2. 162 Chapter 5. Perturbationsdistribution in section 5.5, showing that they are exponentially close to the band.5.2 Spectral bandsWe have seen in section 4.3 that the periodic Sturm-Liouville and Dirac operatorson the half-line have purely absolutely continuous spectrum in the set S of stabilityintervals of the corresponding periodic diﬀerential equation system. In the presentsection it is shown that this property is stable when the coeﬃcients of the operatorare perturbed by the addition of non-periodic terms which satisfy a mild decaycondition at inﬁnity, stipulating essentially that their local average tends to zeroand their oscillations can be controlled. For a 2 × 2 matrix S, we denote by |S| the matrix operator norm, |Sv| |S| = sup . v∈C2 {0} |v|Then for two matrices S1 and S2 , we have |S1 S2 | ≤ |S1 | |S2 |. Moreover, conver-gence of a matrix sequence in the norm sense implies convergence for each entryseparately.Theorem 5.2.1. Let B and W be 2 × 2 matrix-valued functions on [0, ∞) satisfyingthe general hypotheses of section 1.5, and assume that B = B1 + B2 , where B1and W are a-periodic and B2 has the properties ∞ |B2 (t) − B2 (t − a)| dt < ∞, (5.2.1) a x+a lim |B2 | = 0. (5.2.2) x→∞ xLet [λ , λ ] ⊂ S, where S is the stability set of the periodic equation u = J(B1 + λW ) u. (5.2.3)Then there is a constant C > 0 such that |u(x, λ)| < C for all λ ∈ [λ , λ ] and allsolutions u(·, λ) of u = J(B1 + B2 + λW ) u (5.2.4)such that |u(0, λ)| = 1.Proof. Let Φ and Ψ be the canonical fundamental matrices of the periodic equation(5.2.3) and of the perturbed periodic equation (5.2.4), respectively. For j ∈ N, letΨj be the solution of (5.2.4) with initial value Ψj (a(j − 1)) = I; we then setMj := Ψj (aj). Then Φj (x) := Φ(x − a(j − 1)) will serve an analogous purpose forthe unperturbed equation (5.2.3), with Φj (aj) = M , the monodromy matrix of(5.2.3), for all j.
3. 5.2. Spectral bands 163 Rewriting (5.2.4) in the form u = J(B1 + λW ) u + JB2 u,we ﬁnd by the variation of constants formula (1.2.11) that x Ψj (x) = Φj (x) + Φj (x) Φ−1 J B2 Ψj j (x ≥ a(j − 1)). (5.2.5) a(j−1)Denoting in the following by (const.) a uniform constant for all λ ∈ [λ , λ ] —although not always the same constant — we ﬁnd from (5.2.5) that x |Ψj (x)| ≤ |Φj (x)| 1 + |Φ−1 J| |B2 | |Ψj | , j a(j−1)and hence by Gronwall’s lemma the estimate x |Ψj (x)| ≤ (const.) exp (const.) |B2 | , (5.2.6) a(j−1)for x ∈ [a(j − 1), aj]. Using this in combination with (5.2.5) again, we obtain forsuch x, x |Ψj (x) − Φj (x)| ≤ |Φj (x) Φ−1 J| |B2 | |Ψj | j a(j−1) x t ≤ (const.) |B2 (t)| exp (const.) |B2 | dt a(j−1) a(j−1) aj aj ≤ (const.) |B2 | exp (const.) |B2 | a(j−1) a(j−1) →0 (j → ∞) (5.2.7)because of (5.2.2). In particular, taking x = aj, we ﬁnd that |Mj − M | = |Ψj (aj) − Φj (aj)| → 0 (j → ∞) (5.2.8)uniformly for λ ∈ [λ , λ ]. Setting Dj = Tr Mj , we conclude that lim |Dj −D| = 0, j→∞where D is the discriminant of (5.2.3). Since [λ , λ ] ⊂ S, this implies that, for asuﬃciently large J ∈ N and some δ > 0, |Dj (λ)| ≤ 2 − δfor all j > J and λ ∈ [λ , λ ]. The matrix Mj has determinant 1 and hence can be analysed as in our studyof the monodromy matrix in section 1.4. For j > J, we are in Case 3; so Mj has
4. 164 Chapter 5. Perturbationscomplex conjugate eigenvalues μj , μj with |μj | = 1 and corresponding eigenvaluesgiven in terms of the eigenvalues and the entries of Mj by a formula analogous to(4.5.2). In view of the convergence of Mj to the monodromy matrix M in (5.2.8),these eigenvectors converge to those of M , uniformly in [λ , λ ], as j → ∞. Let Ej be the matrix of eigenvectors (4.5.2) for Mj , j > J. Then Ej converges −1to the matrix E of eigenvectors of M , and so Ej converges to E −1 and hence isbounded, uniformly in λ ∈ [λ , λ ]. From μj 0 −1 Mj = E j Ej 0 μjwe obtain Ψ(na) = Mn Mn−1 · · · MJ+1 Ψ(Ja) μn 0 −1 μn−1 0 −1 = En En En−1 En−1 · · · 0 μn 0 μn−1 μJ+1 0 −1 · · · EJ+1 EJ+1 Ψ(Ja) 0 μJ+1 μj 0and, since is unitary, it follows that 0 μj −1 −1 −1 −1 |Ψ(na)| ≤ |En | |En En−1 | |En−1 En−2 | · · · |EJ+2 EJ+1 | |EJ+1 | |Ψ(Ja)|. (5.2.9) −1In order to estimate |Ej Ej−1 |, we observe that Ψj−1 (· − a) is a fundamentalmatrix of u (x) = J(B1 (x) + B2 (x − a) + λW (x)) u(x),and so by the variation of constants formula (1.2.11) Ψj (x) = Ψj−1 (x − a) x + Ψj−1 (x − a) Ψj−1 (t − a)−1 J (B2 (t) − B2 (t − a)) Ψj (t) dt. a(j−1)Using (5.2.6) and the convergence of Ψj−1 to Φj−1 , we can therefore estimate |Mj − Mj−1 | = |Ψj (aj) − Ψj−1 (a(j − 1))| aj ≤ (const.) |B2 (t) − B2 (t − a)| dt a(j−1)and, in view of (4.5.2), also aj |Ej − Ej−1 | ≤ (const.) |B2 (t) − B2 (t − a)| dt. a(j−1)
5. 5.2. Spectral bands 165Hence, observing that −1 −1 −1 |Ej Ej | = |I − Ej (Ej − Ej−1 )| ≤ 1 + |Ej | |Ej − Ej−1 |,we can follow up on (5.2.9), n −1 −1 |Ψ(na)| ≤ |En | |EJ+1 | |Ψ(Ja)| |Ej Ej−1 | j=J+2 aj ≤ (const.) 1 + (const.) |B2 (t) − B2 (t − a)| dt j=J+2n a(j−1) ⎛ ⎞ n aj ≤ (const.) exp ⎝(const.) |B2 (t) − B2 (t − a)| dt⎠ j=J+2 a(j−1) ∞ ≤ (const.) exp (const.) |B2 (t) − B2 (t − a)| dt < ∞. a(j+1)In conjunction with the uniform boundedness of Ψn , this shows that Ψ, and henceany solution u(·, λ) of (5.2.4) with |u(0, λ)| = 1, is bounded uniformly with respectto λ ∈ [λ , λ ]. We remark that the condition (5.2.2) can, in a sense, already be inferred from(5.2.1). Indeed, if we consider the shifted functions B2,n (x) := B2 (x + na) (x ∈ [0, a]; n ∈ N),then we ﬁnd that, for n, m ∈ N with m < n, a a n−1 |B2,n − B2,m | = (B2 (x + (j + 1)a)) − B2 (x + ja)) dx 0 0 j=m n−1 a ≤ |B2 (x + (j + 1)a) − B2 (x + ja)| dx j=m 0 na = |B2 (x + a) − B2 (x)| dx → 0 (m, n → ∞) maby (5.2.1), which shows that B2,n converges to a limit B2,∞ in L1 ([0, a]). We ˜ ˜extend B2,∞ to an a-periodic function on [0, ∞). Then B = B1 + B2 , where ˜ ˜B1 := B1 + B2,∞ and B2 := B2 − B2,∞ satisfy both (5.2.1) and (5.2.2). Note,however, that now S in Theorem 5.2.1 will be the stability set for the periodicequation ˜ u = J(B1 + λW ) u.Theorems 5.2.1 and 4.9.1 give the following statement which shows that the ab-solutely continuous spectral bands of the periodic Dirac operator (see Theorem4.5.4) are preserved under perturbations satisfying a mild decay condition.
