Periodic differential operators

Chapter 5

Perturbations

5.1 Introduction
The periodic Sturm-Liouville or Dirac operator on the whole real line has a purely
absolutely continuous spectrum of band-gap structure; the regular end-point of
the operator restricted to a half-line only introduces a single eigenvalue, if any,
into each spectral gap. In applications, however, one does not always have exact
periodicity of the coefficients, and the question arises how the spectral properties
of the operator change if a non-periodic perturbation is added to the periodic
background potential. In many ways this is analogous to the general question
of the spectrum generated by a more or less localised potential added to a free
Sturm-Liouville or Dirac operator, but here we take as an unperturbed reference
a periodic operator, whose spectral properties are very well known by the results
shown in the preceding chapters.
We begin by noting in section 5.2 that the spectral bands remain intervals
of purely absolutely continuous spectrum under a very mild decay condition on
the perturbation. In section 5.3, we observe that if the perturbation tends to 0
at infinity, then every compact subinterval of an instability interval contains at
most finitely many eigenvalues and no further spectrum. In particular, this means
that the instability intervals, while not devoid of spectrum in general, continue to
be gaps in the essential spectrum. We also derive asymptotics for the distribution
of eigenvalues thus introduced into the gaps in the limit of slow variation (some-
times called the adiabatic limit) of the perturbation. The question of whether an
instability interval as a whole contains a finite or infinite number of eigenvalues
turns out to have a more subtle answer, given in section 5.4. There is a critical
boundary case for perturbations with x−2 asymptotic decay, and the critical cou-
pling constant can be expressed in terms of the derivative of Hill’s discriminant at
the point of transition between instability and stability. In the supercritical case,
where eigenvalues in the gap accumulate at a band edge, we find their asymptotic

B.M. Brown et al., Periodic Differential Operators, Operator Theory: Advances 161
and Applications 230, DOI 10.1007/978-3-0348-0528-5_5, © Springer Basel 2013

162 Chapter 5. Perturbations

distribution in section 5.5, showing that they are exponentially close to the band.

5.2 Spectral bands
We have seen in section 4.3 that the periodic Sturm-Liouville and Dirac operators
on the half-line have purely absolutely continuous spectrum in the set S of stability
intervals of the corresponding periodic differential equation system. In the present
section it is shown that this property is stable when the coefficients of the operator
are perturbed by the addition of non-periodic terms which satisfy a mild decay
condition at infinity, stipulating essentially that their local average tends to zero
and 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 entry
separately.
Theorem 5.2.1. Let B and W be 2 × 2 matrix-valued functions on [0, ∞) satisfying
the general hypotheses of section 1.5, and assume that B = B1 + B2 , where B1
and 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→∞ x

Let [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 all
solutions 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, let
Ψj be the solution of (5.2.4) with initial value Ψj (a(j − 1)) = I; we then set
Mj := Ψj (aj). Then Φj (x) := Φ(x − a(j − 1)) will serve an analogous purpose for
the unperturbed equation (5.2.3), with Φj (aj) = M , the monodromy matrix of
(5.2.3), for all j.

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 for
such 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 a
sufficiently large J ∈ N and some δ > 0,

|Dj (λ)| ≤ 2 − δ

for all j > J and λ ∈ [λ , λ ].
The matrix Mj has determinant 1 and hence can be analysed as in our study
of the monodromy matrix in section 1.4. For j > J, we are in Case 3; so Mj has


complex conjugate eigenvalues μj , μj with |μj | = 1 and corresponding eigenvalues
given 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
−1
to the matrix E of eigenvectors of M , and so Ej converges to E −1 and hence is
bounded, uniformly in λ ∈ [λ , λ ]. From

μj 0 −1
Mj = E j Ej
0 μj

we 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 0
and, 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)
−1
In order to estimate |Ej Ej−1 |, we observe that Ψj−1 (· − a) is a fundamental
matrix 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)


Hence, 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 hence
any solution u(·, λ) of (5.2.4) with |u(0, λ)| = 1, is bounded uniformly with respect
to 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 → ∞)
ma

by (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 periodic
equation
˜
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 Theorem
4.5.4) are preserved under perturbations satisfying a mild decay condition.


