WKB AND SPECTRAL ANALYSIS
OF ONE-DIMENSIONAL SCHRÖDINGER OPERATORS
WITH SLOWLY VARYING POTENTIALS
MICHAEL CHRIST AND ALEXANDER KISELEV
Abstract. Consider a Schrödinger operator on L2 of the line, or of a half line
with appropriate boundary conditions. If the potential tends to zero and is a finite
sum of terms, each of which has a derivative of some order in L1 + Lp for some
exponent p < 2, then an essential support of the the absolutely continuous spectrum
equals R+ . Almost every generalized eigenfunction is bounded, and satisfies certain
WKB-type asymptotics at infinity. If moreover these derivatives belong to Lp with
respect to a weight |x|γ with γ > 0, then the Hausdorff dimension of the singular
component of the spectral measure is strictly less than one.
1. Introduction
The semiclassical WKB method, first proposed in [29, 13, 2], is one of the most
widely used methods for approximating the wave function of a particle, and is a
part of most textbooks on quantum mechanics. The principal requirement for its
applicability is that the electric field potential V (x) vary slowly, allowing certain
terms involving its derivatives to be neglected, provided that the difference E − V (x),
where E is the energy, is bounded away from zero. The literature on mathematically
rigorous applications of the WKB idea is enormous. As an example we refer to [3, 10]
for the construction of modified wave operators for long-range decaying potentials;
see [19] for many further references. A typical requirement, for example, is that the
potential satisfy both |V (x)| ≤ C(1 + |x|)−1/2−² and |Dα V (x)| ≤ C(1 + |x|)−3/2−² for
any |α| = 1; more slowly decaying potentials require stronger decay assumptions for
derivatives of higher order.
In this paper, we extend the scope of the WKB method to analyze the asymptotic behavior at infinity of generalized eigenfunctions in one dimension, in the case
where the potential may decay only very slowly or not at all, but where at least one
derivative decays at a moderate rate, including (1 + |x|)−α for 1/2 < α < 1. We
establish WKB-type asymptotics for almost every energy (with respect to Lebesgue
measure), even though there can occur an everywhere dense set of energies for which
such asymptotics fail to hold; the decay rates hypothesized are very nearly optimal
Date: August 7, 2000. Revised October 16, 2000.
The first author was supported in part by NSF grant DMS-9970660 and performed part of this
research while on appointment as a Miller Research Professor in the Miller Institute for Basic
Research in Science.
The second author was supported in part by NSF grant DMS-9801530.
1
2
MICHAEL CHRIST AND ALEXANDER KISELEV
in this respect. This paper is one of a series of works [5, 6] treating such asymptotics
with unstable parameter dependence.
As a corollary, we deduce the existence of absolutely continuous spectrum, and an
upper bound on the dimension of the singular continuous spectrum. Thus certain
parts of the spectrum are stable under perturbations of the free Hamiltonian (V = 0)
by suitably slowly varying potentials.
Denote by `p (L1 )(R) the Banach space of all (equivalence classes of) measurable
functions from R to R for which the norm
à ∞
!
´p 1/p
X ³ Z k+1
kf k`p (L1 ) =
|f (x)| dx
k=−∞
k
is finite. This Banach space contains L1 + Lp . If p ≤ q, then `p (L1 ) ⊂ `q (L1 ).
Throughout the paper, we assume the following conditions (the hypotheses of Theorems 1.3 and 1.4 are more restrictive). Let n ≥ 0 be a nonnegative integer, and let
p ∈ [1, 2) be an exponent. Let V be a measurable, real-valued function defined on the
real line R. Suppose that V admits a decomposition V = V0 + Vn where1 V0 ∈ `p (L1 ),
Vn is continuous and bounded, and dn Vn /dxn ∈ `p (L1 ), in the sense of distributions.
Define esslimsupx→+∞ V (x) to be lim supx→+∞ Vn (x). Define essliminf x→+∞ V (x),
and the corresponding quantities with +∞ replaced by −∞, likewise. Let H =
−d2 /dx2 + V (x). In the following statement, “almost every” means with respect to
Lebesgue measure.
Theorem 1.1. Under the above hypotheses, for almost every E > esslimsupx→+∞ V (x),
each solution of the generalized eigenfunction equation Hu = Eu is a bounded function of x ∈ [0, +∞).
In addition, for almost every E in this same interval, WKB-type asymptotics
hold in the sense that there exists a generalized eigenfunction u(x, E) satisfying
u(x, E) · e−iΨ(x,E) → 1 as x → +∞, where Ψ is a certain complex-valued function constructed from the potential by an explicit, though somewhat complicated,
recipe dependent on the index n in the hypotheses. The exponent Ψ has relatively
tame behavior, in contrast to u(x, E); ∂Ψ/∂x and all its partial derivatives with
respect to x, E are uniformly bounded for x ∈ [0, ∞) and E in any compact subinterval of (esslimsupx→+∞ V (x), ∞). The complex conjugate of u is a second, linearly
independent solution, with corresponding asymptotics.
2
d
Theorem 1.2. Under the above hypotheses, for the Schrödinger operator H = − dx
2+
2
V (x) acting on L ([0, ∞)), with some self-adjoint boundary condition at the origin, an
essential support for the absolutely continuous spectrum of H is [esslimsupx→+∞ V (x), ∞).
Moreover, the essential spectrum coincides with [essliminf x→+∞ V (x), +∞), and is
purely singular in the interval from essliminf x→+∞ V (x) to esslimsupx→+∞ V (x).
d2
2
Let H = − dx
2 + V (x) acting on L (R), and define A± = esslimsupx→±∞ V (x),
A = max(A+ , A− ), and a = min(A+ , A− ). Then an essential support for the absolutely continuous spectrum of H is [a, +∞]. The absolutely continuous spectrum has
1This
includes P
any potential decomposable as
n
k ≥ 0, and where k=1 Vk is bounded.
Pn
k=0
Vk where dk Vk /dxk ∈ `p (L1 )(R) for each
ONE-DIMENSIONAL SCHRÖDINGER OPERATORS
3
multiplicity two in [A, ∞), and has multiplicity one in [a, A]. The essential spectrum
of H coincides with [essliminf |x|→∞ V (x), ∞), and is purely singular in the interval
from essliminf |x|→∞ V (x) to a.
As is well known [26], the almost everywhere boundedness of the generalized eigenfunctions asserted in Theorem 1.1 implies the assertions of Theorem 1.2 concerning
the absolutely continuous spectrum; see [8] for an alternative approach to this implication. The assertions concerning the absence of absolutely continuous spectrum are
proved as was done for the case n = 1 in [6].
The first related result in one dimension was proved by Weidmann [28], who showed
that if the potential is a sum of an L1 function and a function whose derivative is in L1 ,
then the spectrum on the positive semi-axis is purely absolutely continuous. Later,
Behncke [1] and Stolz [25] established a result similar to our Theorems 1.1,1.2 for
p = 1; the spectrum above esslimsup|x|→∞ V (x) is also purely absolutely continuous
in that case. The price to be paid for a more general class of potentials is that the
WKB asymptotics can actually fail to hold for a dense set of energies having positive
Hausdorff dimension [22], and moreover, the point spectrum can be dense in [0, ∞)
[16, 23].
The conclusions of Theorems 1.1 and 1.2 are false for p > 2; Pearson [18] has
exhibited potentials satisfying V (x) → 0 as |x| → ∞, for which every derivative of
V belongs to Lp for every p > 2, yet the spectrum on the positive semi-axis is purely
singular; see also [12]. Theorem 1.1 remains open for p = 2.
For 1 < p ≤ 2, the conclusion concerning the absolutely continuous spectrum
had been independently conjectured by Molchanov, Novitskii and Vainberg [15], who
proved by a different method that a support for the absolutely continuous spectrum
coincides with (0, ∞), provided that dn V /dxn ∈ L2 , under the supplementary hypothesis that V ∈ Ln+2 [15]. Certain partial results had also been obtained by Killip
[11]. For n = 0, the main conclusion of Theorem 1.2 was obtained by Deift and Killip
[9] under the hypothesis V ∈ L2 + L1 .
For potentials with more rapidly decaying derivatives, our conclusions can be
strengthened. Define p0 = p/(p − 1).
Theorem 1.3. Suppose that n ≥ 0, 1 ≤ p ≤ 2, 0 < γ, and γp0 ≤ 1. Let V be a
measurable, real-valued function defined on R. Suppose that V = V0 + Vn where Vn
is bounded and continuous, and both (1 + |x|)γ V0 and (1 + |x|)γ dn Vn /dxn belong to
`p (L1 ). Then every solution of Hu = Eu is a bounded function of x ∈ R+ , for all
E > esslimsupx→+∞ V (x), except for a set of values of E having Hausdorff dimension
≤ 1 − γp0 .
Moreover, except for a set of energies having Hausdorff dimension ≤ 1 − γp0 , there
exists a generalized eigenfunction satisfying u(x, E) exp(−iΨ(x, E)) → 1 as x → +∞.
Theorem 1.4. Suppose that n ≥ 0, 1 ≤ p ≤ 2, 0 < γ, and γp0 ≤ 1. Let V be
a measurable real-valued function defined on R, satisfying the hypotheses of Theod2
2
rem 1.3. Consider the Schrödinger operator − dx
2 + V (x) acting on L ([0, ∞)), with
some self-adjoint boundary condition at the origin. Then the Hausdorff dimension of
4
MICHAEL CHRIST AND ALEXANDER KISELEV
the singular component of its spectral measure in the interval (lim supx→+∞ V (x), ∞)
does not exceed 1 − γp0 .
For a Schrödinger operator acting instead on L2 (R), the singular component of its
spectral measure has dimension ≤ 1 − γp0 in the interval from min[lim supx→+∞ V (x),
lim supx→−∞ V (x)] to +∞.
For n = 0, this was proved by Remling [21] under a power decay hypothesis V (x) =
O(|x|−α ). Remling [22] also constructed examples for which WKB asymptotics fail
to hold, for a set of energies of dimension equal to 2(1 − α), precisely consistent with
the above bound 1 − γp0 , but it remains unproven that singular continuous spectrum
of positive dimension can actually occur for power decaying potentials. Theorem 1.4
is a consequence of Theorem 1.3; see for instance [8] for a general criterion which
yields this implication.
