ct is integrable ˆπ ct dt =. Let c in 9 be a eriodic function of t with eriod ˆπ let ct be satisfy [ + ct] +/ > c t for all t. o emhasize the deendence solution of 9 on we shall write xt. First we construct a solution y of 9 such that x 2 x + x 2 x = x = xˆπ = xt = At S ϕt 2 x t = + ct At S ϕt 3 has a solution x = C S t for C R which is a eriodic solution. In order to obtain nonvanishing solutions it is necessary that = n n = the eigenfunctions are given by x n = C n S n t. If = 2 ct is 2π eriodic c L 2π this is the case = 2π then the classical results are known see e.g. [3]. However when some otentials are resent in 9 ct the eriodic antieriodic eigenvalues are studied by M. Zhang [9]. It is known that there exist two sequences { k : k Z + } { k : k N} of the reals such that < 2 < 3 4 < 5 6 <... 2 < 3 4 < 5 6 <... both sequences { k : k Z + } } { k : k N tend to + as k +. It is also known that the number of nodes of x k or of x k in [ is finite. Additionally we can give the number of nodes in the two cases. Let x k be the eigenfunction associated with k k = hen the number of nodes of x k in [ for k = 2n is equal to 2n n = 2... for k = 2n is equal to 2n n = he smallest eigenvalue is simle isolated. the roof is similar as in []. Let x k be the eigenfunction associated with k k = hen the number of nodes of x k in [ is 2n for k = 2n n = n also for k = 2n n = Here the smallest eigenvalue is not simle in general. It is enough to take the linear case = 2 with ct const. when = 2. where ϕt At are continuously differentiable on [ determined by the differential equations with notation ϕ t = + ct + A t At = c t Gϕ. 4 + ct c t + ct S ϕt 5 Gϕ = S ϕ S ϕ 2 S ϕ. he conditions on x x determine the values of ϕ A. Inequality guarantees that ϕt is monotonically increasing function of t. From 7 it follows that if ϕt At rovide a solution xt then ϕt + ˆπ At also rovide as a solution xt. All the solutions can be obtained on the range of values ϕ where the range is of length ˆπ. We get from 5 that function with αt = At = A ex αt + cτ S ϕτ dτ is monotone non-increasing tends to a limit A as t. If + cτ S ϕτ dτ = then A = If lim xt =. t + cτ S ϕτ dτ < then A > solution xt oscillates where its amlitude tends to a ositive value. Issue Volume

4 INERNAIONAL JOURNAL OF MAHEMAICAL MODELS AND MEHODS IN APPLIED SCIENCES 4 INERNAIONAL JOURNAL OF MAHEMAICAL MODELS AND MEHODS IN APPLIED SCIENCES If xt is eriodic with eriod ˆπ then we get from 2 3 with conditions that xˆπ = x 6 x ˆπ = x Aˆπ = A 7 ϕˆπ ϕ = 2k ˆπ where k is a ositive integer. If xt is antieriodic then for the solution we have conditions hence where k is a ositive integer. Problem 9 with xˆπ = x 8 x ˆπ = x Aˆπ = A 9 ϕˆπ ϕ = 2k ˆπ xˆπ = x x ˆπ = x has countable infinity of values 2... k... accumulating at similarly for roblem 9 with xˆπ = x has countable infinity of values x ˆπ = x 2... k... accumulating at for each nonnegative integer = k with ϕˆπ k ϕ k = 2k ˆπ = k with ϕˆπ k ϕ k = 2k ˆπ for the roof see [9]. Now we gain more information regarding the distribution of the arameter. We give estimates for large eigenvalues: heorem 4 Let ct c t c t be bounded eriodic functions with eriod ˆπ. hen for k k concerning the solutions of resectively 2k 2k = O k ν 2k 2k = O k ν 2k 2k = O k 2 ν 2k 2k = O k ν hold for large values of k with { 2 if < < 2 ν = + if 2. Proof: For t = we get a Volterra tye integral equation for ϕ ϕt = ϕ cτ dτ 2 Gϕ dτ. + cτ Since c t ct are bounded if is large enough then K = const. + cτ ϕt = ϕ + As for sufficiently large so that t S ϕ = S ϕ + t ϕ + S ϕ = S G ϕ = G ϕ + By an iteration we find that + ϕt = ϕ + Gϕ dτ < K + cτ dτ + O + cτ dτ τ + cτ G ϕ + For ˆπ eriodic solution we get + cτ dτ + O + O 22 + cτ dτ + O cτ dτ +O. 2 x = xˆπ x = x ˆπ Aˆπ = A 2k ˆπ = = + + cs ds dτ ϕˆπ ϕ = 2k ˆπ k + cτ dτ 24 k + cτ G ϕτ k dτ k + cτ S ϕτ k dτ. 25 he values of ϕ k k are unknown. As ϕt k is determined from 2 then ϕ k k can be determined from for every k. Alying we obtain estimates on k k. heorem 5. Let ct be eriodic with eriod ˆπ let M be a uniform bound for c c c c. hen the eigenvalues belonging to the roblem 9-P 9-AP when 3 satisfy the inequalities 2k > 2k 2k > 2k 2k > 2k 2k > 2k rovided that they are greater than constant Λ defined by Issue Volume

5 INERNAIONAL JOURNAL OF MAHEMAICAL MODELS AND MEHODS IN APPLIED SCIENCES 5 INERNAIONAL JOURNAL OF MAHEMAICAL MODELS AND MEHODS IN APPLIED SCIENCES Λ = max M + + M C + C2M + C3M 3 M + M where C = C C 2 = C 2 C 3 = C 3. For the roof we refer [3]. Remark. he bound obtained for the Hill s equation equation 9 with = 2 by H. Hochstadt [] is better than our bound. he reason is that in the linear case we are able to use trigonometric formulas but if 2 then these formulas do not exist for the generalized trigonometric functions. Acknowledgements: he research was suorted by Hungarian National Foundation for Scientific Research OKA K 662. REFERENCES [] A. Anane J-L. Lions Simlicité et isolation de la remière valeur rore du -lalacien avec oids = Simlicity isolation of first eigenvalue of the -Lalacian with weight Comtes rendus de l Académie des sciences. Série Mathématique 35: [2] I. Bihari An asymtotic statement concerning the solutions of the differential equation x + atx = Studia Sci. Math. Hungar [3] G. Bognar Lower bound for the eigenvalues of quasilinear Hill s equation Proc. of the Conf. on Differential Difference Equations Alications Edited by R. P. Agarwal K. Pereira ISBN [4] L.W. Caserson Solvable Hill equation Phys. Rev. A [5] E. A. Coddington N. Levinson heory of Ordinary Differential Equations McGraw Hill New York 955. [6] O. Dosly P.Rehák Half-linear Differential Equations North-Holl Mathematics Studies vol. 22. Elsevier Amsterdam 25. [7] M. S. P. Eastham he Sectral heory of Periodic Differential Equations Scottish Academic Press Edinburgh London 973. [8] Á. Elbert A half-linear second order differential equation Coll. Math. Soc. János Bolyai 3. Qualitative theory of differential equations Szeged [9] G. W. Hill: On the art of the motion of the lunar erigee which is a function of the mean motions of the sun the moon Acta Math [] H. Hochstadt Asymtotic estimates for the Sturm-Liouville sectrum Commun. Pure Al. Math [] H. Hochstadt Estimates on the stability intervals for Hill s equation Proc. Amer. Math. Soc [2] V. Krylov A. Rebane A.G. Kalintsev H. Schwoerer U. P. Wild Secondharmonic generation of amlified femtosecond i: sahire laser ulses Otic Lett [3] W. Magnus S. Winkler Hill s Equation Dover Publ. Inc [4] M. Otani Existence nonexistence of nontrivial solutions of some nonlinear degenerate ellitic equations J. Funct. Anal [5] E. Sidick A. Knosen A. Dienes Ultrashort-ulse second harmonic generation I. ransform limited fundamental ulses J. Ot. Soc. Amer. B [6] H. Steudel C. Figueira de Morisson Faria M. G. A. Paris A. M. Kamchatnov O. Steuernagel Second harmonic generation: the solution for an amlitude-modulated initial ulse Otics Communications [7] K. akayama Note on solvable Hill equations Phys. Rev. A [8] S. M. Wu C. C. Shih Construction of solvable Hill equations Phys. Rev. A [9] M. Zhang he rotation number aroach to eigenvalues of the onedimensional -Lalacian with eriodic otentials J. London Math. Soc Issue Volume