6. 166 Chapter 5. PerturbationsCorollary 5.2.2. Let p1 , p2 and q be locally integrable, a-periodic real-valued func-tions on [0, ∞). Moreover, let p1 , p2 and q be locally integrable real-valued functions ˜ ˜ ˜satisfying ∞ x+a |˜1 (t) − p1 (t − a)| dt < ∞, p ˜ lim |˜1 | = 0 p a x→∞ x(and similarly for p2 , q ). Then, for any α ∈ [0, π) the one-dimensional Dirac ˜ ˜operator d Hα = −iσ2 + (p1 + p1 )σ3 + (p2 + p2 )σ1 + (q + q ) ˜ ˜ ˜ dxwith boundary condition (4.3.1) has purely absolutely continuous spectrum in thestability set S of the periodic Dirac equation (1.5.4).Proof. Let [λ , λ ] ⊂ S. Then by Theorem 5.2.1 there exists a constant C suchthat for all λ ∈ [λ , λ ], all solutions of the perturbed periodic equation −iσ2 u + ((p1 + p1 )σ3 + (p2 + p2 )σ1 + q + q ) u = λ u ˜ ˜ ˜with |u(0)| = 1 are bounded: |u(x)| < C (x ≥ 0). By the same reasoning as at theend of the proof of Lemma 4.5.2, this also implies that |u(x)| > 1/C (x ≥ 0). Inparticular, x k(x) ≤ |u|2 ≤ k(x) (x ≥ 0) C4 0with k(x) := C 2 x. Theorem 4.9.1 now shows that [λ , λ ] is an interval of purelyabsolutely continuous spectrum of Hα . For the perturbed Hill equation, the required lower bound on the growth ofthe square-integral of solutions y is slightly more diﬃcult to obtain. Nevertheless,we have the following analogue of Corollary 5.2.2.Corollary 5.2.3. Let p > 0, w > 0 and q be locally integrable, a-periodic real-valuedfunctions on [0, ∞). Moreover, let p and q be locally integrable real-valued functions ˜ ˜such that p + p > 0, ˜ ∞ x+a |˜(t) − q (t − a)| dt < ∞, q ˜ lim |˜| = 0 q a x→∞ xand ∞ x+a 1 p ˜ p ˜ p˜ (t) − (t − a) dt < ∞, lim = 0. a p(t) p + p ˜ p+p˜ x→∞ x p(p + p) ˜Then, for any α ∈ [0, π) the one-dimensional Sturm-Liouville operator 1 d d Hα = − ((p + p) ) + (q + q ) ˜ ˜ w dx dx
7. 5.2. Spectral bands 167with boundary condition y(0) cos α − (py )(0) sin α = 0has purely absolutely continuous spectrum in the stability set S of the periodicSturm-Liouville equation (1.5.2).Proof. Let [λ , λ ] ⊂ S. Then, as in the proof of Corollary 5.2.2, we can useTheorem 5.2.1 to ﬁnd a constant C > 0 such that, for all λ ∈ [λ , λ ], all solutionsof the perturbed periodic Sturm-Liouville system 1 0 p+p˜ u (x, λ) = u(x, λ) (5.2.10) q + q − λw ˜ 0with |u(0, λ)| = 1 satisfy 1/C ≤ |u(x, λ)| ≤ C for all x ≥ 0. Let y be a real-valued solution of the perturbed Sturm-Liouville equation −((p + p) y ) + (q + q ) y = λ w y ˜ ˜ ysuch that |y(0)|2 + |(py )(0)|2 = 1; then u = will be a solution of (5.2.10) pywith the required property. Now for n ∈ N, let Ψn and Φn be deﬁned as in theproof of Theorem 5.2.1. Then, for x ∈ [(n − 1)a, na], u(x) = Ψn (x) u((n − 1)a) = Φn (x) u((n − 1)a) + (Ψn (x) − Φn (x)) u((n − 1)a)and therefore by Minkowski’s inequality na na 2 |y|2 w ≥ |[Φn u((n − 1)a)]1 | (n−1)a (n−1)a na 2 − |[(Ψn − Φn ) u((n − 1)a)]1 | . (n−1)aAs |u((n − 1)a)| < C for all n and Ψn − Φn → 0 uniformly on the interval ofintegration as n → ∞ by (5.2.7), the last term tends to 0 in this limit. For theﬁrst term on the right-hand side, we observe that [Φn u((n − 1)a)]1 is a real-valuedsolution of the periodic equation with |Φn ((n − 1)a) u((n − 1)a)| = |u((n − 1)a)| ≥ 1/C,and so by Lemma 4.5.3 we have na 2 C |[Φn u((n − 1)a)]1 | w ≥ (n−1)a C2with a constant C which only depends on [λ , λ ]. Hence we see that, for suﬃciently large x > 0, (4.9.2) will be satisﬁed withk(x) = C 2 x and c = C /2C 4 . The assertion now follows by Theorem 4.9.1.
8. 168 Chapter 5. Perturbations5.3 Gap eigenvaluesWe now turn to the instability intervals of the periodic system. As we have seenin section 4.5, the essential spectrum of the unperturbed periodic operator on thehalf-line has gaps coinciding with the instability set I, with each gap containingeither no spectrum at all or only a single eigenvalue. We shall now show that, whena perturbation is added which tends to 0 at ∞, the qualitative picture remainsunchanged; indeed, each instability interval contains only discrete eigenvalues andthus is still a gap in the essential spectrum. Every compact subinterval of an in-stability interval contains at most a ﬁnite number of eigenvalues. However, thequestion whether the whole instability interval contains only ﬁnitely many eigen-values, or eigenvalues which accumulate at one or both of its end-points, is moresubtle and will be considered in section 5.4. Regarding the number of eigenvalues in a given subinterval of an instabilityinterval, we then observe that, in the adiabatic or homogenisation limit where theperturbation (which is assumed to be continuous) varies on a very long scale com-pared to the period, the number of eigenvalues generally increases asymptoticallylinearly in the scaling parameter and has a limit density which can be convenientlyexpressed in terms of the rotation number of the periodic equation.Theorem 5.3.1. Let [λ , λ ] ⊂ I, where I is the instability set of Hill’s equation(1.5.3). Moreover, let α ∈ [0, π) and let q be a locally integrable, real-valued function ˜on [0, ∞) such that q (x) ˜ lim = 0. x→∞ w(x)Then the perturbed periodic Sturm-Liouville operator 1 d d Hα = − (p ) + q + q ˜ w dx dxhas at most ﬁnitely many eigenvalues and no other spectrum in [λ , λ ].Proof. As I is open, there exists δ > 0 such that [λ − δ, λ + δ] ⊂ I. Let x0 bean integer multiple of a such that |˜(x)| q ≤δ (x ≥ x0 ), w(x)and set q(x) + q˜ if x ∈ [0, x0 ), q± (x) := q(x) ∓ δ w(x) if x ∈ [x0 , ∞).Now for λ ∈ {λ , λ }, let θ(x, λ) (x ≥ 0) be the solution of the initial-value problemfor the Pr¨fer equation u 1 θ (x, λ) = cos2 θ(x, λ) + (λ w(x) − q(x) − q (x)) sin2 θ(x, λ), ˜ θ(0, λ) = α, p(x) (5.3.1)
9. 5.3. Gap eigenvalues 169and similarly θ± (x, λ) (x ≥ 0) the solutions of 1 θ± (x, λ) = cos2 θ± (x, λ) + (λ w(x) − q± (x)) sin2 θ± (x, λ), θ± (0, λ) = α. p(x) (5.3.2)Then, since q+ ≤ q + q ≤ q+ throughout, comparison of (5.3.1) with (5.3.2) and ˜Theorem 2.3.1 (a) show that θ− (x, λ) ≤ θ(x, λ) ≤ θ+ (x, λ) (5.3.3)for all x ≥ 0; the three functions are identical on [0, x0 ]. Let n ∈ Z be the index, according to the enumeration of Theorem 2.4.1, ofthe instability interval in which both λ − δ and λ + δ lie. Then, noting that forx ≥ x0 the coeﬃcient of the last term in (5.3.2) is λ w(x) − q± (x) = (λ ± δ) w(x) − q(x)and thus (5.3.2) is the Pr¨fer equation for the periodic equation (1.5.3) with uspectral parameter λ ± δ, we can apply (2.4.1) to ﬁnd that nπ θ+ (x, λ ) = θ(x0 , λ ) + (x − x0 ) + O(1), a nπ θ− (x, λ ) = θ(x0 , λ ) + (x − x0 ) + O(1) aasymptotically for x → ∞. Hence, using (5.3.3), we conclude that θ(x, λ ) − θ(x, λ ) ≤ θ+ (x, λ ) − θ− (x, λ ) = θ(x0 , λ ) − θ(x0 , λ ) + O(1);in particular, the diﬀerence remains bounded as x → ∞. The ﬁniteness of the totalspectral multiplicity of Hα in [λ , λ ] now follows by Theorem 4.8.3. In the case of the Dirac operator, the following analogue holds for generalmatrix-valued perturbations.Theorem 5.3.2. Let [λ , λ ] ⊂ I, where I is the instability set of the periodic Diracequation (1.5.4). Moreover, let α ∈ [0, π), and let p1 , p2 and q be locally integrable, ˜ ˜ ˜real-valued functions on [0, ∞) such that lim p1 (x) = lim p2 (x) = lim q (x) = 0. ˜ ˜ ˜ (5.3.4) x→∞ x→∞ x→∞Then the perturbed periodic Dirac operator d Hα = −iσ2 + (p1 + p1 ) σ3 + (p2 + p2 ) σ1 + (q + q ) ˜ ˜ ˜ dxhas at most ﬁnitely many eigenvalues and no other spectrum in [λ , λ ].
10. 170 Chapter 5. PerturbationsProof. We proceed in analogy to the proof of Theorem 5.3.1. Again, there is δ > 0such that [λ − δ, λ + δ] ⊂ I. The Pr¨fer equation (2.2.2) for the perturbed Dirac usystem takes the form T sin θ ˜ sin θ θ = B(x) + B(x) + λ I , cos θ cos θwhere −p1 − q −p2 ˜ −˜1 − q −˜2 p ˜ p B= , B= . −p2 p1 − q −˜2 p p1 − q ˜ ˜Hypothesis (5.3.4) ensures that there is x0 > 0 such that the pointwise operator ˜ ˜norm of the perturbation matrix B satisﬁes |B(x)| ≤ δ for all x ≥ x0 . Conse-quently, for such x the matrices ˜ δ I ± B(x)are positive semideﬁnite. Hence T T sin θ sin θ sin θ ˜ sin θ (B(x) + (λ − δ) I) ≤ B(x) + B(x) + λ I cos θ cos θ cos θ cos θ T sin θ sin θ ≤ (B(x) + (λ + δ) I) cos θ cos θfor any θ and x ≥ x0 . Theorem 2.3.1 (a) then implies that the Pr¨fer angle θ of uthe perturbed equation with initial value θ(0) = α can be estimated above andbelow by the Pr¨fer angles θ± of the equation where the perturbation matrix B u ˜is replaced with the constant matrix ±δ I on [x0 , ∞), in analogy to (5.3.3). Theremainder of the proof is exactly as for Theorem 5.3.1. The key idea of Theorems 5.3.1 and 5.3.2 is to use the monotonicity of Pr¨fer uangles under perturbations to eventually replace the perturbation with a constantand then apply the growth asymptotic of the Pr¨fer angle for the periodic equation, uas obtained in section 2.4. The same idea can be adapted to estimating how manyeigenvalues appear in any subinterval of an instability interval under the inﬂuenceof a continuous perturbation in the limit of slow variation. More precisely, givena continuous function q which serves as a template, we consider perturbations ˜of the form q (x/c), where c is a dilation parameter which tends to inﬁnity in ˜the limit. Clearly, the local modulus of continuity of the perturbation decreasestowards zero as c increases, which means that the perturbation changes ever moreslowly on the length scale deﬁned by the period a. This limit is related to theadiabatic limit in quantum mechanics, which refers to perturbations changingslowly in time compared to the dynamic time scale of the unperturbed system,and to the homogenisation limit, in which microscopic material properties, hererepresented by the periodic background, are treated by averaging in contrast tothe macroscopic structures.