Corollary 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 )
˜ ˜ ˜
dx
with boundary condition (4.3.1) has purely absolutely continuous spectrum in the
stability 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 such
that 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 the
end of the proof of Lemma 4.5.2, this also implies that |u(x)| > 1/C (x ≥ 0). In
particular,
x
k(x)
≤ |u|2 ≤ k(x) (x ≥ 0)
C4 0

with k(x) := C 2 x. Theorem 4.9.1 now shows that [λ , λ ] is an interval of purely
absolutely continuous spectrum of Hα .
For the perturbed Hill equation, the required lower bound on the growth of
the 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-valued
functions 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→∞ x

and
∞ 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


with boundary condition
y(0) cos α − (py )(0) sin α = 0
has purely absolutely continuous spectrum in the stability set S of the periodic
Sturm-Liouville equation (1.5.2).
Proof. Let [λ , λ ] ⊂ S. Then, as in the proof of Corollary 5.2.2, we can use
Theorem 5.2.1 to find a constant C > 0 such that, for all λ ∈ [λ , λ ], all solutions
of the perturbed periodic Sturm-Liouville system
1
0 p+p˜
u (x, λ) = u(x, λ) (5.2.10)
q + q − λw
˜ 0
with |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
˜ ˜
y
such that |y(0)|2 + |(py )(0)|2 = 1; then u = will be a solution of (5.2.10)
py
with the required property. Now for n ∈ N, let Ψn and Φn be defined as in the
proof 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)a

As |u((n − 1)a)| < C for all n and Ψn − Φn → 0 uniformly on the interval of
integration as n → ∞ by (5.2.7), the last term tends to 0 in this limit. For the
first term on the right-hand side, we observe that [Φn u((n − 1)a)]1 is a real-valued
solution 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 C2
with a constant C which only depends on [λ , λ ].
Hence we see that, for sufficiently large x > 0, (4.9.2) will be satisfied with
k(x) = C 2 x and c = C /2C 4 . The assertion now follows by Theorem 4.9.1.


5.3 Gap eigenvalues
We now turn to the instability intervals of the periodic system. As we have seen
in section 4.5, the essential spectrum of the unperturbed periodic operator on the
half-line has gaps coinciding with the instability set I, with each gap containing
either no spectrum at all or only a single eigenvalue. We shall now show that, when
a perturbation is added which tends to 0 at ∞, the qualitative picture remains
unchanged; indeed, each instability interval contains only discrete eigenvalues and
thus 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, the
question whether the whole instability interval contains only finitely many eigen-
values, or eigenvalues which accumulate at one or both of its end-points, is more
subtle and will be considered in section 5.4.
Regarding the number of eigenvalues in a given subinterval of an instability
interval, we then observe that, in the adiabatic or homogenisation limit where the
perturbation (which is assumed to be continuous) varies on a very long scale com-
pared to the period, the number of eigenvalues generally increases asymptotically
linearly in the scaling parameter and has a limit density which can be conveniently
expressed 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 dx
has 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 be
an 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 problem
for 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)

5.3. Gap eigenvalues 169

and 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, of
the instability interval in which both λ − δ and λ + δ lie. Then, noting that for
x ≥ 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
u
spectral 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)
a
asymptotically 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 total
spectral 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 general
matrix-valued perturbations.
Theorem 5.3.2. Let [λ , λ ] ⊂ I, where I is the instability set of the periodic Dirac
equation (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 )
˜ ˜ ˜
dx
has at most finitely many eigenvalues and no other spectrum in [λ , λ ].