Theorem 1.1 can be viewed as a nonlinear analogue of a basic property of the
Fourier transform. Menshov, Paley, and Zygmund
showed (in
¯ R x −iλy
¯ different versions)
p
¯
that if 1 ≤ p < 2 and V ∈ L (R), then supx 0 e
V (y) dy ¯ is finite for almost
every λ ∈ R; the nonlinear analogue is the almost everywhere boundedness of the
generalized eigenfunctions of the Schrödinger operator with potential V . Now if
instead, V is a bounded function for which dn V /dxn ∈ Lp , then again the conclusion
of Menshov, Paley, and Zygmund is valid, and is a direct consequence of the case
n = 0 by the relation dn\
V /dxn (λ) = in λn Vb (λ) (or integration by parts). Sums of
finitely many functions V are equally easily handled. For the nonlinear analogue, we
know of no such simple way to deduce the case n > 0 from n = 0, nor to conclude
that V1 + V2 can be handled if V1 , V2 can be separately treated.
γ
p
1
Likewise, Theorem 1.3 is related
R y −ixλto the fact that if (1 + |x|) · f ∈ ` (L )(R) for
some p ∈ [1, 2], then limy→∞ 0 e
f (x) dx exists for all λ except for an exceptional
set of Hausdorff dimension ≤ min(1 − γp0 , 0). This fact follows from a simpler version
of our analysis.
In an earlier paper [6], we proved Theorems 1.1 and 1.2 for n = 1, developing
a general method for treating certain kinds of multilinear expansions on which the
present paper is also based. The principal new ingredient needed to make our multilinear machinery applicable for n > 1 is a sufficiently good WKB-type approximation
exp[iΨ(x, E)] to the generalized eigenfunctions. A second novelty is that whereas for
n = 1 we had been led [6] to multilinear operators mapping functions of x to functions of E, we now obtain operators mapping functions of (x, E) to functions of E;
however, this second point had also arisen in [6], in the proof of a somewhat artificial
supplementary theorem, in which V itself was allowed to depend on E. In the present
paper we discuss the new points in detail, and merely outline the less novel remainder
of the argument.
2. Simplifying a first-order system
Our first step is to write an ordinary differential equation −g 00 + U g = 0 as a
first-order system, and to record certain transformations that bring the system into
a nearly diagonalized form. Let φ(x) be a complex-valued function to be determined
ONE-DIMENSIONAL SCHRÖDINGER OPERATORS
5
later. Introduce the quantity
E(x) = iφ00 − (φ0 )2 − U .
Write the equation for g as y 0 = M y where
µ ¶
g
and
(2.1)
y(x) =
g0
Substitute y = Az where
µ
A=
µ
M=
eiφ
e−iφ̄
iφ0 eiφ −iφ̄0 e−iφ̄
¶
0 1
.
U 0
¶
.
Then z 0 = Bz where B = A−1 M A − A−1 A0 . We have
¶
µ 0 −iφ
−ie−iφ
−1
0
0 −1 φ̄ e
A = (φ + φ̄ )
φ0 eiφ̄
ieiφ̄
µ 0 iφ
¶
iφ e
−iφ̄0 e−iφ̄
MA =
U eiφ
U e−iφ̄
µ
¶
0
0
0
MA − A =
−Eeiφ −Ēe−iφ̄
and therefore
µ
¶
iE
iEe−i(φ+φ̄)
B = (φ + φ̄ )
.
−iEei(φ+φ̄)
−iE
Let ρ be another auxiliary function to be specified later, and substitute z = Λu
where
µ ρ
¶
e 0
Λ=
0 eρ̄
to obtain
0
0 −1
u0 = Du
(2.2)
with D = Λ−1 BΛ − Λ−1 Λ0 . Since
0 −1
BΛ = (2 Re φ )
we have
µ
iEeρ
iĒe−i(2 Re φ)+ρ̄
·
−iEei(2 Re φ)+ρ
−iĒeρ̄
,
¶
iE − (2 Re φ0 )ρ0
iĒ exp(−2i Re φ − 2i Im ρ)
.
·
−iE exp(2i Re φ + 2i Im ρ)
−iĒ − (2 Re φ0 )ρ̄0
µ
0 −1
D = (2 Re φ )
Choosing
(2.3)
¶
Z
x
ρ(x) = i
0
E
2 Re (φ0 )
eliminates the diagonal entries, yielding
µ
¶
0
iĒ exp(−2i Re φ − 2i Im ρ)
0 −1
D = (2 Re φ )
.
−iE exp(2i Re φ + 2i Im ρ)
0
6
MICHAEL CHRIST AND ALEXANDER KISELEV
More succinctly,
(2.4)
Ã
D=
0
−iE
2 Re φ0
eih
iE
2 Re φ0
where
(2.5)
!
0
Z
x
h(x) = 2 Re φ +
0
is purely real-valued.
Define
(2.6)
e−ih
Re E
Re φ0
Z
Ψ = φ − iρ = φ(x) +
1
2
x
0
E
.
Re φ0
Then
iΨ0 =
(2.7)
−φ00 − iU + i|φ0 |2
.