11. 5.3. Gap eigenvalues 171 Speciﬁcally for the perturbed periodic Sturm-Liouville equation we have thefollowing result.Theorem 5.3.3. Let [λ , λ ] ⊂ I, where I is the instability set of Hill’s equation(1.5.2) with w = 1. Moreover, let α ∈ [0, π) and let q be a continuous real-valued ˜function on [0, ∞) with lim q (r) = 0. ˜ r→∞ Then the number of eigenvalues in [λ , λ ] of the perturbed periodic Sturm-Liouville operator d d Hα = − (p ) + q(x) + q (x/c) ˜ dx dxhas asymptotic ∞ c N[λ ,λ ] ∼ k(λ − q (r)) − k(λ − q (r)) dr ˜ ˜ (c → ∞), (5.3.5) πa 0where k is the rotation number of the unperturbed equation (1.5.2).Proof. Let δ > 0 be so small that [λ − δ, λ + δ] ⊂ I. Then there is r0 > 0 suchthat |˜(r)| ≤ δ for all r ≥ r0 . Let θ(·, λ) be the solution of the initial-value problem q(5.3.1) for the perturbed Pr¨fer equation, with spectral parameter λ ∈ {λ , λ }. u Now let m ∈ N and consider a dissection of the interval [0, r0 ] into m parts,i.e. division points 0 = s0 < s1 < · · · < sm = r0 . For j ∈ {1, . . . , m}, let ˜− qj = sup q (s), ˜ ˜+ qj = inf q (s), ˜ s∈[sj−1 ,sj ] s∈[sj−1 ,sj ] ±and let θj (·, λ) be the solutions of the Pr¨fer equations u ± 1 ± (θj ) (x, λ) = ˜± ± cos2 θj (x, λ) + (λ − qj (x) − q(x)) sin2 θj (x, λ), (5.3.6) p(x) ±with initial condition θj (csj−1 , λ) = θ(csj−1 , λ). Then by Sturm comparison(Corollary 2.3.2), we ﬁnd that − − + + θj (csj , λ) − θj (csj−1 , λ) ≤ θ(csj , λ) − θ(csj−1 , λ) ≤ θj (csj , λ) − θj (csj−1 , λ). (5.3.7)On the other hand, the equations (5.3.6) have a-periodic coeﬃcients and are infact the Pr¨fer equations for the unperturbed periodic Sturm-Liouville equation u q±with spectral parameter shifted by −˜j . Therefore the asymptotics (2.4.9) (withk = nπ in the instability interval In ) apply, giving ± ± csj − csj−1 ˜± θj (csj , λ) − θj (csj−1 ) = k(λ − qj ) + O(1) (5.3.8) ain the limit c → ∞. Also, by the same reasoning as in the proof of Theorem 5.3.1, we ﬁnd thatfor λ ∈ {λ , λ } and x > cr0 , nπ θ(x, λ) − θ(cr0 , λ) = (x − cr0 ) + O(1), a
12. 172 Chapter 5. Perturbationsthe remainder staying bounded as x → ∞, where n is the number of the instabilityinterval such that [λ , λ ] ⊂ In ; we here use the fact that λ − δ, λ + δ ∈ In . Now appealing to the Relative Oscillation Theorem 4.8.3, we ﬁnd that thenumber of eigenvalues of Hα in (λ , λ ] has the asymptotic N(λ ,λ ] 1 lim = lim (θ(cr0 , λ ) − θ(cr0 , λ )) c→∞ c c→∞ πc m 1 = lim (θ(csj , λ ) − θ(csj−1 , λ )) − (θ(csj , λ ) − θ(csj−1 , λ )) . c→∞ πc j=1Hence, using the estimates (5.3.7) and the asymptotics (5.3.8), we conclude that m 1 ˜− ˜+ k(λ − qj ) − k(λ − qj ) (sj − sj−1 ) πa j=1 m N(λ ,λ ] 1 ≤ lim ≤ ˜+ ˜− k(λ − qj ) − k(λ − qj ) (sj − sj−1 ). c→∞ c πa j=1The statement of Theorem 5.3.3 now follows by observing that the sums on eitherside are lower and upper Riemann sums corresponding to the given dissection forthe integral r0 k(λ − q (r)) − k(λ − q (r)) dr ˜ ˜ 0and that k(λ − q (r)) = k(λ − q (r)) = nπ if r ≥ r0 . ˜ ˜ For the perturbed periodic Dirac system, the situation is a bit more compli-cated, mostly due to the fact that perturbations often apply to the matrix coeﬃ-cients p1 and p2 in practice, so the simple estimate using upper and lower Riemannsums, as used in the proof of Theorem 5.3.3 above, needs to be replaced with lesstight matrix operator norm estimates. For example, the angular momentum termarising from the separation in spherical polar coordinates of a three-dimensionalradially periodic Dirac operator has the form σ1 k/r, with r the radial variable,and thus can be considered a perturbation of p2 . The angular momentum term isalso singular at 0. In the the following we shall focus on problems with one regularend-point; see the notes for the doubly singular case. Moreover, the integral for the asymptotic density of eigenvalues will alsoinvolve values of the rotation number of the periodic equation where not only thespectral parameter, but also the coeﬃcients p1 and p2 are shifted by a constant.Speciﬁcally, we shall denote by k(λ, c1 , c2 ) the rotation number of the periodicDirac equation −iσ2 u + p1 σ3 u + p2 σ1 u + q u = (λ − c1 σ3 − c2 σ1 ) u, (5.3.9)where p1 , p2 , q are real-valued, locally integrable and a-periodic, and λ, c1 , c2 ∈ R.As c1 σ3 + c2 σ1 + (|c1 | + |c2 |) I ≥ 0, (|c1 | + |c2 |) I − c1 σ3 − c2 σ1 ≥ 0
13. 5.3. Gap eigenvalues 173in the sense of positive semideﬁnite matrices, Sturm comparison (Corollary 2.3.2)and the continuity of the rotation number as a function of the spectral parametershow that k is jointly continuous in all three variables.Theorem 5.3.4. Let [λ , λ ] ⊂ I, where I is the instability set of the periodic Diracequation (1.5.4). Let α ∈ [0, π) and let q , p1 , p2 be continuous real-valued functions ˜ ˜ ˜on [0, ∞) with lim q (r) = lim p1 (r) = lim p2 (r) = 0. ˜ ˜ ˜ r→∞ r→∞ r→∞ Then the number of eigenvalues in [λ , λ ] of the perturbed periodic Diracoperator d Hα = −iσ2 + (p1 (x) + p1 (x/c)) σ3 + (p2 (x) + p2 (x/c)) σ1 + q(x) + q (x/c) ˜ ˜ ˜ dxhas asymptotic ∞ c N[λ ,λ ] ∼ k(λ − q (r), p1 (r), p2 (r)) − k(λ − q (r), p1 (r), p2 (r)) dr ˜ ˜ ˜ ˜ ˜ ˜ πa 0 (5.3.10)as c → ∞.Proof. Let δ > 0 be so small that λ > λ + 2δ and [λ − δ, λ + δ] ∈ I. Then thereis r0 > 0 such that |˜(r)| + |˜1 (r)| + |˜2 (r)| < δ for r ≥ r0 . Moreover, there is a q p pbound M > 0 such that |˜(r)|, |˜1 (r)|, |˜2 (r)| ≤ M for all r ≥ 0. q p p Let > 0. As k is uniformly continuous on K = [λ − δ − M, λ + δ + M ] × [−M, M ]2 , ˜ ˜ ˜ ˜there is δ ∈ (0, δ] such that, for (λ, c1 , c2 ), (λ, c1 , c2 ) ∈ K, ˜ ˜ ˜ ˜ ˜ |λ − λ|, |c1 − c1 |, |c2 − c2 | < δ ⇒ |k(λ, c1 , c2 ) − k(λ, c1 , c2 )| < . ˜ ˜Since p1 , p2 and q are uniformly continuous on [0, r0 ], there exists γ > 0 such that, ˜ ˜ ˜for x, y ∈ [0, r0 ], q ˜ p ˜ p ˜ ˜ |x − y| < γ ⇒ |˜(x) − q (y)|, |˜1 (x) − p1 (y)|, |˜2 (x) − p2 (y)| < δ/3. Now consider a dissection of [0, r0 ] into m subintervals with dissection points0 = s0 < s1 < · · · < sm = r0 such that |sl − sj−1 | < γ. Choose sj ∈ [sj−1 , sj ] and ˆset c1,j = p1 (ˆj ), ˜ s c2,j = p2 (ˆj ), ˜ s c3,j = q (ˆj ), ˜sfor each j ∈ {1, . . . m}. Then on [sj−1 , sj ], p p ˜ ˜ (˜1 − c1,j )σ3 + (˜2 − c2,j )σ1 + q − c3,j + δ I ≥ 0, ˜ δ I − (˜1 − c1,j )σ3 − (˜2 − c2,j )σ1 − (˜ − c3,j ) ≥ 0 p p q (5.3.11)