Proof. We proceed in analogy to the proof of Theorem 5.3.1. Again, there is δ > 0
such that [λ − δ, λ + δ] ⊂ I. The Pr¨fer equation (2.2.2) for the perturbed Dirac
u
system takes the form
T
sin θ ˜ sin θ
θ = B(x) + B(x) + λ I ,
cos θ cos θ

where
−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 θ

for any θ and x ≥ x0 . Theorem 2.3.1 (a) then implies that the Pr¨fer angle θ of
u
the perturbed equation with initial value θ(0) = α can be estimated above and
below 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). The
remainder 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
u
angles under perturbations to eventually replace the perturbation with a constant
and then apply the growth asymptotic of the Pr¨fer angle for the periodic equation,
u
as obtained in section 2.4. The same idea can be adapted to estimating how many
eigenvalues appear in any subinterval of an instability interval under the influence
of a continuous perturbation in the limit of slow variation. More precisely, given
a 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 decreases
towards zero as c increases, which means that the perturbation changes ever more
slowly on the length scale defined by the period a. This limit is related to the
adiabatic limit in quantum mechanics, which refers to perturbations changing
slowly in time compared to the dynamic time scale of the unperturbed system,
and to the homogenisation limit, in which microscopic material properties, here
represented by the periodic background, are treated by averaging in contrast to
the macroscopic structures.


Specifically for the perturbed periodic Sturm-Liouville equation we have the
following 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 dx
has asymptotic
∞
c
N[λ ,λ ] ∼ k(λ − q (r)) − k(λ − q (r)) dr
˜ ˜ (c → ∞), (5.3.5)
πa 0

where 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 such
that |˜(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 in
fact 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) (with
k = 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)
a
in the limit c → ∞.
Also, by the same reasoning as in the proof of Theorem 5.3.1, we find that
for λ ∈ {λ , λ } and x > cr0 ,
nπ
θ(x, λ) − θ(cr0 , λ) = (x − cr0 ) + O(1),
a


the remainder staying bounded as x → ∞, where n is the number of the instability
interval 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 the
number 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=1

Hence, 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=1

The statement of Theorem 5.3.3 now follows by observing that the sums on either
side are lower and upper Riemann sums corresponding to the given dissection for
the integral
r0
k(λ − q (r)) − k(λ − q (r)) dr
˜ ˜
0
and 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 Riemann
sums, as used in the proof of Theorem 5.3.3 above, needs to be replaced with less
tight matrix operator norm estimates. For example, the angular momentum term
arising from the separation in spherical polar coordinates of a three-dimensional
radially 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 is
also singular at 0. In the the following we shall focus on problems with one regular
end-point; see the notes for the doubly singular case.
Moreover, the integral for the asymptotic density of eigenvalues will also
involve values of the rotation number of the periodic equation where not only the
spectral 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 periodic
Dirac 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


in 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 parameter
show 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 Dirac
equation (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 Dirac
operator

d
Hα = −iσ2 + (p1 (x) + p1 (x/c)) σ3 + (p2 (x) + p2 (x/c)) σ1 + q(x) + q (x/c)
˜ ˜ ˜
dx
has 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 there
is r0 > 0 such that |˜(r)| + |˜1 (r)| + |˜2 (r)| < δ for r ≥ r0 . Moreover, there is a
q p p
bound 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 points
0 = s0 < s1 < · · · < sm = r0 such that |sl − sj−1 | < γ. Choose sj ∈ [sj−1 , sj ] and
ˆ
set
c1,j = p1 (ˆj ),
˜ s c2,j = p2 (ˆj ),
˜ s c3,j = q (ˆj ),
˜s
for 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)


in the sense of positive semidefinite matrices. Let θ(·, λ) be the solution of the
Pr¨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 find
that
˜ ˜ ˜ ˜
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)
a
in 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

5.4. Critical coupling constants 175

and 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=1

As in the proof of Theorem 5.3.3, the Pr¨fer angles remain bounded indepen-
u
dently of c on [cr0 , ∞). Furthermore, the above Riemann sums converge to the
corresponding integrals due to the uniform continuity of the integrand, and the
integrand of (5.3.10) vanishes on [cr0 , ∞). Thus, the Relative Oscillation Theorem
4.8.3 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 πa
As > 0 was arbitrary, the statement of Theorem 5.3.4 follows.