2 Re φ0
Indeed,
(iφ0 + ρ0 ) · 2 Re φ0 = iφ0 · 2 Re φ0 + i[−(φ0 )2 + iφ00 − U ]
= iφ0 · 2 Re φ0 − i(φ0 )2 − φ00 − iU
= −φ00 − iU + i(Re φ0 + i Im φ0 )(2 Re φ0 )
− i[(Re φ0 )2 − (Im φ0 )2 + 2i Re φ0 · Im φ0 ]
= −φ00 − iU + i[2(Re φ0 )2 − (Re φ0 )2 + (Im φ0 )2 ] .
A solution u of u0 = Du gives rise to a solution y of y 0 = M y by the substitution
y = AΛu, that is,
µ iΨ
¶
e
e−iΨ̄
(2.8)
y=
u.
iφ0 eiΨ −iφ̄0 e−iΨ̄
Let M, E, Ψ, D be related to U, φ as above.
Lemma 2.1. Let a potential U be given. Suppose that there exists a continuous
complex-valued function φ such that log | Re φ0 | is a bounded function on [0, ∞). Then
the function Ψ defined by (2.6) has bounded imaginary part.
If in addition φ0 and each solution of u0 = Du are bounded on [0, ∞), then each
solution of y 0 = M y is likewise bounded.
d
d
Proof. (2.7) implies that dx
Re (iΨ) = − 21 dx
log | Re (φ0 )| , whence the first conclusion.
The second then follows from (2.8).
¤
3. Splitting the potential
Let V be as in Theorem 1.1. Fix an auxiliary function η ∈ C0∞ (R) having compact
support, identically equal to one in some neighborhood of the origin, and whose
inverse Fourier transform is real-valued. Decompose
c (ξ) = Vb (ξ) · η(ξ).
(3.1)
V = W + Ṽ where W
ONE-DIMENSIONAL SCHRÖDINGER OPERATORS
7
Then W, Ṽ are real-valued. This decomposition will reduce matters to the analysis
of sums of only two types of potentials.
The proof of the next elementary observation is left to the reader.
Lemma 3.1. W ∈ C ∞ ∩ L∞ , and for every k ≥ 1, dk W (x)/dxk → 0 as |x| → ∞.
Moreover dk W/dxk ∈ Lp for every k ≥ n, while Ṽ ∈ `p (L1 ).
The following routine refinement will be needed; we include a proof for the reader’s
convenience.
Lemma 3.2. For each 1 ≤ k < n, dk W/dxk ∈ Lqk ∩ L∞ where qk = pn/k.
Proof. Let 1 ≤ k < n. We claim that if f is bounded and f (n) ∈ Lp (R), then
f (k) ∈ Lqk . To begin, observe that there exists C < ∞ such that for any smooth
function f and any interval I,
(3.2)
kf (k) kL∞ (I) ≤ C|I|−k kf kL∞ (I) + C|I|n−k−1 kf (n) kL1 (I) .
By scaling, it suffices to establish this for intervals of length one. Suppose that the
right-hand side is ≤ 1, and the left-hand side is large. f (n−1) may be decomposed
as a polynomial of degree zero, plus a function whose supremum over I is a priori
bounded. Iterating this, f (k) may be decomposed as a polynomial of degree ≤ n − k,
plus a function whose supremum over I is a priori bounded. Since the L∞ norm of
f (k) is large, this last polynomial must be large in L∞ (I) norm. Consequently f (k−1)
may be decomposed as a polynomial of degree ≤ n − k + 1 with large norm in L∞ (I),
plus a function whose L∞ (I) norm is a priori bounded. Iterating this, we eventually
conclude that f itself equals a large polynomial of bounded degree, plus a function
whose supremum norm is small; hence kf kL∞ (I) is large, a contradiction.
Fix 1 < p < ∞, let q = np/k, let f be any C n function with f (n) ∈ Lp , and
normalize so that kf kL∞ = 1. Any point x ∈ R belongs to some interval I satisfying
(3.3)
|I|1−n = kf (n) kL1 (I) ;
for if I is taken to be centered at x then the right-hand side is a nondecreasing function
of |I|, while the left-hand side decreases to zero as |I| → ∞. Choose and fix a covering
{Ij } of R consisting of such intervals, such that no point of R belongs to more than
two intervals Ij . By (3.2) and the normalization, kf (k) kL∞ (Ij ) ≤ C|I|n−k−1 kf (n) kL1 (Ij )
for each Ij . By (3.2), (3.3), and Hölder’s inequality, for each J = Ij ,
kf (k) kLq (J) ≤ C|J|1/q kf (k) kL∞ (J) ≤ C|J|1/q |J|n−k−1 kf (n) kL1 (J)
k/n
≤ C|J|1/q |J|n−k−1 kf (n) kL1 (J) · |J|(1−n)(n−k)/n
≤ C|J|1/q |J|n−k−1 |J|(1−n)(n−k)/n |J|
p−1 k
p n
k/n
k/n
kf (n) kLp (J) = Ckf (n) kLp (J) .
Raising this to the power q and summing over j completes the proof.
The result can alternatively be proved via complex interpolation.
¤
8
MICHAEL CHRIST AND ALEXANDER KISELEV
4. Higher-order WKB approximations
We next show how to construct useful approximations exp(iΨ), exp(−iΨ̄) to the
generalized eigenfunctions associated to V , by constructing the auxiliary function φ
of §2. Consider a Schrödinger equation −g 00 + (W − λ2 )g = 0 with some potential
W (x). If g = eiΦ , this equation becomes (Φ0 )2 − iΦ00 + W − λ2 = 0. Writing F = Φ0 ,
this becomes
F 2 − iF 0 + W − λ2 = 0.
