Asymptotic distribution of eigenvalues of Hill's equation with ...
Asymptotic distribution of eigenvalues of Hill's equation with ... Asymptotic distribution of eigenvalues of Hill's equation with ...
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2 T. Nagasawa and H. OhruiWe improve this result as follows.Theorem 1.2Let Q be a real valued L 1 (R/πZ)-function andThen it holds that∣ √ λ 2n−1 − 2n∣ = o(n −1 ),∫ π0Q(x) dx =0.∣ √ λ 2n − 2n∣ = o(n −1 ) as n →∞.Indeed, we can show the following, from which Theorem 1.2 is easily derived.Theorem 1.3 Let λ be λ 2n−1 or λ 2n ,andput √ λ − 2n = d. Under the sameassumption as in Theorem 1.2, it holds that4nd + d 2 = o(1), (4nd + d 2 ) 2 −|¯Q 4n | 2 = O(n −1 ) as n →∞,where¯Q n = 1 π∫ π0Q(x)e −inx dx.Remark ∫ 1.1 The advantage <strong>of</strong> our theorem is appeared in the caseπQ(x) dx ≠0. Put0Since our <strong>equation</strong> is¯Q(= ¯Q 0 )= 1 π∫ π0Q(x) dx,y ′′ +(λ + ¯Q + ˜Q(x))y =0<strong>with</strong>˜Q(x) =Q(x) − ¯Q.∫ π0˜Q(x) dx =0,Theorem 1.1 implies that the eigenvalue near 2n behaves like√λ + ¯Q − 2n = O(n −1 ) as n →∞.Combining this <strong>with</strong>√λ + ¯Q = √ (λ 1+ ¯Q )2λ + O(λ−2 ) ,we get
ASYMPTOTIC DISTRIBUTION OF EIGENVALUES OF HILL’S EQUATION 3√ ¯Qλ − 2n = −2 √ λ + O(n−1 ).√Both terms in the right hand side are O(n −1 ), and therefore we know onlyλ =2n + O(n −1 ). By using Theorem 1.2, we can determine the next termto 2n as√ ¯Qλ =2n −4n + o(n−1 ).Remark 1.2 Under higher regularity <strong>of</strong> Q, the more precise asymptotics havebeen already known. For example, see [6, Theorems 2.12 and 2.13]. The readerscan refer [4, 5, 7] and references cited therein for asymptotic formulae <strong>of</strong><strong>eigenvalues</strong> for Hill’s <strong>equation</strong> <strong>with</strong> discontinuous coefficient.2. On √ λ 2n−1 , √ λ 2n ∼ 2nIn this section we comment on the fact(2.1)√λ2n−1 =2n + o(1),√λ2n =2n + o(1) as n →∞.This was shown in [6] under the assumptionQ ∈ L ∞ (R/πZ),∫ π0Q(x) dx =0.Borg [2, pp.88ff.] had investigated the problem under Q ∈ L 2 (R/πZ). Here weassert thatTheorem 2.1The asymptotics (2.1) holds underQ ∈ L 1 (R/πZ),∫ π0Q(x) dx =0.Since Hill’s <strong>equation</strong> has very long history, we find about 600 papers onMathSciNet, some <strong>of</strong> them are not easy to access now. The authors could notstudy all <strong>of</strong> them, and are not sure whether the above theorem has been alreadyknown or not. Even if it is not new, they would like to give the pro<strong>of</strong> here forthe sake <strong>of</strong> completeness. Since it is almost parallel <strong>with</strong> that in [6], we mentiononly its outline.Pro<strong>of</strong>.Let y 1 (·,λ)andy 2 (·,λ) be solutions <strong>of</strong> (1.1) satisfying(2.2)y 1 (0,λ)=1,y ′ 1(0,λ)=0,