5.4 Critical coupling constants
We now turn to the question whether a perturbed periodic Sturm-Liouville or
Dirac operator has a finite or infinite total number of eigenvalues in an instability
interval of the unperturbed periodic equation. The results of section 5.3 have shown
that any compact subinterval of an instability interval contains at most finitely
many eigenvalues. Therefore it only remains to settle the question whether or
not eigenvalues accumulate at an end-point of the instability interval. The answer
depends on the rate of decay of the perturbation. We shall see in the following that
the critical decay rate is x−2 , and that the exact value of the asymptotic constant
at this scale is crucial.
We begin by considering the relative oscillation of a real-valued solution of
the perturbed periodic equation
˜
w = J(B + B + λW ) w (5.4.1)


compared 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 × 2
matrix-valued function, but there will be further restrictions later. By the general
assumption that B and W are symmetric, Tr(J(B + λW )) = 0 throughout; so we
can consider linearly independent, R2 -valued solutions u and v of (1.5.1) whose
Wronskian W (u, v) = 1. Let Ψ = (u, v) be the fundamental matrix formed from
these 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 γ

with a non-zero amplitude function a and a relative angle function γ; from (1.5.1)
ˆ
and (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 γ
Multiplying 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 γ

in 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 difference
between the Pr¨fer angles of w and of the solution u of the unperturbed equation,
u
as the next lemma shows.
Lemma 5.4.1. Let u, v : [0, ∞) → R2 be solutions of (1.5.1) with Wronskian
W (u, v) = 1, and let w : [0, ∞) → R2 be a non-trivial solution of (5.4.1). Let θ
and θ1 be Pr¨fer angles of w and u, respectively, and let γ be defined as in (5.4.2)
u
and such that θ(0) − θ1 (0) and γ(0) − π lie in the same interval [nπ, (n + 1)π] with
2
n ∈ Z.
Then |γ − (θ − θ1 + π )| < π.
2

Proof. Let R, R1 , R2 be the Pr¨fer radii of w, u and v and θ2 a Pr¨fer angle of v.
u u
Then (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 )


which 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 2
Now considering that 1 = W (u, v) = R1 R2 sin(θ1 − θ2 ), we see that the first two
factors on the right-hand side are positive, and that cot(θ1 −θ2 ) is locally absolutely
continuous. Therefore γ is related to θ − θ1 + π by a Kepler transformation, and
2
the assertion follows by Theorem 2.2.1.
In the following we assume that, for x ≥ 1, the perturbation is of the specific
form
˜ 1 ˆ
B(x) = 2 (B + β(x)), β(x) = o(1) (x → ∞), (5.4.5)
x
ˆ
with a constant real symmetric matrix B. As the only condition on β is that it
tends 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 be
the 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 question
whether γ is unbounded or not, and γ is continuous, it is clearly sufficient to
consider the differential equation (5.4.3) for γ on the interval [1, ∞), where it
takes the form
1 ˆ
γ = (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)),
x
whereupon (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


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
x
If 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 an
indicator as γ of the asymptotic boundedness or otherwise of the difference of
Pr¨fer angles of w and u. We now observe that F1 and F2 are a-periodic and
u
therefore the analysis of the differential equation (5.4.9) can be much simplified
by 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 a
as 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→∞


˜
Clearly φ is locally absolutely continuous, and using (5.4.9) and integrating
by parts we obtain
x+a
˜ 1
φ (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 x

In 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 keeping
the same asymptotic order for the remainder term.
˜
The differential equation for φ in Lemma 5.4.2 has asymptotically constant
coefficients. This makes it possible to find a simple criterion to decide whether or
not its solutions are globally bounded.
Lemma 5.4.3. Let C1 , C2 ∈ R, and let h : [1, ∞) → R be locally integrable with
h(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 )2
we can rewrite

˜ C1 + C2 + 1 + (C1 − C2 )2
C1 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)


By the hypothesis on h, and since we assume that C1 C2 = 1/4, there exists a
point 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 discriminant
of the system
ˆ
u = J (B + cB + λW ) u, (5.4.14)
ˆ
where 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).
a2
Proof. 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


where Φ = Φ(·, 0).
Now let Ψ = (u, v) be the fundamental matrix of the unperturbed periodic
equation, as considered above in (5.4.2), with a-periodic or a-semi-periodic u and
v 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 matrices
is 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 ±1

Further, 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 1

Also
ˆ
uT Bu ˆ
uT Bv
ˆ
ΨT B Ψ = . (5.4.19)
T ˆ T ˆ
v Bu v Bv
Taking (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 of
0.
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.