(4.1)
Throughout this discussion we assume that the real-valued function λ2 − W (x)
is uniformly bounded below by some fixed strictly positive number, and moreover
that the quantities Fk0 , to be introduced shortly, are uniformly small. It will be
a consequence of our construction that this is the case, under the hypotheses of
Theorem p
1.1, for all potentials W and all k to which this discussion is applied. The
notation λ2 − W (x) + iFk0 (x) thus refers, without ambiguity, to that branch of
the square root function close to the positive square root of the positive quantity
λ2 − W (x).
We approximately solve the preceding equation by recursion: set F0 (x, λ) = λ and
for k ≥ 0,
p
(4.2)
Fk+1 = λ2 − W + iFk0 .
Define the error
(4.3)
Ek = Fk2 − iFk0 + W − λ2 .
Substituting the recursion formula for Fk2 gives the alternative expression
(4.4)
0
Ek = iFk−1
− iFk0 .
A different expression will be more useful for our purpose. (4.3) can
pbe rewritten as
0
2
2
iFk = Fk + (W − λ ) − Ek ; substituting this into (4.2) gives Fk+1 = Fk2 − Ek . Thus
from (4.4) we deduce
q
E
d
d
pk
(4.5)
Ek+1 = i (Fk − Fk2 − Ek ) = i
.
dx
dx Fk + Fk2 − Ek
The first few functions Fk , Ek are:
F0 = λ
E0 = W
√
F1 = λ2 − W
i
E1 = (λ2 − W )−1/2 W 0
2
·
¸1/2
i 2
2
−1/2
0
F2 = (λ − W ) − (λ − W )
W
2

!−1 
Ã
r
d  0 2
i

E2 = − 21
W (λ − W )−1 1 + 1 − (λ2 − W )−3/2 W 0
dx
2
ONE-DIMENSIONAL SCHRÖDINGER OPERATORS
9
In our application, the function φ0 of §2 will be chosen to equal Fn for some n. Our
asymptotic expression for generalized eigenfunctions will thus be
(4.6)
u(x, λ) ∼ exp(iΨ(x, λ))
where
Ψ0n = Fn −
(4.7)
Ṽ + En
.
2 Re Fn
The first two functions Ψ0n are:
Ψ00 = λ − (2λ)−1 V
Ψ01 = (λ2 − W )1/2 − 12 (λ2 − W )−1/2 Ṽ − 4i (λ2 − W )−1 W 0 .
Ψ02 is already rather complicated.
In the next lemma, W is a bounded, continuous real-valued function of x ∈ R,
and K is an arbitrary compact subinterval of (0, ∞), which is to remain fixed, for
the remainder of the proof of Theorem 1.1; all assertions are uniform in λ ∈ K. We
will always assume that supx max [W (x), 0] is as small as may be desired, and that
dk W/dxk is likewise small in L∞ ∩ Lqk for 1 ≤ k < n and in Lp for k ≥ n. In our
eventual application, this can be achieved by replacing the original potential V by a
suitable potential that equals V on [N, ∞) for sufficiently large N .
Lemma 4.1. Suppose that W ∈ L∞ , that dk W/dxk ∈ Lp (R, dx) for every k ≥ n,
and that the supremum of max [W (x), 0] is sufficiently small. Then Fn ∈ L∞ , and
Re (Fn − λ) ≥ −δ where δ > 0 may be taken to be as small as desired. The remainder
term En belongs to Lp (R, dx), and moreover ∂ m En /∂λm ∈ Lp (R, dx) for every m,
uniformly in λ ∈ K.
Proof. Write W (s) = ∂ s W/∂xs . It is a direct consequence of the recursion (4.2) that
0
(k−1)
each Fk (x, λ) may be expressed
(x).
√ as a smooth function of λ, W (x), W (x), . . . , W
2
Moreover, for k ≥ 1, Fk − λ − W will be as small as may be desired in supremum
norm, provided that W 0 , . . . , W (k−1) and maxx (W (x), 0) are all sufficiently small in
the senses detailed above. Q
P
We say that a monomial m (W (m) )dm has weight d =
m dm · m/n. Such a
monomial belongs to Lr (R, dx), where r−1 = d/p, by Lemma 3.2.
Like Fk+1 , Ek may be expressed as a smooth function of λ, W (x), W 0 (x), . . . , W (k) (x).
More precisely, Ek can be expressed as a finite sum of terms H(W, W 0 , . . . , W (k−1) ) ·
P (W, W 0 , . . . , W (k) ) where H is a smooth function in a neighborhood of (−∞, ²) ×
{0, 0, . . . , 0} ⊂ R × Rk−1 for a fixed constant ² = ²(K) > 0, and P is a monomial
of weight exactly k/n. In particular, there can be at most one factor of the highestorder derivative W (k) . This description of Ek follows by induction on k from the
above description of Fk , together with the recursion (4.5). Consequently En ∈ Lp ,
by Lemma 3.2 and Hölder’s inequality.
¤
One consequence is that the real parts of both Fk and Fk2 will be bounded below
by a fixed strictly positive constant.