14. 174 Chapter 5. Perturbationsin the sense of positive semideﬁnite matrices. Let θ(·, λ) be the solution of thePr¨fer equation for the perturbed periodic Dirac equation (cf. (2.2.7)), u θ (x, λ) = λ − q(x) − q (x/c) + (p1 (x) + p1 (x/c)) cos 2θ(x, λ) ˜ ˜ − (p2 (x) + p2 (x/c)) sin 2θ(x, λ) ˜ ˜with initial value θ(0, λ) = α. For j ∈ {1, . . . , m}, let θj,k (·, λ) with k ∈ {1, 2}be the solutions on [csj−1 , csj ] of the periodic Pr¨fer equation corresponding to u(5.3.9), θj,k (x, λ) = λ − q(x) − c3,j + (p1 (x) + c1,j ) cos 2θj,k (x, λ) − (p2 (x) + c2,j ) sin 2θj,k (x, λ)with initial values θj,1 (csj−1 , λ) = θ(csj−1 , λ ), θj,2 (csj−1 , λ) = θ(csj−1 , λ ).Then, using the estimates (5.3.11) in Sturm comparison (Corollary 2.3.2), we ﬁndthat ˜ ˜ ˜ ˜ θj,1 (csj , λ − δ) − θj,1 (csj−1 , λ − δ) ≤ θ(csj , λ ) − θ(csj−1 , λ ) ˜ ˜ ˜ ˜ ≤ θj,1 (csj , λ + δ) − θj,1 (csj−1 , λ + δ), ˜ ˜ ˜ ˜ θj,2 (csj , λ − δ) − θj,2 (csj−1 , λ − δ) ≤ θ(csj , λ ) − θ(csj−1 , λ ) ˜ ˜ ˜ ˜ ≤ θj,2 (csj , λ + δ) − θj,2 (csj−1 , λ + δ).From (2.4.9) we have for μ ∈ R, ˜ ˜ c (sj − sj−1 ) θj,k (csj , μ) − θj,k (scj−1 , μ) = k(μ − c3,j , c1,j , c2,j ) + O(1) ain the limit c → ∞. Therefore we ﬁnd that 1 lim (θ(cr0 , λ ) − θ(cr0 , λ )) c→∞ πc m 1 ˜ ˜ ˜ ˜ ≥ lim (θj,2 (csj , λ − δ) − θj,2 (csj−1 , λ − δ)) c→∞ π c j=1 ˜ ˜ ˜ ˜ − (θj,1 (csj , λ + δ) − θj,1 (csj−1 , λ + δ)) m 1 ˜ ˜ = (sj − sj−1 ) k(λ − δ − c3,j , c1,j , c2,j ) − k(λ + δ − c3,j , c1,j , c2,j ) πa j=1 m 1 ≥ (sj − sj−1 ) k(λ − c3,j , c1,j , c2,j ) − k(λ − c3,j , c1,j , c2,j ) − 2 πa j=1
15. 5.4. Critical coupling constants 175and analogously 1 lim (θ(cr0 , λ ) − θ(cr0 , λ )) c→∞ πc m 1 ˜ ˜ ˜ ˜ ≤ lim (θj,2 (csj , λ + δ) − θj,2 (csj−1 , λ + δ)) c→∞ π c j=1 ˜ ˜ ˜ ˜ − (θj,1 (csj , λ − δ) − θj,1 (csj−1 , λ − δ)) m 1 ˜ ˜ = (sj − sj−1 ) k(λ + δ − c3,j , c1,j , c2,j ) − k(λ − δ − c3,j , c1,j , c2,j ) πa j=1 m 1 ≤ (sj − sj−1 ) k(λ − c3,j , c1,j , c2,j ) − k(λ − c3,j , c1,j , c2,j ) + 2 . πa j=1As in the proof of Theorem 5.3.3, the Pr¨fer angles remain bounded indepen- udently of c on [cr0 , ∞). Furthermore, the above Riemann sums converge to thecorresponding integrals due to the uniform continuity of the integrand, and theintegrand of (5.3.10) vanishes on [cr0 , ∞). Thus, the Relative Oscillation Theorem4.8.3 gives ∞ 1 2 r0 (k(λ − q (r), p1 (r), p2 (r)) − k(λ − q (r), p1 (r), p2 (r)) dr − ˜ ˜ ˜ ˜ ˜ ˜ πa 0 πa N(λ ,λ ] ≤ lim c→∞ c ∞ 1 2 r0 ≤ (k(λ − q (r), p1 (r), p2 (r)) − k(λ − q (r), p1 (r), p2 (r)) dr + ˜ ˜ ˜ ˜ ˜ ˜ . πa 0 πaAs > 0 was arbitrary, the statement of Theorem 5.3.4 follows.5.4 Critical coupling constantsWe now turn to the question whether a perturbed periodic Sturm-Liouville orDirac operator has a ﬁnite or inﬁnite total number of eigenvalues in an instabilityinterval of the unperturbed periodic equation. The results of section 5.3 have shownthat any compact subinterval of an instability interval contains at most ﬁnitelymany eigenvalues. Therefore it only remains to settle the question whether ornot eigenvalues accumulate at an end-point of the instability interval. The answerdepends on the rate of decay of the perturbation. We shall see in the following thatthe critical decay rate is x−2 , and that the exact value of the asymptotic constantat this scale is crucial. We begin by considering the relative oscillation of a real-valued solution ofthe perturbed periodic equation ˜ w = J(B + B + λW ) w (5.4.1)
16. 176 Chapter 5. Perturbationscompared to a real-valued solution of the periodic equation (1.5.1). At the mo- ˜ment, we only assume that the perturbation B is a locally integrable, real 2 × 2matrix-valued function, but there will be further restrictions later. By the generalassumption that B and W are symmetric, Tr(J(B + λW )) = 0 throughout; so wecan consider linearly independent, R2 -valued solutions u and v of (1.5.1) whoseWronskian W (u, v) = 1. Let Ψ = (u, v) be the fundamental matrix formed fromthese solutions. Then we combine the idea of variation of constants (cf. Proposition 1.2.2)with that of the Pr¨fer transformation (2.2.1), writing u sin γ w = Ψa ˆ (5.4.2) cos γwith a non-zero amplitude function a and a relative angle function γ; from (1.5.1) ˆand (5.4.1) we obtain ˜ ˆ sin γ sin γ cos γ J BΨa = Ψa ˆ + Ψaγ ˆ . cos γ cos γ − sin γ T cos γMultiplying from the left with , this gives − sin γ T T cos γ sin γ sin γ sin γ γ = Ψ−1 J B Ψ ˜ = ˜ ΨT B Ψ ; (5.4.3) − sin γ cos γ cos γ cos γin the last step we used the identity Ψ−1 = −JΨT J (5.4.4)which is easily veriﬁed for any 2 × 2 matrix of determinant 1 by direct calculation. The relative angle variable γ serves as a suitable proxy for the diﬀerencebetween the Pr¨fer angles of w and of the solution u of the unperturbed equation, uas the next lemma shows.Lemma 5.4.1. Let u, v : [0, ∞) → R2 be solutions of (1.5.1) with WronskianW (u, v) = 1, and let w : [0, ∞) → R2 be a non-trivial solution of (5.4.1). Let θand θ1 be Pr¨fer angles of w and u, respectively, and let γ be deﬁned as in (5.4.2) uand such that θ(0) − θ1 (0) and γ(0) − π lie in the same interval [nπ, (n + 1)π] with 2n ∈ Z. Then |γ − (θ − θ1 + π )| < π. 2Proof. Let R, R1 , R2 be the Pr¨fer radii of w, u and v and θ2 a Pr¨fer angle of v. u uThen (5.4.2) can be rewritten as sin γ R2 cos θ2 −R2 sin θ2 sin θ R2 sin(θ − θ2 ) A =R =R , cos γ −R1 cos θ1 R1 sin θ1 cos θ −R1 sin(θ − θ1 )
17. 5.4. Critical coupling constants 177which gives R2 sin(θ − θ1 + θ1 − θ2 ) tan γ = − R1 sin(θ − θ1 ) R2 π = sin(θ1 − θ2 ) tan(θ − θ1 + ) − cot(θ1 − θ2 ) . R1 2Now considering that 1 = W (u, v) = R1 R2 sin(θ1 − θ2 ), we see that the ﬁrst twofactors on the right-hand side are positive, and that cot(θ1 −θ2 ) is locally absolutelycontinuous. Therefore γ is related to θ − θ1 + π by a Kepler transformation, and 2the assertion follows by Theorem 2.2.1. In the following we assume that, for x ≥ 1, the perturbation is of the speciﬁcform ˜ 1 ˆ B(x) = 2 (B + β(x)), β(x) = o(1) (x → ∞), (5.4.5) x ˆwith a constant real symmetric matrix B. As the only condition on β is that ittends to 0 at inﬁnity, this assumption is purely asymptotic and does not impose ˜any restrictions, beyond the general hypotheses, on B in any compact interval. Moreover, we assume that in (1.5.1) λ is an end-point of an instability inter-val, u is a corresponding periodic or semi-periodic solution, and we take v to bethe solution arising from u by Rofe-Beketov’s formula v = f Ju + gu (5.4.6)with real-valued functions f, g (cf. Theorem 1.9.1). As we are studying the questionwhether γ is unbounded or not, and γ is continuous, it is clearly suﬃcient toconsider the diﬀerential equation (5.4.3) for γ on the interval [1, ∞), where ittakes the form 1 ˆγ = (u sin γ + gu cos γ − f Ju cos γ)T (B + β(x)) x2 × (u sin γ + gu cos γ + f Ju cos γ) T 1 1 1 1 = cos2 γ (tan γ + g) u + f Ju ˆ (B + β(x)) (tan γ + g) u + f Ju . x x x x (5.4.7)As g is locally absolutely continuous, we can perform the Kepler transformation 1 tan φ(x) = (tan γ(x) + g(x)), xwhereupon (2.2.12) gives the diﬀerential equation for φ, 1 φ = − sin φ cos φ + g cos2 φ x 1 1 + cos2 φ (u tan φ + f Ju)T (B + β(x)) (u tan φ + f Ju) ˆ x x