If c/ccrit > 1, then λ(n) is an accumulation point of eigenvalues in In of the
perturbed periodic Sturm-Liouville operator
d d
Hα = − (p ) + q(x) + q (x);
˜
dx dx
if 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 be
expressed 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 if
c/ccrit > 1.
Now let u be a periodic or semi-periodic Floquet solution of the unperturbed
system 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, the
difference of the Pr¨fer angles of u and of w is globally bounded if c/ccrit < 1 and
u
globally unbounded if c/ccrit > 1.
On the other hand, if z is a non-trivial solution of (5.4.1) with spectral
parameter λ ∈ In , then by the reasoning in the proof of Theorem 5.3.1, its Pr¨fer
u
angle differs by no more than a globally bounded error from that of a solution of
the unperturbed equation with the same spectral parameter λ. The Pr¨fer angles
u
of solutions of the unperturbed equation with spectral parameter in the closure In
of the instability interval all have the same asymptotics (2.4.1), with only bounded
errors.
Hence we conclude that the difference of the Pr¨fer angles of w and of z is
u
globally bounded if c/ccrit < 1 and globally unbounded if c/ccrit > 1. The assertion
of 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 at
the two end-points of the same instability interval, at least one of the end-points is
always in the subcritical case in the situation of Theorem 5.4.5. Hence eigenvalues
can only accumulate at either the upper or the lower end of the gap in the essential
spectrum, depending on the sign of c. By appeal to the comparison principle for
Pr¨fer angles (Corollary 2.3.2), it is easy to see that perturbations which decay at
u


a faster rate than x−2 only produce finitely many eigenvalues in any gap, whereas
perturbations of fixed sign and slower decay rate than x−2 always generate an
infinity 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 statement
holds. As in section 5.3, the situation is complicated by the possibility of perturbing
the matrix coefficients p1 and p2 . We denote by D(λ, c1 , c2 ) the Hill discriminant
of the periodic Dirac system (5.3.9). We emphasise that the critical constant in
the next theorem plays a somewhat different role from that defined in Theorem
5.4.5.
Theorem 5.4.6. Let λ(n) be an end-point of an instability interval In of the periodic
Dirac 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 x2
with 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 the
perturbed periodic Dirac operator

d
Hα = −iσ2 + (p1 (x) + p1 (x)) σ3 + (p2 (x) + p2 (x)) σ1 + q(x) + q (x);
˜ ˜ ˜
dx

if ccrit < 1, then λ(n) is not an accumulation point of eigenvalues of Hα .
ˆ
Proof. The matrix B of (5.4.5) takes the form

ˆ −ˆ1 − q −ˆ2
p ˆ p
B=
−ˆ2
p p1 − q
ˆ ˆ

and, comparing (5.4.14) and (5.3.9), we see that D(c) = D(λ(n) − cˆ, cˆ1 , cˆ2 ) and
˜ q p p
consequently

∂ ∂ ∂
˜
|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 for
the difference of the Pr¨fer angles of the solutions u of the unperturbed periodic
u
equation and w of the perturbed periodic equation.


Inside the instability interval In , all the Pr¨fer angles of all solutions of the
u
unperturbed equation have the same asymptotic (2.4.1) as u, with only bounded
errors, and this also holds for the perturbed equation by the same comparison
argument as in the proof of Theorem 5.3.2.
Thus the difference of the Pr¨fer angles of solutions of the perturbed periodic
u
Dirac equation at λ(n) and at some λ ∈ In is globally bounded if ccrit < 1 and
globally unbounded if ccrit > 1, and the assertion of Theorem 5.4.6 follows by the
Relative Oscillation Theorem 4.8.3.