4 T. Nagasawa and H. Ohrui(2.3)Puty 2 (0,λ)=0,y ′ 2(0,λ)=1.Δ(λ) =y 1 (π, λ)+y 2 ′ (π, λ),then λ is an eigenvalue if and only if Δ(λ) = 2 ([6, 3]). Putθ(λ) =−Δ(λ)+22(1 − cos π √ λ) .Since both numerator and denominator are analytic in λ, in order to show (2.1)we show∫(2.4)1 dlog θ(λ) dλ =02πi C ndλfor a closed curve C n ⊂ C containing{λ ∈ C ||λ| ≦ (2n +1) 2 ,λ∈ R}in its interior, provided n is sufficient large. It is easy to see1θ(λ) − 1=2(1 − cos π √ λ)∫ π0Y (x, λ)Q(x) dx,whereTakeY (x, λ) = sin √ λ(π − x){√ y 1 (x, λ)+ cos √ }λ(π − x) y 2 (x, λ).λC n ={λ ∈ C ∣ √ }λ ∈ C 1,n ∪ C 2,n ∪ C 3,n ,whereC 1,n ={ξ − i(2n +1)∈ C | 0 ≦ ξ ≦ 2n +1},C 2,n ={2n +1+iη ∈ C ||η| ≦ 2n +1},C 3,n ={ξ + i(2n +1)∈ C | 2n +1≧ ξ ≧ 0}.From now on λ is on C n .Thenwehave2e −π|I√ λ| ( 1 − cos π √ )λ =1+o(1) as n →∞.Therefore we know(2.5)providedθ(λ) − 1=o(1) as n →∞
(2.6)ASYMPTOTIC DISTRIBUTION OF EIGENVALUES OF HILL’S EQUATION 5∫ π0( )Y (x, λ)Q(x) dx = o e π|I√ λ|as n →∞holds. (2.5) implies (2.4) for large n. Thus what we should show is (2.6).By successive approximation, the functions y i are given byy i (x, λ) =∞∑j=0y i,j (x, λ) =−√ 1 ∫ xλWe can prove by inductiony i,j (x, λ), y 1,0 (x, λ) =cos √ λx, y 2,0 (x, λ) = sin √ λx√λ,0{sin √ }λ(x − x 1 ) Q(x 1 )y i,j−1 (x 1 ,λ) dx 1 (j ≧ 1).|y i,j (x, λ)| ≦ e|I√ λ|x(∫ x) jj! ∣ √ λ∣ i+j−1 |Q(x 1 )| dx 10for x ≧ 0. It follows from this thatfor every λ ∈ C n . Therefore∫ π0∞∑y i,j (x, λ) converges uniformly in x ∈ [0,π]j=0Y (x, λ)Q(x) dx =∞∑j=0∫ π0Y j (x, λ)Q(x) dxholds, whereY j (x, λ) = sin √ λ(π − x){√ y 1,j (x, λ)+ cos √ }λ(π − x) y 2,j (x, λ).λSince Y 0 is independent <strong>of</strong> x,∫ π0Y 0 (x, λ)Q(x) dx =0.From above estimates we get∫ πλ| ∣ Y j (x, λ)Q(x) dx∣ ≦ 2eπ|I√ ‖Q‖j+1L 10(j +1)! ∣ √ λ∣ j+1for j ≧ 1. Hereafter L p means L p (0,π). Consequently⎧ ⎛ ⎞⎫∫ π⎨∣ Y (x, λ)Q(x) dx∣ ≦ λ|2eπ|I√0⎩ exp ⎝ ‖Q‖ L1∣ √ ⎠ − 1 − ‖Q‖ ⎬L 1λ∣∣ √ λ∣⎭ ,and (2.6) follows.□
6 T. Nagasawa and H. Ohrui3. Pro<strong>of</strong> <strong>of</strong> Theorem 1.3In this section λ is λ 2n−1 or λ 2n , and put√λ − 2n = d.The functions y i satisfyy i ′′ +(2n) 2 y i = −(4nd + d 2 + Q(x))y i .Taking their initial conditions into account, we havesin 2nx(3.1)y 1 =cos2nx + Ky 1 , y 2 = + Ky 2 ,2nwhere K is a bounded operator from L ∞ (0,π) into itself defined by(Ky)(x) =− 1 ∫ x{sin 2n(x − x 1 )}(4nd + d 2 + Q(x 1 ))y(x 1 ) dx 1 .2n 0Since d = o(1) as n →∞,itiseasytosee‖K‖ L∞ →L ∞ ≦ 1 {π|4nd + d 2 }| + ‖Q‖ L 1 = o(1).2nFor n so large that ‖K‖ L∞ →L ∞‖y 1 ‖ L ∞ ≦The derivative <strong>of</strong> y 2 isy ′ 2 =cos2nx −< 1, we have11 −‖K‖ L∞ →L ∞ , ‖y 2 ‖ L ∞ ≦∫ x0Therefore the condition Δ(λ) =2isequivalentto∫ π012n(1 −‖K‖ L∞ →L ∞).{cos 2n(x − x 1 )} (4nd + d 2 + Q(x 1 ))y 2 (x 1 ) dx 1 .U n (x)(4nd + d 2 + Q(x)) dx =0,wheresin 2nxU n (x) =2ny 1(x) − (cos 2nx) y 2 (x).Putting (3.1) into this twice, we havesin 2nxU n (x) =2n(Ky 1)(x) − (cos 2nx)(Ky 2 )(x)= − 1 ∫ x{sin 24n 2 2n(x − x 1 ) } (4nd + d 2 + Q(x 1 )) dx 1+0sin 2nx2n(K2 y 1 )(x) − (cos 2nx)(K 2 y 2 )(x).