18. 178 Chapter 5. Perturbations 1 = − sin φ cos φ + g cos2 φ + sin2 φ uT (B + β(x)) u ˆ x 1 ˆ ˆ + 2 sin φ cos φ f ((Ju)T (B + β(x)) u + uT (B + β(x)) Ju) x 1 + 3 cos2 φ f 2 (Ju)T (B + β(x)) (Ju). ˆ xIf we introduce the a-periodic functions F1 := g , ˆ F2 := uT B u, G := uT β u, (5.4.8)and use the fact that f = −|u|−2 (see (1.9.8)) is a-periodic and therefore bounded,we can rewrite the above diﬀerential equation for φ more brieﬂy in the form 1 φ = F1 cos2 φ − sin φ cos φ + (F2 + G) sin2 φ + O(x−2 ) (x → ∞). x (5.4.9) Since φ and γ are connected by a Kepler transformation, φ is as good anindicator as γ of the asymptotic boundedness or otherwise of the diﬀerence ofPr¨fer angles of w and u. We now observe that F1 and F2 are a-periodic and utherefore the analysis of the diﬀerential equation (5.4.9) can be much simpliﬁedby averaging φ over a period interval.Lemma 5.4.2. Let F1 , F2 : [0, ∞) → R be locally integrable and a-periodic, G :[1, ∞) → R locally integrable with lim G(x) = 0 and φ : [1, ∞) → R a locally x→∞absolutely continuous function such that (5.4.9) holds. Then the averaged function x+a ˜ 1 φ(x) := φ (x ≥ 1) a x ˜is locally absolutely continuous, lim |φ(x) − φ(x)| = 0, and x→∞ x+a 1 1 ˜ φ (x) = C1 cos2 φ − sin φ cos φ + C2 + ˜ ˜ ˜ G sin2 φ + O(x−2 ) ˜ x a x 1 aas x → ∞, where Cj = a 0 Fj , j ∈ {1, 2}.Proof. By the Mean Value Theorem for integrals, for each x ≥ 1 there is an ˜x ∈ [x, x + a] such that φ(x) = φ(x ). Hence for all t ∈ [x, x + a], t a 1 1 ˜ |φ(t) − φ(x)| = φ ≤ (|F1 | + + |F2 |) + o(x−1 ) = O(x−1 ) (5.4.10) x x 0 2 ˜(x → ∞), and in particular lim |φ(x) − φ(x)| = 0. x→∞
19. 5.4. Critical coupling constants 179 ˜ Clearly φ is locally absolutely continuous, and using (5.4.9) and integratingby parts we obtain x+a˜ 1φ (x) = φ a x x+a x+a 1 1 2 2 =− (F1 cos φ − sin φ cos φ + (F2 + G) sin φ) a t t x x+a x+a 1 1 1 − (F1 cos2 φ − sin φ cos φ + (F2 + G) sin2 φ) dt + O( ) a x t2 t x2 x+a 1 = (F1 cos2 φ − sin φ cos φ + (F2 + G) sin2 φ) + O(x−2 ). ax xIn view of ⎫ | sin2 z1 − sin2 z2 | ⎬ | cos2 z1 − cos2 z2 | ≤ |z1 − z2 | (z1 , z2 ∈ R) (5.4.11) ⎭ | sin z1 cos z1 − sin z2 cos z2 | ˜and the estimate (5.4.10), φ can be replaced with φ in the integral while keepingthe same asymptotic order for the remainder term. ˜ The diﬀerential equation for φ in Lemma 5.4.2 has asymptotically constantcoeﬃcients. This makes it possible to ﬁnd a simple criterion to decide whether ornot its solutions are globally bounded.Lemma 5.4.3. Let C1 , C2 ∈ R, and let h : [1, ∞) → R be locally integrable withh(x) = o(x−1 ) (x → ∞). Let φ : [1, ∞) → R be a locally absolutely continuous ˜function such that 1 φ (x) = (C1 cos2 φ(x) − sin φ(x) cos φ(x) + C2 sin2 φ(x)) + h(x) ˜ ˜ ˜ ˜ ˜ (5.4.12) x ˜(x ≥ 1). Then φ is bounded if C1 C2 < 1/4 and unbounded if C1 C2 > 1/4.Proof. Choosing the constant φ0 ∈ R such that −1 C1 − C2 sin 2φ0 = , cos 2φ0 = , 1 + (C1 − C2 )2 1 + (C1 − C2 )2we can rewrite ˜ C1 + C2 + 1 + (C1 − C2 )2C1 cos2 φ − sin φ cos φ + C2 sin2 φ = ˜ ˜ ˜ ˜ cos 2(φ − φ0 ). 2 2 ˜Then the function ψ = φ − φ0 satisﬁes 1 ˜ ψ (r) = C1 + C2 + 1 + (C1 − C2 )2 cos 2(φ(x) − φ0 ) + h(x) (x ≥ 1). 2x (5.4.13)
20. 180 Chapter 5. PerturbationsBy the hypothesis on h, and since we assume that C1 C2 = 1/4, there exists apoint x0 > 1 such that |4xh(x)| ≤ 1 + (C1 − C2 )2 − |C1 + C2 |for all x ≥ x0 . Now assume that C1 C2 < 1/4, which is equivalent to |C1 + C2 | < 1 + (C1 − C2 )2 .Then for x ≥ x0 , the right-hand side of (5.4.13) is strictly positive if ψ = 0 (mod π)and strictly negative if ψ = π (mod π), and hence ψ(x) is trapped in the interval 2(nπ, nπ + 3π ), where n ∈ Z is such that ψ(x0 ) lies in this interval. Hence ψ, and 2 ˜consequently φ, are globally bounded. In the case C1 C2 > 1/4, which is equivalent to |C1 + C2 | > 1 + (C1 − C2 )2 , 1 x |ψ(x) − ψ(x0 )| ≥ |C1 + C2 | − 1 + (C1 − C2 )2 log →∞ (x → ∞), 4 x0 ˜and so ψ and φ are unbounded. The critical product C1 C2 can be conveniently expressed in terms of the prop- ˜erties of the discriminant of the periodic equation. Let D(c) be the discriminantof the system ˆ u = J (B + cB + λW ) u, (5.4.14) ˆwhere B is the constant matrix of (5.4.5) and we assume as before that λ is an ˜end-point of an instability interval of the unperturbed equation. Thus D(0) = ±2.Then we have the following.Lemma 5.4.4. The constants C1 and C2 of Lemma 5.4.2 satisfy 1 ˜ C 1 C2 = − |D| (0). a2Proof. Let Φ(x, c) be the canonical fundamental matrix of (5.4.14), so that ˆ Φ (x, c) = J (B(x) + cB + λW (x)) Φ(x, c), Φ(0, c) = I. ∂Then ∂c Φ(x, 0) is the solution of the initial-value problem ∂Φ ∂Φ ˆ ∂Φ (x, 0) = J (B(x) + λW (x)) (x, 0) + J B Φ(x, 0), (0, 0) = 0. ∂c ∂c ∂c ˜Using the variation of constants formula (1.2.11), (1.2.15) and D(c) = Tr Φ(a, c),we hence ﬁnd ∂D˜ a (0) = Tr Φ(a) J ˆ ΦT (s) B Φ(s) ds , (5.4.15) ∂c 0
21. 5.4. Critical coupling constants 181where Φ = Φ(·, 0). Now let Ψ = (u, v) be the fundamental matrix of the unperturbed periodicequation, as considered above in (5.4.2), with a-periodic or a-semi-periodic u andv as in (5.4.6). Then Ψ(x) = Φ(x) Ψ(0) and therefore Tr(Φ(a) Φ(s)−1 J B Φ(s)) = Tr(Ψ(a) Ψ(s)−1 J B Ψ(s) Ψ(0)−1 ) ˆ ˆ −1 ˆ = Tr(Ψ(0) Ψ(a) J Ψ(s)T B Ψ(s)), (5.4.16)bearing in mind identity (5.4.4) and the fact that the trace of a product of matricesis invariant under cyclic permutation. Since u(a) = ±u(0), ±1 v2 (0)v1 (a) − v1 (0)v2 (a) Ψ(0)−1 Ψ(a) = . (5.4.17) 0 ±1Further, f (a) = f (0) = |u(0)|−2 and so v(0) = f (0) J u(0), v(a) = ±(f (0) J u(0) + g(a) u(0)),which together with (5.4.17) gives 1 g(a) Ψ(0)−1 Ψ(a) = ± . (5.4.18) 0 1Also ˆ uT Bu ˆ uT Bv ˆ ΨT B Ψ = . (5.4.19) T ˆ T ˆ v Bu v BvTaking (5.4.15), (5.4.16), (5.4.18), (5.4.19) and (5.4.8) together, we arrive at ˜ ∂D a a a (0) = ∓g(a) ˆ uT B u = ∓ F1 F2 , ∂c 0 0 0 ˜ ˜and the assertion follows because |D(c)| = ±D(c) for c in a neighbourhood of0. These considerations give the following criterion for the ﬁniteness of the num-ber of gap eigenvalues for the perturbed Hill’s equation.Theorem 5.4.5. Let λ(n) be an end-point of an instability interval In of Hill’s equa-tion (1.5.2) with w = 1. Moreover, let α ∈ [0, π) and let q be a locally integrable, ˜real-valued function on [0, ∞) such that q (x) ∼ x2 (x → ∞) with constant c. Let ˜ c a2 ccrit = , 4|D| (λ(n) )where D is the discriminant (1.5.6) of Hill’s equation.