5.5 Eigenvalue asymptotics
We have seen in Theorem 5.4.5 for the perturbed Hill’s equation and in Theorem
5.4.6 for the perturbed periodic Dirac equation that eigenvalues accumulate at an
end-point of an instability interval of the unperturbed equation if the perturbation
has x−2 decay with supercritical asymptotic constant. In the present section, we
conclude the study of perturbed periodic problems by deriving the asymptotic
distribution of the eigenvalues near their accumulation point. We shall use the
oscillation techniques of section 5.4, but the reference equation will be the periodic
equation 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 fixed
reference point inside the instability interval. We wish to count the eigenvalues
between λ and λ, where λ is a point between λ and λ(n) , and derive the leading
asymptotics for this count in the limit λ → λ(n) . Since we assume the supercritical
case, we know that the eigenvalue count will tend to infinity in the limit. By
Theorems 5.3.1 or 5.3.2, it will be finite between any two points inside the same
instability interval, and so the leading asymptotic is clearly independent of the
choice of λ .
Lemma 5.5.1. Let λ(n) be an end-point of an instability interval In of the periodic
equation (1.5.1). Then, for λ ∈ In , the Floquet exponent with positive real part
satisfies

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.

5.5. Eigenvalue asymptotics 185

Proof. 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))2

for λ ∈ In by (1.4.6), we see from (5.5.4) that (Re μ(λ))2 = O(|λ − λ(n) |), and
hence
|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 instability
interval 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 without
loss 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 conclude
that (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 by
Φ(·, 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].


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
u
of a real-valued solution, with θ(0, λ) = α; here α ∈ [0, π) is the parameter of
the boundary condition at 0. By the Relative Oscillation Theorem 4.8.3, we only
need to estimate θ(x, λ) − θ(x, λ(n) ) in the limit as x → ∞ in order to count the
eigenvalues between λ and λ up to a bounded error.
˜
Since the perturbation B(x) tends to 0 as x → ∞, it follows by Sturm
Comparison, as in the proof of Theorem 5.3.2, that θ(x, λ) grows regularly as nπx , a
up 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 point
r(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 be
determined 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.
ˆ
Indeed, B = 0 would mean that F2 = 0 in (5.4.8) and hence C2 = 0 in Lemma
5.4.2. Then C1 C2 = 0 < 1/4 in Lemma 5.4.3; this is the subcritical case without
accumulation of eigenvalues at λ(n) and not of interest here.
ˆ
Lemma 5.5.2. Let B be a non-zero symmetric 2 × 2 matrix with real entries and
β 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))
x2

and
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)


as 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) . Taking
into account the regular growth behaviour of θ(r(λ), λ ), we conclude that, up to
an error bounded uniformly in λ ∈ In , the number of eigenvalues between λ and
λ is given by
1 nr(λ)
θ(r(λ), λ) − . (5.5.6)
π a
On the other hand, we know from Theorem 2.4.1 that the Pr¨fer angle θ0 (x, λ) of
u
a 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),
a
with 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 of
the perturbed equation with spectral parameter λ compared to a solution of the
unperturbed equation with the same spectral parameter. By Lemma 5.4.1, this
difference can be estimated, up to a universally bounded error, by the growth of
a 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 not
be periodic or semi-periodic, but a Floquet solution with Floquet multiplier of
modulus |eμ(λ) | > 1.
For the coefficients f and g in (5.4.6) — which now depend on λ, too —, we
have

f (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 Floquet
solution 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 and
g (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 Kepler
transformation
e2 Re μ(λ)x/a
tan ψ(x) = (tan γ(x) + g(x))
x


and 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)
x
Defining 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)
ˆ

we 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 x2

The remainder term is uniform in λ ∈ In ; indeed, the combinations of f , u and
the exponential in the last two terms of (5.5.7) are a-periodic, hence bounded, in
x and continuous in λ.
We shall now apply an averaging procedure analogous to that of Lemma
5.4.2. There is the essential difference that (5.5.8) has a non-decaying leading
term; however, we are here concerned with the limit λ → λ(n) , and the x decay
enters only indirectly as r(λ) → ∞ in that limit.
We now define
˜ 1 x+a
ψ(x) = ψ (x ≥ 1)
a x
and 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