ASYMPTOTIC DISTRIBUTION OF EIGENVALUES OF HILL’S EQUATION 7Therefore it holds that0=∫ π0= − 14n 2 ∫ π+∫ π0U n (x)(4nd + d 2 + Q(x)) dx0It is easy to see∫ x0{ sin 2nx2nUsing the condition{sin 2 2n(x − x 1 ) } (4nd + d 2 + Q(x))(4nd + d 2 + Q(x 1 )) dx 1 dx}(K2 y 1 )(x) − (cos 2nx)(K 2 y 2 )(x) (4nd + d 2 + Q(x)) dx.∫ π ∫ x0∫ π0∫ π ∫ x0sin 2 2n(x − x 1 ) dx 1 dx = π24 .Q(x) dx =0,wehave{sin 2 2n(x − x 1 ) } (Q(x 1 )+Q(x)) dx 1 dx =0,00∫ π ∫ xHence d satisfies∣ (4nd + d 2 ) 2 −|¯Q 4n | 2∣ ∣=16n 2∣ π 20∫ π00{sin 2 2n(x − x 1 ) } Q(x 1 )Q(x) dx 1 dx = − π24 | ¯Q 4n | 2 .{ }sin 2nx2n(K2 y 1 )(x) − (cos 2nx)(K 2 y 2 )(x) (4nd + d 2 + Q(x)) dx∣(≦ 16n2‖y1 ‖ L ∞π 2 ‖K‖2 L ∞ →L ∞ 2n≦ 16(π|4nd + d 2 | + ‖Q‖ L 1nπ 2 1 −‖K‖ L ∞ →L ∞≦ C ( )π|4nd + d 2 | + ‖Q‖ L 1 {(4nd + d 2 ) 2 +1 } .n(3.2)Since we have already known d = o(1), it holds) 3(π|4nd+ ‖y 2 ‖ L ∞)+ d 2 )| + ‖Q‖ L 1π|4nd + d 2 | + ‖Q‖ L 1 = o(n).Combining this and ¯Q 4n = o(1) (by Riemann-Lebesgue’s Lemma) <strong>with</strong> (3.2), weobtain the assertion <strong>of</strong> Theorem 1.3□
8 T. Nagasawa and H. OhruiReferences[ 1 ] Borg, G., Über die Stabilität gewisser Klassen von linearen Differentialgleichungen, Ark.Mat. Astr. Fys. 31A (1) (1944), 1–3.[ 2 ] Borg, G., Eine Umkehrung der Strum-Liouvilleschen Eigenwertaufgabe. Bestimmungder Differentialgleichung durch die Eigenwerte, Acta Math. 78 (1946), 1–96.[ 3 ] Coddington, E. A. & N. Levinson, “Theory <strong>of</strong> Ordinary Differential Equations”,McGraw-Hill, New York, 1955.[ 4 ] Karaca, I. Y., On Hill’s <strong>equation</strong> <strong>with</strong> a discontinuous coefficient, Int.J.Math.Math.Sci. 25 (2003), 1599–1614.[ 5 ] Karaca, I. Y., <strong>Asymptotic</strong> formulas for <strong>eigenvalues</strong> <strong>of</strong> a Hill’s <strong>equation</strong> <strong>with</strong> piecewiseconstant coefficient, Math.Proc.R.Ir.Acad.109 (1) (2009), 19–34.[ 6 ] Mugnus, W. & S. Winkler, “Hill’s Equation”, Interscience Tracts Pure Appl. Math. 20,Interscience Publ., New York · London · Sydney, 1966.[ 7 ] Yaslan, I. & G. S. Guseinov, On Hill’s <strong>equation</strong> <strong>with</strong> piecewise constant coefficient, TurkishJ. Math. 21 (4) (1997), 461–474.Takeyuki NagasawaHirotaka OhruiDepartment <strong>of</strong> MathematicsGraduate School <strong>of</strong> Science and EngineeringSaitama University255 Shimo-OkuboSakura-ku, Saitama 338-8570, Japane-mail: tnagasaw@rimath.saitama-u.ac.jps08mm101@mail.saitama-u.ac.jp