10
MICHAEL CHRIST AND ALEXANDER KISELEV
5. Combining Ingredients
Fix n ≥ 1 and assume that V satisfies the hypotheses of Theorem 1.1 with this
index n. For E > A+ = esslimsupx→+∞ V (x), rewrite V (x) − E = [V (x) − A+ ] − λ2
where λ > 0. We will henceforth replace V by V − A+ , and may thus assume that
esslimsupx→+∞ V (x) = 0. By modifying this new V only on an interval (−∞, N ],
we may then assume that esslimsupx→+∞ V (x) is smaller than any preassigned positive quantity. Such a modification has no effect on the asymptotic behavior of the
generalized eigenfunctions, as x → +∞.
Now we combine the splitting V = W + Ṽ of the potential, the generalized WKB
approximation of the preceding section, and the computations in §2. Decompose
V = W + Ṽ as in (3.1). Then max(0, supx W (x)) may likewise be taken to be
arbitrarily small.
Let Fn (x, λ) be constructed from W , by iterating the recursion (4.2) to order n,
and let En = Fn2 − iFn0 + (W − λ2 ). As in (4.7), define
!
Z xÃ
Ṽ + En
(5.1)
Ψ(x, λ) =
Fn −
(s, λ) ds .
2 Re Fn
0
Rx
Also define φ(x, λ) = 0 Fn .
Ψ has bounded imaginary part. Indeed, by Lemma 4.1, log | Re φ0 (x, λ)| = log | Re Fn |
is a bounded function of x, for each λ ∈ K, provided that V is sufficiently small, in the
sense described in the first paragraph of this section. By Lemma 2.1, exp(iΨ(x, λ))
is therefore a bounded function of x ∈ R+ , uniformly for every λ ∈ K.
The two functions exp(±iΨ) are linearly independent, and the same holds for any
two perturbations of them that are sufficiently small in the supremum norm near
+∞. Indeed, the main constituent Fn of ∂Ψ/∂x has a real part that is bounded
below by a positive constant. The other part, (Ṽ + En )/2 Re Fn , tends to zero in
the sense that its L1 norm on an arbitrary interval [N, N + 1] approaches zero as
N → ∞. Linear independence thus follows from an elementary argument.
The remainder of the paper is devoted to the proof of the following result, which
was mentioned in the Introduction but not formulated precisely there.
Theorem 5.1. Under the hypotheses of Theorem 1.1, for almost every λ > 0 = A+ ,
there exists a generalized eigenfunction u(x, λ) such that
(5.2)
u(x, λ)e−iΨ(x,λ) → 1
as x → +∞. u and its complex conjugate are linearly independent.
Under the stronger hypotheses of Theorem 1.3, the set of all λ for which (5.2) fails
to hold has Hausdorff dimension less than or equal to 1 − γp0 .
Because Ψ has bounded imaginary part, (5.2) implies boundedness of all generalized eigenfunctions associated to the spectral parameter λ2 , and hence Theorem 5.1
implies the main conclusions of the theorems formulated in the Introduction.
To make use of the results in §2, set U = V − λ2 . Then
E(x, λ) = iφ00 − (φ0 )2 − (V − λ2 ) = −En (x, λ) − Ṽ (x) .
ONE-DIMENSIONAL SCHRÖDINGER OPERATORS
11
We have ∂ m E/∂λm ∈ `p (L1 )(R, dx) for every m ≥ 0, uniformly for each λ in any
compact subset of R, and the same goes for E/ Re φ0 = E/ Re Fn . Define
(5.3)
F(x, λ) = −iE/2 Re Fn .
We know that F(·, λ) ∈ `p (L1 ) uniformly for all λ in any compact subset K of (0, ∞),
and moreover the same goes for ∂λr F(·, λ) for all r. According to (2.5),
Z x
Re E
h(x, λ) = 2 Re Fn +
.
0 Re Fn
In order to demonstrate the first conclusion of Theorem 1.1, it suffices to show
that forµalmost
every λ, there exists a C2 –valued solution u of u0 = Du such that
¶
1
u(x) −
→ 0 as x → +∞. We will deduce this from analytic machinery built
0
up in earlier papers [6, 7]. In the final section of the paper, we will indicate the
modifications needed to treat all λ outside a set of appropriately bounded Hausdorff
dimension, under the stronger hypotheses of Theorem 1.3.
6. A nonlinear Hausdorff-Young inequality
Fix n, let 1 ≤ p ≤ 2, and a compact subinterval K ⊂ (0, ∞). Let h(x, λ) be defined
as in §5. Consider the operator
Z
Sf (λ) =
eih(x,λ) f (x) dx.
R
Lemma 6.1. Let V satisfy the hypotheses of Theorem 1.1, for a certain n ≥ 0 and
p < ∞, with sufficiently small norms. Let s ≤ 2, and let q = s/(s−1) be the exponent
conjugate to s. Then there exists C < ∞ such that for any f ∈ L1 (R),
(6.1)
kSf kLq (K,dλ) ≤ Ckf k`s (L1 )(R) .