22. 182 Chapter 5. Perturbations If c/ccrit > 1, then λ(n) is an accumulation point of eigenvalues in In of theperturbed periodic Sturm-Liouville operator d d Hα = − (p ) + q(x) + q (x); ˜ dx dxif c/ccrit < 1, then λ(n) is not an accumulation point of eigenvalues of Hα .Proof. The perturbation matrix in the system (5.4.1) takes the form ˜ −˜ 0 q 1 B= = (−c + o(1)) W, 0 0 x2 ˆ ˜and so B = −cW . Hence the derivative of the discriminant D of (5.4.14) can beexpressed in terms of the derivative of D with respect to the spectral parameter, |D| (0) = −c |D| (λ(n) ). ˜From Lemma 5.4.4 we see that c |D| (λ(n) ) 1 c C1 C2 = 2 = , a 4 ccrit ˜and so the function φ of Lemma 5.4.3 is bounded if c/ccrit < 1 and unbounded ifc/ccrit > 1. Now let u be a periodic or semi-periodic Floquet solution of the unperturbedsystem with spectral parameter λ(n) and w a solution of the perturbed system(5.4.1) with λ = λ(n) , as in Lemma 5.4.1. Then, by Lemmas 5.4.1 and 5.4.2, thediﬀerence of the Pr¨fer angles of u and of w is globally bounded if c/ccrit < 1 and uglobally unbounded if c/ccrit > 1. On the other hand, if z is a non-trivial solution of (5.4.1) with spectralparameter λ ∈ In , then by the reasoning in the proof of Theorem 5.3.1, its Pr¨fer uangle diﬀers by no more than a globally bounded error from that of a solution ofthe unperturbed equation with the same spectral parameter λ. The Pr¨fer angles uof solutions of the unperturbed equation with spectral parameter in the closure Inof the instability interval all have the same asymptotics (2.4.1), with only boundederrors. Hence we conclude that the diﬀerence of the Pr¨fer angles of w and of z is uglobally bounded if c/ccrit < 1 and globally unbounded if c/ccrit > 1. The assertionof Theorem 5.4.5 now follows by the Relative Oscillation Theorem 4.8.3. We remark that, since the derivative of the discriminant has opposite sign atthe two end-points of the same instability interval, at least one of the end-points isalways in the subcritical case in the situation of Theorem 5.4.5. Hence eigenvaluescan only accumulate at either the upper or the lower end of the gap in the essentialspectrum, depending on the sign of c. By appeal to the comparison principle forPr¨fer angles (Corollary 2.3.2), it is easy to see that perturbations which decay at u
23. 5.4. Critical coupling constants 183a faster rate than x−2 only produce ﬁnitely many eigenvalues in any gap, whereasperturbations of ﬁxed sign and slower decay rate than x−2 always generate aninﬁnity of eigenvalues in each gap. The behaviour in the borderline case c = ccrit depends on higher-order asymp-totics of the perturbation, as we explain in the Chapter notes. For the perturbed periodic Dirac operator, the following analogous statementholds. As in section 5.3, the situation is complicated by the possibility of perturbingthe matrix coeﬃcients p1 and p2 . We denote by D(λ, c1 , c2 ) the Hill discriminantof the periodic Dirac system (5.3.9). We emphasise that the critical constant inthe next theorem plays a somewhat diﬀerent role from that deﬁned in Theorem5.4.5.Theorem 5.4.6. Let λ(n) be an end-point of an instability interval In of the periodicDirac equation (1.5.4), and let α ∈ [0, π). Let q , p1 and p2 be locally integrable, ˜ ˜ ˜real-valued functions on [0, ∞) such that qˆ p1 ˆ p2 ˆ q (x) ∼ ˜ , p1 (x) ∼ ˜ , p2 (x) ∼ ˜ (x → ∞) x2 x2 x2with constants q , p1 , p2 . Let ˆ ˆ ˆ 4 ∂ ∂ ∂ ccrit = ˆ |D|(λ(n) , 0, 0) − p1 q ˆ |D|(λ(n) , 0, 0) − p2 ˆ |D|(λ(n) , 0, 0) . a2 ∂λ ∂c1 ∂c2 If ccrit > 1, then λ(n) is an accumulation point of eigenvalues in In of theperturbed periodic Dirac operator d Hα = −iσ2 + (p1 (x) + p1 (x)) σ3 + (p2 (x) + p2 (x)) σ1 + q(x) + q (x); ˜ ˜ ˜ dxif ccrit < 1, then λ(n) is not an accumulation point of eigenvalues of Hα . ˆProof. The matrix B of (5.4.5) takes the form ˆ −ˆ1 − q −ˆ2 p ˆ p B= −ˆ2 p p1 − q ˆ ˆand, comparing (5.4.14) and (5.3.9), we see that D(c) = D(λ(n) − cˆ, cˆ1 , cˆ2 ) and ˜ q p pconsequently ∂ ∂ ∂ ˜ |D| (0) = − |D|(λ(n) , 0, 0) q + ˆ |D|(λ(n) , 0, 0) p1 + ˆ |D|(λ(n) , 0, 0) p2 . ˆ ∂λ ∂c1 ∂c2 ˜By Lemma 5.4.4, the function φ of Lemma 5.4.3 is globally bounded if ccrit < 1,and globally unbounded if ccrit > 1. By Lemmas 5.4.1 and 5.4.2, this also holds forthe diﬀerence of the Pr¨fer angles of the solutions u of the unperturbed periodic uequation and w of the perturbed periodic equation.
24. 184 Chapter 5. Perturbations Inside the instability interval In , all the Pr¨fer angles of all solutions of the uunperturbed equation have the same asymptotic (2.4.1) as u, with only boundederrors, and this also holds for the perturbed equation by the same comparisonargument as in the proof of Theorem 5.3.2. Thus the diﬀerence of the Pr¨fer angles of solutions of the perturbed periodic uDirac equation at λ(n) and at some λ ∈ In is globally bounded if ccrit < 1 andglobally unbounded if ccrit > 1, and the assertion of Theorem 5.4.6 follows by theRelative Oscillation Theorem 4.8.3.5.5 Eigenvalue asymptoticsWe have seen in Theorem 5.4.5 for the perturbed Hill’s equation and in Theorem5.4.6 for the perturbed periodic Dirac equation that eigenvalues accumulate at anend-point of an instability interval of the unperturbed equation if the perturbationhas x−2 decay with supercritical asymptotic constant. In the present section, weconclude the study of perturbed periodic problems by deriving the asymptoticdistribution of the eigenvalues near their accumulation point. We shall use theoscillation techniques of section 5.4, but the reference equation will be the periodicequation with spectral parameter inside the instability interval, not at the end-point. More precisely, let λ(n) be an end-point of a stability interval and λ a ﬁxedreference point inside the instability interval. We wish to count the eigenvaluesbetween λ and λ, where λ is a point between λ and λ(n) , and derive the leadingasymptotics for this count in the limit λ → λ(n) . Since we assume the supercriticalcase, we know that the eigenvalue count will tend to inﬁnity in the limit. ByTheorems 5.3.1 or 5.3.2, it will be ﬁnite between any two points inside the sameinstability interval, and so the leading asymptotic is clearly independent of thechoice of λ .Lemma 5.5.1. Let λ(n) be an end-point of an instability interval In of the periodicequation (1.5.1). Then, for λ ∈ In , the Floquet exponent with positive real partsatisﬁes Re μ(λ) = |D (λ(n) )| |λ − λ(n) | + o( |λ − λ(n) |) (λ → λ(n) ), (5.5.1)and the corresponding eigenvector v(λ) of the monodromy matrix satisﬁes v(λ) = v(λ(n) ) + O( |λ − λ(n) |) (λ → λ(n) ). (5.5.2)For the corresponding Floquet solution u(x, λ), we have u(x, λ) = u(x, λ(n) ) + O( |λ − λ(n) |) (λ → λ(n) )uniformly in x ∈ [0, a], where u(·, λ(n) ) is a-periodic or a-semi-periodic.
25. 5.5. Eigenvalue asymptotics 185Proof. The monodromy matrix M (λ) is analytic in λ. In particular M (λ) = M (λ(n) ) + M (λ(n) ) (λ − λ(n) ) + o(λ − λ(n) ), (5.5.3)and similarly D(λ) = D(λ(n) ) + D (λ(n) ) (λ − λ(n) ) + o(λ − λ(n) ) (5.5.4)for the discriminant; here |D(λ(n) )| = 2. Since |D(λ)| = 2 cosh Re μ(λ) ≥ 2 + (Re μ(λ))2for λ ∈ In by (1.4.6), we see from (5.5.4) that (Re μ(λ))2 = O(|λ − λ(n) |), andhence |D(λ)| = 2 cosh Re μ(λ) = 2 + (Re μ(λ))2 + o(|λ − λ(n) |),which together with (5.5.4) yields (5.5.1). Since we assume that λ(n) separates a stability interval from an instabilityinterval and hence is not a point of coexistence, at least one of φ12 (λ), φ21 (λ) in φ11 (λ) φ12 (λ) M (λ) = φ21 (λ) φ22 (λ)is non-zero at λ(n) and so, by continuity, also close to λ(n) in In . Assuming withoutloss of generality that φ21 (λ) = 0, we have the eigenvector eμ(λ) − φ22 (λ) v(λ) = φ21 (λ)of M (λ) for eigenvalue eμ(λ) . By (5.5.1), eμ(λ) = sgn D(λ(n) ) eRe μ(λ) = sgn D(λ(n) ) + O( |λ − λ(n) |).Bearing in mind the asymptotics of φ21 (λ) and φ22 (λ) from (5.5.3), we concludethat (5.5.2) holds, where sgn D(λ(n) ) − φ22 (λ(n) ) v(λ(n) ) = φ21 (λ(n) )is an eigenvector of M (λ(n) ). The statement about the Floquet solution now follows since, denoting byΦ(·, λ) the canonical fundamental matrix of (1.5.1), u(x, λ) = Φ(x, λ) v(λ) = (Φ(x, λ(n) ) + O(|λ − λ(n) |)) (v(λ(n) ) + O( |λ − λ(n) |) = u(x, λ(n) ) + O( |λ − λ(n) |)uniformly in x ∈ [0, a].