for 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 purpose
of counting eigenvalues up to bounded error. Defining the continuous functions
a
1
Cj (λ) = Fj (t, λ) dt (λ ∈ In ; j ∈ {1, 2}),
a 0

we 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)
x2
In view of (5.5.1), the first term on the right-hand side of (5.5.10) is of order
O( |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 the
estimate
1 1 1
Re μ(λ) Ounif + Ounif = O( |λ − λ(n) |) + Ounif .
x x2 x2
Similarly, e−2 Re μ(λ)x/a uT βu = ounif (1). Thus we can write (5.5.10) more briefly
in the form
1 1
ψ (x) = (C1 (λ) cos2 ψ−sin ψ cos ψ+C2 (λ) sin2 ψ)+O( |λ − λ(n) |)+ounif
˜ ˜ ˜ ˜ ˜ .
x x
The 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.


Lemma 5.5.3. Let Λ be a closed interval with end-point λ(n) , and assume that
C1 , 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 absolutely
continuous 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) ).
4
Proof. In analogy to the beginning of the proof of Lemma 5.4.3, we choose a
continuous 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))2

and 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)


Then, by Theorem 2.2.1,
⎛ ⎞
Γ− (λ) ⎝ Γ+ (λ) G(x, λ) ⎠
ω (x, λ) =
˜ +
Γ+ (λ) 2x cos 2 ω(x, λ) + Γ− (λ) sin2 ω(x, λ)
Γ+ (λ)

sgn(C1 (λ) + C2 (λ))
= 4 C1 (λ) C2 (λ) − 1
2x
Γ− (λ) Γ+ (λ)
+ cos2 ω (x, λ) +
˜ sin2 ω (x, λ) G(x, λ).
˜
Γ+ (λ) Γ− (λ)
(5.5.15)

By (5.5.14), we can estimate

r(λ)
Γ− (λ) Γ+ (λ)
cos2 ω (x, λ) +
˜ sin2 ω (x, λ) G(x, λ) dx
˜
1 Γ+ (λ) Γ− (λ)
ˆ
|B| + sup |β(x)|
x≥r1 (λ) r(λ)
≤ S(λ) c |λ − λ(n) | + S(λ) h,
|λ − λ(n) | 1

Γ− Γ+ ˆ
where S = Γ+ + Γ− . Both S(λ) and |B| + sup |β(x)| remain bounded as
x≥r1 (λ)
λ → λ(n) , and by l’Hospital’s rule
x
h1
lim = lim x h(x) = 0,
x→∞ log x x→∞

and so
r(λ)
h = o(log r(λ)) (λ → λ(n) ).
1

From (5.5.11),

4 C1 (λ) C2 (λ) − 1 = 4 C1 (λ(n) ) C2 (λ(n) ) − 1 + O( |λ − λ(n) |) (λ → λ(n) ),

and

1
log r(λ) = − log |λ − λ(n) | + log ˆ
|B| + sup |β(x)| + |λ − λ(n) |
2 x≥r1 (λ)

1
=− log |λ − λ(n) | + O(1) (λ → λ(n) ).
2


Thus, 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 periodic
Sturm-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’s
equation (1.5.2) with w = 1, and let λ ∈ In . Moreover, let α ∈ [0, π) and let
q 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 perturbed
periodic Sturm-Liouville operator
d d
Hα = − (p ) + q(x) + q (x)
˜
dx dx
has asymptotic
1 c
N (λ) ∼ − 1 log |λ − λ(n) | (λ → λ(n) ). (5.5.16)
4π ccrit

Theorem 5.5.5. Let λ(n) be an end-point of an instability interval In of the periodic
Dirac 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 x2
with 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 ∂c2
Then the number N (λ) of eigenvalues between λ and λ of the perturbed periodic
Dirac operator
d
Hα = −iσ2 + (p1 (x) + p1 (x)) σ3 + (p2 (x) + p2 (x)) σ1 + q(x) + q (x);
˜ ˜ ˜
dx