Proof. First consider the case s = 2. Let ζ be a real-valued, nonnegative, smooth auxiliary function that is strictly positive on K and is supported in a small neighborhood
of K. Consider
Z ¯Z
¯2
¯
¯
ih(x,λ)
f (x) dx¯ ζ(λ) dλ
¯ e
K
R
¸
¸ ·Z
Z ·Z
ih(x,λ)
−ih(y,λ) ¯
f (y) dy ζ(λ) dλ
e
=
e
f (x) dx
R
K
R
ZZ
Z
=
ei(h(x,λ)−h(y,λ)) ζ(λ) dλ f (x)f¯(y) dx dy.
R×R
K
The hypotheses and earlier lemmas imply that |h(x, λ) − h(y, λ)| is bounded above
and below by positive constants times |x − y|, provided that |x − y| is sufficiently
large, uniformly in λ.
We claim that the inner integral is bounded by C(1 + |x − y|)−2 ; the conclusion
(6.1) then follows directly from this, for q = 2. When |x − y| is bounded, the estimate
is trivial. When |x − y| is large, multiply and divide by ∂λ ((h(x, λ) − h(y, λ)), and
integrate by parts with respect to λ in the inner integral, integrating exp(i(h(x, λ) −
12
MICHAEL CHRIST AND ALEXANDER KISELEV
h(y, λ)) · ∂λ (h(x, λ) − h(y, λ)). Multiply and divide, then integrate by parts once
more, to conclude the proof for s = 2. The case s = 1 is trivial, and the general case
then follows by interpolation.
¤
In our application, S will act on F(x, λ), which itself depends on λ. This will
necessitate some modification; see the proof of Proposition 7.1.
7. Summation of the solution series
In order to prove Theorem 5.1, we now combine the WKB-type Ansatz developed
in §§2-5 with Lemma 6.1 and with the machinery developed in [6, 7]. We seek to solve
¡
¢t
the equation u0 = Du, and to find a solution such that u(x) → 1 0 as x → +∞.
Writing the equation as
µ ¶ Z ∞
1
u(x) =
−
D(y)u(y) dy ,
0
x
we obtain the formal series solution
µ ¶
1
(7.1) u(x) =
0
Z
Z
∞
X
k
+
(−1)
···
k=1
µ ¶
1
D(t1 )D(t2 ) · · · D(tk )
dtk · · · dt2 dt1 .
0
x≤t1 ≤t2 ···≤tk <∞
Introduce multilinear operators
Z
(7.2)
Tm (f1 , . . . , fm )(x, λ) =
m
Y
exp(i(−1)m−k h(tk , λ)) fk (tk ) dtk
x≤t1 ≤t2 ≤···≤tm k=1
for each m ≥ 1. Then the series solution is formally
µ ¶ µ
¶
P∞
1
T2m (F2m,1 , . . . , F2m,2m )
m=1
P
u(x) =
+
∞
0
m=0 T2m+1 (F2m+1,1 , . . . , F2m+1,2m+1 )
where each Fm,j equals either F(x, λ) or its complex conjugate; the precise rule is
of no consequence for our estimates. Even the operator T1 is highly nonlinear, since
both h and F are nonlinear functions of the potential V .
It suffices to show that for almost every λ ∈ K, these two series converge, and
define bounded functions of x ∈ R. The proof below, together with arguments in
[6], then demonstrates that the sum of the vector-valued series does define a solution
¡
¢t
of u0 = Du such that u(x) → 1 0 as x → +∞, hence gives rise to a bounded
solution, not identically vanishing, of the original generalized eigenfunction equation
Hf = λ2 f .
This type of result was treated in our analysis [6] of the case n = 1. The principal
new twist here is that the functions F on which our multilinear operators act, now
depend on λ. This situation is much like that considered in Theorem 1.3 of [6], where
V itself was allowed to depend on λ.
ONE-DIMENSIONAL SCHRÖDINGER OPERATORS
Let
Ω=
1
X
13
k∂ρr F(·, ρ)kp`p (L1 ) .
r=0
As proved in [6], there exist sets Ejm ⊂ R, indexed by 1 ≤ m < ∞, 1 ≤ j ≤ 2m ,
satisfying:
• R = ∪j Ejm for every m.
• Ejm ∩ Ejm0 = ∅ for every j < j 0 .
• If j < j 0 , x ∈ Ejm , and x0 ∈ Ejm0 , then x < x0 .
m+1
m+1
• For every m, j, Ejm = E(2j−1)
∪ E2j
.
r
m p
• For every m, j, k∂ρ F(·, ρ)χj k`p (L1 ) ≤ 2−m Ω for r = 0, 1.
We denote by χE the characteristic function of the set E, and introduce the special
m
m
notation χm
j for the characteristic function of the interval Ej . Fix such sets Ej , for
the remainder of §7.
Define a multilinear operator Mn , acting on n functions gk of (x, λ), by
Z
n
Y
£
¤
0
Mn (g1 , . . . , gn )(x, x , λ) =
gk (tk , λ) dtk .
x≤t1 ≤···≤tn ≤x0 k=1
In the special case when there is a single function g such that each gk is an element
of the set {g, ḡ}, we write simply Mn (g)(x, x0 , λ). In our application, g(x, λ) will
essentially be equal to eih · F . Define
Mn∗ (g1 , . . . gn )(λ) = sup |Mn (g1 , . . . gn )(x, x0 , λ)|
x≤x0 ∈R
Mn∗ (g)(λ) = sup |Mn (g)(x, x0 , λ)| ,
x≤x0 ∈R
(7.3)
G̃(g, λ) =
1 X
∞
X
r=0 m=1
à 2m Z
X ¯¯
m
¯
j=1
Ejm
¯2
¯
g(x, λ) dx¯
!1/2
.