26. 186 Chapter 5. Perturbations We now study the perturbed equation (5.4.1) on [0, ∞), where the pertur-bation takes the form (5.4.5) for x ≥ 1. For all λ, let θ(x, λ) be the Pr¨fer angle uof a real-valued solution, with θ(0, λ) = α; here α ∈ [0, π) is the parameter ofthe boundary condition at 0. By the Relative Oscillation Theorem 4.8.3, we onlyneed to estimate θ(x, λ) − θ(x, λ(n) ) in the limit as x → ∞ in order to count theeigenvalues between λ and λ up to a bounded error. ˜ Since the perturbation B(x) tends to 0 as x → ∞, it follows by SturmComparison, as in the proof of Theorem 5.3.2, that θ(x, λ) grows regularly as nπx , aup to a bounded error, as soon as x is so large that |B(x)| ≤ |λ − λ(n) |. Therefore ˜we need not keep track of θ(x, λ) for all x ≥ 0, but only up to a λ-dependent pointr(λ), after which the diﬀerence θ(x, λ) − θ(x, λ ) will be bounded uniformly in λ.The leading asymptotic of the number of eigenvalues between λ and λ will bedetermined by the growth of θ(r(λ), λ) − θ(r(λ), λ ) as λ → λ(n) . ˆ Note that we can assume without loss of generality that the matrix B deter-mining the asymptotic behaviour of the perturbation B ˜ is non-zero in the following. ˆIndeed, B = 0 would mean that F2 = 0 in (5.4.8) and hence C2 = 0 in Lemma5.4.2. Then C1 C2 = 0 < 1/4 in Lemma 5.4.3; this is the subcritical case withoutaccumulation of eigenvalues at λ(n) and not of interest here. ˆLemma 5.5.2. Let B be a non-zero symmetric 2 × 2 matrix with real entries andβ a symmetric 2 × 2 matrix-valued function on [0, ∞) with lim β(x) = 0. x→∞ Then for λ ∈ In there is r(λ) ≥ 1 such that 1 ˆ (B(x) + β(x)) ≤ |λ − λ(n) | (x ≥ r(λ)) x2and ˆ |B| r(λ) ∼ (λ → λ(n) ). (5.5.5) |λ − λ(n) |Proof. First let ˆ |B| r1 (λ) := + 1, |λ − λ(n) |then set ˆ |B| + sup |β(x)| x≥r1 (λ) r(λ) := + 1 ≥ r1 (λ). |λ − λ(n) |Then for x ≥ r(λ), 1 ˆ ˆ |B| + |β(x)| 2 (B + β(x)) ≤ |λ − λ(n) | ≤ |λ − λ(n) |, x ˆ |B| + sup |β(x)| x≥r1 (λ)
27. 5.5. Eigenvalue asymptotics 187as required. Moreover, sup |β(x)| x≥r1 (λ) 1 0 ≤ r(λ) − r1 (λ) + 1 ≤ +1=o ˆ |λ − λ(n) | 2 |B| + 1 |λ − λ(n) |(λ → λ(n) ), which proves (5.5.5). Clearly, r(λ) as given in the above lemma tends to ∞ as λ → λ(n) . Takinginto account the regular growth behaviour of θ(r(λ), λ ), we conclude that, up toan error bounded uniformly in λ ∈ In , the number of eigenvalues between λ andλ is given by 1 nr(λ) θ(r(λ), λ) − . (5.5.6) π aOn the other hand, we know from Theorem 2.4.1 that the Pr¨fer angle θ0 (x, λ) of ua real-valued solution of the unperturbed periodic equation (1.5.1), with θ0 (0, λ) ∈[0, π), also satisﬁes nπx θ0 (x, λ) = + Ounif (1), awith error term bounded uniformly in λ ∈ In . Thus the diﬀerence of angles in(5.5.6) can be read as the relative rotation, up to the point r(λ), of a solution ofthe perturbed equation with spectral parameter λ compared to a solution of theunperturbed equation with the same spectral parameter. By Lemma 5.4.1, thisdiﬀerence can be estimated, up to a universally bounded error, by the growth ofa solution γ of (5.4.3). We now follow the general approach of section 5.4 from (5.4.5) onwards,but with the diﬀerence that the solution u of the unperturbed equation will notbe periodic or semi-periodic, but a Floquet solution with Floquet multiplier ofmodulus |eμ(λ) | > 1. For the coeﬃcients f and g in (5.4.6) — which now depend on λ, too —, wehavef (x) = −|u(x, λ)|−2 , g (x) = |u(x, λ)|−4 u(x, λ)T (JA(x, λ) − A(x, λ)J)u(x, λ)from (1.9.8) and (1.9.9); here A(x, λ) = J(B(x) + λW (x)). As u(x, λ) is a Floquetsolution with multiplier eμ(λ) and λ ∈ In , (1.4.7) shows that e− Re μ(λ)x/a u(x, λ)is a-periodic or a-semi-periodic. Therefore the functions f (x) e2 Re μ(λ)x/a andg (x) e2 Re μ(λ)x/a are a-periodic. In the light of this observation, we now start from (5.4.7), perform the Keplertransformation e2 Re μ(λ)x/a tan ψ(x) = (tan γ(x) + g(x)) x
28. 188 Chapter 5. Perturbationsand obtain the diﬀerential equation for ψ, 2 Re μ(λ) ψ = sin ψ cos ψ a 1 2 Re μ(λ)x/a + e g (x) cos2 ψ − sin ψ cos ψ x + e−2 Re μ(λ)x/a uT (B + β(x)) u sin2 ψ ˆ 1 ˆ ˆ + f (x) ((Ju)T (B + β(x))u + uT (B + β(x))Ju) sin ψ cos ψ x2 1 + 3 e2 Re μ(λ)x/a f 2 (x) (Ju)T (B + β(x))Ju cos2 ψ. ˆ (5.5.7) xDeﬁning the a-periodic functions F1 (x, λ) = g (x) e2 Re μ(λ)x/a , F2 (x, λ) = e−2 Re μ(λ)x/a uT (x, λ) B u(x, λ) ˆwe can rewrite (5.5.7) in the form Re μ(λ) 1 ψ = sin 2ψ + F1 (x, λ) cos2 ψ − sin ψ cos ψ + F2 (x, λ) sin2 ψ a x e−2 Re μ(λ)x/a T ˆ 1 + u (B + β(x)) u sin2 ψ + Ounif . (5.5.8) x x2The remainder term is uniform in λ ∈ In ; indeed, the combinations of f , u andthe exponential in the last two terms of (5.5.7) are a-periodic, hence bounded, inx and continuous in λ. We shall now apply an averaging procedure analogous to that of Lemma5.4.2. There is the essential diﬀerence that (5.5.8) has a non-decaying leadingterm; however, we are here concerned with the limit λ → λ(n) , and the x decayenters only indirectly as r(λ) → ∞ in that limit. We now deﬁne ˜ 1 x+a ψ(x) = ψ (x ≥ 1) a xand apply the Mean Value Theorem for integrals, which gives, for each x ≥ 1, an ˜x ∈ [x, x + a] such that ψ(x) = ψ(x ). We then obtain t ˜ |ψ(t) − ψ(x)| = ψ x a 1 1 1 ≤ Re μ(λ) + (|F1 (t, λ)| + + |F2 (t, λ)|) dt + ounif x 0 2 x 1 = Re μ(λ) + Ounif (5.5.9) x
29. 5.5. Eigenvalue asymptotics 189for all t ∈ [x, x + a]. By (5.5.1) and (5.5.5), this implies that 1 ˜|ψ(r(λ), λ)− ψ(r(λ), λ)| ≤ Re μ(λ)+Ounif = O( |λ − λ(n) |) (λ → λ(n) ). r(λ) ˜ ˜Also, |ψ(0, λ) − ψ(0, λ)| = O(1), and so ψ can be used instead of ψ for the purposeof counting eigenvalues up to bounded error. Deﬁning the continuous functions a 1 Cj (λ) = Fj (t, λ) dt (λ ∈ In ; j ∈ {1, 2}), a 0we ﬁnd from (5.5.8) that ˜ 1 x+a ψ (x) = ψ a x Re μ(λ) x+a 1 = 2 sin 2ψ + (C1 (λ) cos2 ψ − sin ψ cos ψ + C2 (λ) sin2 ψ) ˜ ˜ ˜ ˜ a x x x+a 1 + e−2 Re μ(λ)t/a u(t, λ)T β(t)u(t, λ) sin2 ψ(t, λ) dt xa x x+a 1 + F1 (t, λ) (cos2 ψ − cos2 ψ) ˜ xa x − (sin ψ cos ψ − sin ψ cos ψ) + F2 (t, λ) (sin2 ψ − sin2 ψ) dt ˜ ˜ ˜ 1 + Ounif . (5.5.10) x2In view of (5.5.1), the ﬁrst term on the right-hand side of (5.5.10) is of orderO( |λ − λ(n) |). For the last integral in (5.5.10), we use (5.4.11) and (5.5.9) to-gether with the boundedness, uniformly in λ ∈ In , of F1 and F2 to obtain theestimate 1 1 1 Re μ(λ) Ounif + Ounif = O( |λ − λ(n) |) + Ounif . x x2 x2Similarly, e−2 Re μ(λ)x/a uT βu = ounif (1). Thus we can write (5.5.10) more brieﬂyin the form 1 1ψ (x) = (C1 (λ) cos2 ψ−sin ψ cos ψ+C2 (λ) sin2 ψ)+O( |λ − λ(n) |)+ounif ˜ ˜ ˜ ˜ ˜ . x xThe asymptotics of Lemma 5.5.1 for u(x, λ) and μ(λ), along with A(x, λ) = A(x, λ(n) ) + (λ − λ(n) )JW,give Cj (λ) = Cj (λ(n) ) + O( |λ − λ(n) |) (λ → λ(n) ; j ∈ {1, 2}). (5.5.11) ˜We can now deduce the asymptotics of φ from the following lemma.