5.6. Chapter notes 193

has asymptotic
1 √
N (λ) ∼ ccrit − 1 log |λ − λ(n) | (λ → λ(n) ).
4π

5.6 Chapter notes
§5.2 Corollary 5.2.3 was shown by Stolz [175] for the one-dimensional Schr¨dinger
o
operator, 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 the
ensuing conditions on the perturbations of the coefficients of the Dirac and Sturm-
Liouville operators — are satisfied by absolutely integrable functions, functions of
bounded variation multiplied with a-periodic functions, and linear combinations
of these. The results of this section carry over to the case where the left-hand
end-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) is
then a two-dimensional extension of the direct sum of one-dimensional restrictions
of the two part-interval operators with the boundary condition strengthened to
the 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 multiplicity
of the spectrum of the operator on (a, b) differs by no more than 4 from the sum
of the spectral multiplicities of the part-interval operators.
Theorem 5.3.3 is due to Sobolev [170], who considers a strong-coupling limit
of 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 is
singular at 0 but can be shown to generate no eigenvalues in any given compact
λ-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 accurate
indication 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) explains
the appearance of dense point spectrum in radially periodic higher-dimensional
Schr¨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
q
announced by Rofe-Beketov [150]: all sufficiently distant gaps contain at most 2
eigenvalues, 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
˜


in [21]. The existence of eigenvalues in spectral gaps of perturbed Schr¨dinger
o
operators was also shown by Deift and Hempel [32] in a more general setting; see
also [62].
§5.4 Theorem 5.4.5 goes back to Rofe-Beketov (first announced in [151, Theorems
9, 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-Liouville
case by relating it to a different Sturm-Liouville equation by means of an implicit
variable transformation. The oscillation properties of the resulting equation can
then 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 of
finiteness or infinity of the gap spectrum in the limiting case c = ccrit of Theorem
5.4.5. If q (x) ∼ cx2 + x2 (log x)2 , then λ(n) is an accumulation point of eigenvalues
˜ crit c

if 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
˜ crit

a whole hierarchy of asymptotic terms with associated critical coupling constants,
see [121], analogous to the classical oscillation criterion of Kneser [111] as refined
by Weber, Hartman and Hille (cf. [79, Chapter XI, Exercise 1.2]) — this is the
special 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 and
q is constant, then ccrit = −1/4.
This corresponds to the constant in Hardy’s inequality. Indeed, separation
of variables in polar coordinates of the Laplacian −Δ in R2 gives rise to a direct
sum decomposition of this operator with half-line Sturm-Liouville operators
2 1
d2 − 4
− +
dr2 r2
as terms, where ∈ {0, 1, 2, . . . } and r is the radial variable; and = 0 is just the
borderline case of Theorem 5.4.5 — which must be subcritical since the Laplacian
has no eigenvalues. A curious phenomenon appears when a non-constant radially
periodic potential q(r) is added to the Laplacian [160]: if λ0 is the infimum of
y
the essential spectrum, the Floquet solution y corresponding to u = can be
y
chosen strictly positive, and d’Alembert’s formula (1.9.3) can be used instead of
the Rofe-Beketov formula (5.4.6), giving F1 = 1/y 2 . Then
−c a a
1
C1 C2 = y2 ,
a2 0 0 y2
and it follows by the Cauchy-Schwarz inequality that ccrit ≥ − 1 with equality
4
if and only if y 2 and y −2 are linearly dependent, i.e. if q is constant. Hence the
two-dimensional Schr¨dinger operator with non-constant rotationally symmetric
o
potential 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

5.6. Chapter notes 195

criterion 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], see
also [172]. The critical case also gives rise to a ladder of asymptotic scales for the
perturbations, as for the Sturm-Liouville equation.
§5.5 Theorem 5.5.4 was proven in [161] by the method shown in section 5.5. The
analogous statement for the perturbed periodic Dirac operator, Theorem 5.5.5, is
new.
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
d2
potential 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 that
this 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 independent
of c, but there is a critical constant ccrit = 1/4 in the sense of Theorem 5.4.5. For
this 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 subcritical
case.
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, Zelenko
derived an m-term asymptotic series for the number of eigenvalues close to an
end-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 0

For 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
o
asymptotics, generalising the former by the inclusion of a periodic background,
the latter by allowing critical decay of the perturbation.

Periodic differential operators

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