30. 190 Chapter 5. PerturbationsLemma 5.5.3. Let Λ be a closed interval with end-point λ(n) , and assume thatC1 , C2 : Λ → R are continuous and satisfy (5.5.11) and 4 C1 (λ(n) ) C2 (λ(n) ) > 1. (5.5.12) ˜ ˜Moreover, let ψ : [1, ∞)×Λ → R be a function such that ψ(·, λ) is locally absolutelycontinuous for each λ ∈ Λ and 1 ˜ ψ (x, λ) = C1 (λ) cos2 ψ(x, λ) − sin ψ(x, λ) cos ψ(x, λ) + C2 (λ) sin2 ψ(x, λ) ˜ ˜ ˜ ˜ x + G(x, λ) (5.5.13)with |G(x, λ)| ≤ c |λ − λ(n) | + h(x) (x ≥ 1, λ ∈ Λ), (5.5.14)where c > 0 is a constant, h > 0 and lim x h(x) = 0. Also assume that r(λ) has x→∞asymptotic growth (5.5.5). Then 1 ˜ ˜|ψ(r(λ), λ) − ψ(1, λ)| ∼ 4 C1 (λ(n) ) C2 (λ(n) ) − 1 log |λ − λ(n) | (λ → λ(n) ). 4Proof. In analogy to the beginning of the proof of Lemma 5.4.3, we choose acontinuous ψ0 : Λ → R such that −1 C1 (λ) − C2 (λ) sin 2ψ0 (λ) = , cos 2ψ0 (λ) = , 1 + (C1 (λ) − C2 (λ))2 1 + (C1 (λ) − C2 (λ))2and rewrite (5.5.13) as Γ+ (λ) Γ− (λ) ω (x, λ) = cos2 ω(x, λ) + sin2 ω(x, λ) + G(x, λ), 2x Γ+ (λ) ˜where ω(x, λ) = ψ(x, λ) − ψ0 (λ), ω denotes the derivative with respect to x, and Γ± (λ) = C1 (λ) + C2 (λ) ± 1 + (C1 (λ) − C2 (λ))2 .By (5.5.12) and continuity, |C1 (λ) + C2 (λ)| > 1 + (C1 (λ) − C2 (λ))2 and hence Γ− (λ) >0 Γ+ (λ)for λ suﬃciently close to λ(n) , and for such λ we can perform the Kepler transfor-mation Γ− (λ) tan ω = arctan ˜ tan ω . Γ+ (λ)
32. 192 Chapter 5. PerturbationsThus, integrating (5.5.15) we obtain the asymptotic ω (r(λ), λ) − ω (1, λ) ˜ ˜ sgn(C1 (λ) + C2 (λ)) = 4 C1 (λ) C2 (λ) − 1 log r(λ) + o(log r(λ)) 2 sgn(C1 (λ(n) ) + C2 (λ(n) )) ∼− 4 C1 (λ(n) ) C2 (λ(n) ) − 1 log |λ − λ(n) |. 4 We have thus proved the following statements about perturbed periodicSturm-Liouville and Dirac operators in the supercritical case.Theorem 5.5.4. Let λ(n) be an end-point of an instability interval In of Hill’sequation (1.5.2) with w = 1, and let λ ∈ In . Moreover, let α ∈ [0, π) and letq be a locally integrable, real-valued function on [0, ∞) of asymptotic q (x) ∼ x2˜ ˜ c(x → ∞) with constant c such that c/ccrit > 1, where a2 ccrit = , 4|D| (λ(n) )and D is the discriminant (1.5.6) of Hill’s equation. Then the number N (λ) of eigenvalues between λ and λ of the perturbedperiodic Sturm-Liouville operator d d Hα = − (p ) + q(x) + q (x) ˜ dx dxhas asymptotic 1 c N (λ) ∼ − 1 log |λ − λ(n) | (λ → λ(n) ). (5.5.16) 4π ccritTheorem 5.5.5. Let λ(n) be an end-point of an instability interval In of the periodicDirac equation (1.5.4), and let λ ∈ In and α ∈ [0, π). Let q , p1 and p2 be locally ˜ ˜ ˜integrable, real-valued functions on [0, ∞) of asymptotic qˆ p1 ˆ p2 ˆ q (x) ∼ ˜ , p1 (x) ∼ ˜ , p2 (x) ∼ ˜ (x → ∞) x2 x2 x2with constants q , p1 , p2 such that ˆ ˆ ˆ 4 ∂ ∂ ∂ccrit = 2 ˆ |D|(λ(n) , 0, 0) − p1 q ˆ |D|(λ(n) , 0, 0) − p2 ˆ |D|(λ(n) , 0, 0) > 1. a ∂λ ∂c1 ∂c2Then the number N (λ) of eigenvalues between λ and λ of the perturbed periodicDirac operator d Hα = −iσ2 + (p1 (x) + p1 (x)) σ3 + (p2 (x) + p2 (x)) σ1 + q(x) + q (x); ˜ ˜ ˜ dx
33. 5.6. Chapter notes 193has asymptotic 1 √ N (λ) ∼ ccrit − 1 log |λ − λ(n) | (λ → λ(n) ). 4π5.6 Chapter notes§5.2 Corollary 5.2.3 was shown by Stolz [175] for the one-dimensional Schr¨dinger ooperator, i.e. w = p = 1, p = 0; the proof of Theorem 5.2.1 follows his idea. It ˜is worth noting Stolz’s remark that the conditions (5.2.1) and (5.2.2) — and theensuing conditions on the perturbations of the coeﬃcients of the Dirac and Sturm-Liouville operators — are satisﬁed by absolutely integrable functions, functions ofbounded variation multiplied with a-periodic functions, and linear combinationsof these. The results of this section carry over to the case where the left-handend-point is singular, in particular to the full-line operator, see [175].§5.3 The results of this section extend analogously to the situation with two sin-gular end-points by Glazman’s decomposition method [67, Section 7]. If the Sturm-Liouville or Dirac operator is given on an interval (a, b) with both a and b singular,we can choose a point c ∈ (a, b) and consider the two boundary-value problems on(a, c] and on [c, ∞) with some boundary conditions at c. The operator on (a, b) isthen a two-dimensional extension of the direct sum of one-dimensional restrictionsof the two part-interval operators with the boundary condition strengthened tothe condition that y(c) = (py )(c) = 0 for Sturm-Liouville and u(c) = 0 for Dirac.Using the spectral representation, one can hence deduce that the total multiplicityof the spectrum of the operator on (a, b) diﬀers by no more than 4 from the sumof the spectral multiplicities of the part-interval operators. Theorem 5.3.3 is due to Sobolev [170], who considers a strong-coupling limitof a perturbation of inverse power decay, which is clearly equivalent to the slow-variation limit. Theorem 5.3.4 can be found in [164] for the speciﬁc case of per-turbations of the type of the angular momentum term; there the perturbation issingular at 0 but can be shown to generate no eigenvalues in any given compactλ-interval when considered on (0, c] with suﬃciently small c. Numerical evidence suggests that, in the case of the angular momentum per-turbation, the asymptotic densities (5.3.5) and (5.3.10) can give a very accurateindication of the distribution of eigenvalues in a gap even for small scaling con-stants c [25], [163]. The growing density of eigenvalues in each gap with increas-ing angular momentum quantum number (corresponding to increasing c) explainsthe appearance of dense point spectrum in radially periodic higher-dimensionalSchr¨dinger and Dirac operators [83], [159]. o A more precise count of eigenvalues in distant gaps of Hill’s equation un-der the assumption that the perturbation satisﬁes R (1 + |x|)˜(x) dx < ∞ was qannounced by Rofe-Beketov [150]: all suﬃciently distant gaps contain at most 2eigenvalues, at least one eigenvalue if R q = 0, and exactly one eigenvalue if in ˜addition q ≥ 0. See [63] for further details; a reﬁnement of these results was given ˜
34. 194 Chapter 5. Perturbationsin [21]. The existence of eigenvalues in spectral gaps of perturbed Schr¨dinger ooperators was also shown by Deift and Hempel [32] in a more general setting; seealso [62].§5.4 Theorem 5.4.5 goes back to Rofe-Beketov (ﬁrst announced in [151, Theorems9, 10], [152, Theorems 4, 5]; details in [154], [153] including extensions to almost pe-riodic equations); he treated the diﬀerential equation (5.4.3) in the Sturm-Liouvillecase by relating it to a diﬀerent Sturm-Liouville equation by means of an implicitvariable transformation. The oscillation properties of the resulting equation canthen be studied using a criterion of Taam, Hille and Wintner [122, Theorem B].The approach presented here [161] can also be applied to decide the question ofﬁniteness or inﬁnity of the gap spectrum in the limiting case c = ccrit of Theorem5.4.5. If q (x) ∼ cx2 + x2 (log x)2 , then λ(n) is an accumulation point of eigenvalues ˜ crit cif c/ccrit > 1 and not an accumulation point of eigenvalues if c/ccrit < 1 [161,Proposition 4] (in particular, the case q (x) = cx2 is subcritical). This extends to ˜ crita whole hierarchy of asymptotic terms with associated critical coupling constants,see [121], analogous to the classical oscillation criterion of Kneser [111] as reﬁnedby Weber, Hartman and Hille (cf. [79, Chapter XI, Exercise 1.2]) — this is thespecial case of Theorem 5.4.5 where the periodic coeﬃcients are constant and λ(n)is the inﬁmum of the essential spectrum (in particular, n = 0). If p = w = 1 andq is constant, then ccrit = −1/4. This corresponds to the constant in Hardy’s inequality. Indeed, separationof variables in polar coordinates of the Laplacian −Δ in R2 gives rise to a directsum decomposition of this operator with half-line Sturm-Liouville operators 2 1 d2 − 4 − + dr2 r2as terms, where ∈ {0, 1, 2, . . . } and r is the radial variable; and = 0 is just theborderline case of Theorem 5.4.5 — which must be subcritical since the Laplacianhas no eigenvalues. A curious phenomenon appears when a non-constant radiallyperiodic potential q(r) is added to the Laplacian [160]: if λ0 is the inﬁmum of ythe essential spectrum, the Floquet solution y corresponding to u = can be ychosen strictly positive, and d’Alembert’s formula (1.9.3) can be used instead ofthe Rofe-Beketov formula (5.4.6), giving F1 = 1/y 2 . Then −c a a 1 C1 C2 = y2 , a2 0 0 y2and it follows by the Cauchy-Schwarz inequality that ccrit ≥ − 1 with equality 4if and only if y 2 and y −2 are linearly dependent, i.e. if q is constant. Hence thetwo-dimensional Schr¨dinger operator with non-constant rotationally symmetric opotential always has inﬁnitely many eigenvalues below the essential spectrum. We note that results similar to Theorem 5.4.5 in a slightly more general ¨setting can be found in [109]. Gesztesy and Unal [65] extended Kneser’s oscillation
35. 5.6. Chapter notes 195criterion and its relation to Hardy’s inequality to allow background potentials,including the periodic case. The relative oscillation result underlying Theorem 5.4.6 appears in [162], seealso [172]. The critical case also gives rise to a ladder of asymptotic scales for theperturbations, as for the Sturm-Liouville equation.§5.5 Theorem 5.5.4 was proven in [161] by the method shown in section 5.5. Theanalogous statement for the perturbed periodic Dirac operator, Theorem 5.5.5, isnew. It is of interest to compare the asymptotics of eigenvalues in instability inter-vals with the semiclassical asymptotics. In the absence of a periodic background d2potential in the Sturm-Liouville equation, i.e. if q = 0 in Hα = − dx2 + q(x) + q (x), ˜the semiclassical formula for the number of eigenvalues below λ, ∞ 1 N (λ) ∼ (λ − q (x))+ dx ˜ π 0(where (λ − q (x))+ = max{0, λ − q(x)}) is known to hold true in a variety of ˜situations ([135], [185, Chapter VII], [78], [124], [157]). However, it is clear thatthis formula cannot be true in the case q (x) ∼ c/x2 (x → ∞), because the ﬁniteness ˜or otherwise of the limit of the above semiclassical integral as λ → 0 is independentof c, but there is a critical constant ccrit = 1/4 in the sense of Theorem 5.4.5. Forthis case, J¨rgens [104] has given the adjusted asymptotics o ∞ 1 1 N (λ) ∼ λ − q (x) − ˜ dx (λ → 0). π 1 4x2 +Note that here the integrand for λ < 0 has compact support in the subcriticalcase. For the perturbed periodic Sturm-Liouville operator with non-constant pe-riodic potential of period 1 and perturbation q ˜ −x−b , 0 < b < 1/n, Zelenkoderived an m-term asymptotic series for the number of eigenvalues close to anend-point λ(n) of an instability interval [208, Thm. 2]; for m = 1 it has the form ∞ |D (λ(n) )| N (λ) ∼ (λ − λ(n) − q (x))+ dx ˜ (λ → λ(n) ). π 0For the same reason as above, this formula must fail when b = 0. For comparison,we note that for q (x) = c/x2 + O(1/x2+ ) with > 0 and supercritical c, (5.5.16) ˜can be rewritten as ∞ |D (λ(n) )| ccrit N (λ) ∼ q (x) − ˜ − |λ − λ(n) | dx (λ → λ(n) ). πa 1 x2 +Thus this asymptotic formula closes the gap between J¨rgens’ and Zelenko’s oasymptotics, generalising the former by the inclusion of a periodic background,the latter by allowing critical decay of the perturbation.
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