It will be useful to regard G̃ as a linear operator. To do this, introduce the Banach
space B consisting of all complex-valued sequences a = a(m, j) indexed by 1 ≤ m <
³P
´1/2
P
2
∞ and 1 ≤ j ≤ 2m , for which m m
|a(m,
j)|
< ∞. Then G̃(g, λ) equals
j
R
the norm in B of the sequence { E m g(x, λ) dx}.
j
In Proposition 4.2 of [7] and in the proof of Theorem 1.3 of that reference it is
shown2 that
n
Y
∗
n
Mn (g1 , . . . , gn )(λ) ≤ C
(7.4)
G̃(gk , λ)
k=1
(7.5)
Mn∗ (g)(λ) ≤ C n
G̃(g, λ)n
√
n!
√
bound with a factor of 1/ n! also appears in the elementary theory of Volterra integral
equations [27] p. 12; it appears not to be closely related to (7.5).
2A
14
MICHAEL CHRIST AND ALEXANDER KISELEV
√
for some universal constant C < ∞. Moreover, there is likewise a factor of 1/ n! on
the right-hand side of (7.4), if the number of distinct functions gk is bounded by any
fixed constant independent of n. One formal consequence of (7.5) is that the series
solution u of u0 = Du satisfies
sup |u(x, λ)| ≤ C exp(C G̃(λ)2 )
(7.6)
x
where G̃ is as defined in (7.3), with g(x, λ) = exp(ih(x, λ)) · F (x, λ).
Proposition 7.1. Let n, V, p be as in the hypotheses of Theorem 1.1. Let F be as
defined above, and suppose that for each s, k, Fs,k equals either F, or F̄. Then for
each m ≥ 1, for almost every λ ∈ K,
Z
s
hY
i
±ih(tk ,λ)
lim
e
F
(t
,
λ)
dt
s,k k
k
0
x →∞
x≤t1 ≤···≤ts ≤x0
k=1
exists for every x, and
¯
¯
¯
¯
Z
s
i
h
Y
¯
¯
∗
±ih(t
,λ)
k
Ts (λ) = sup ¯¯
e
Fs,k (tk , λ) dtk ¯¯
x ¯
¯
k=1
x≤t1 ≤···≤ts <∞
P
∗
is finite. Finally, ∞
s=0 Ts (λ) is finite for almost every λ ∈ K.
The plus and minus signs in the exponents are not specified; these assertions are valid
for all choices of signs, with uniform bounds.
− 1) > 2 be the
Proof. Fix any compact subinterval K of (0, ∞), and let q =
R p/(p
±ih(x,λ)
exponent conjugate to p. By Lemma 6.1, the mapping f 7→ R e
f (x) dx maps
p
1
q
` (L ) boundedly to L (K). Therefore by Proposition 3.3 of [6]Rand the remark following the proof of Theorem 1.1 of [7], the Lq (K, B) norm of { E m e±ih(x,λ) f (x) dx}
j
is majorized by a fixed constant times the `p (L1 ) norm of f , provided that the col−m
lection of sets Ejm is adapted to f , in the sense that kf · χm
kf k`p (L1 ) .
j k`p (L1 ) ≤ 2
Taking f (x) first equal to F(x, ρ) and thenR equal to ∂ρ F(x, ρ), we conclude that
for r = 0, 1, the Lq (K, B, dλ) norm of {∂ρr E m e±ih(x,λ) F(x, ρ) dx} is majorized by
j
CkFk`p (LR1 ) + Ck∂ρ Fk`p (L1 ) , hence by a finite constant, uniformly for all ρ ∈ K.
Thus ∂ρr E m e±ih(x,λ) F(x, ρ) dx ∈ Lq (K × K, B, dλ dρ) for r = 0, 1. Therefore by the
j
R
one-dimensional Sobolev embedding theorem, K 3 ρ 7→ E m e±ih(x,λ) F(x, ρ) dx is a
j
continuous B–valued function for almost every λ ∈ K, and the supremum over ρ of
its B–norm belongs to Lq (K, dλ). Thus we conclude that
(7.7)
G̃(F(·, λ), λ) ∈ Lq (K, dλ).
¿From (7.5), it follows that the supremum over all pairs x, x0 of
Z
Y
¯
¯
¯
e±ih(tk ,λ) Fs,k (x, λ) dtk ¯
x≤t1 ≤···≤ts ≤x0
ONE-DIMENSIONAL SCHRÖDINGER OPERATORS
15
is finite for almost every λ ∈ K. Existence of the limit, as x0 → ∞, then follows as
in the proof of Proposition
4.1 of [6]. Summability with respect to s holds, because
√
of the factor of 1/ n! in (7.5), as expressed in (7.6).
¤
8. A bound on the set of exceptional energies
Assume that V satisfies the hypotheses of Theorem 1.3 for some 1 < p ≤ 2, γ > 0.
We seek an upper bound on the Hausdorff dimension of the set of all λ for which the
WKB-type asymptotics fail to hold. Suppose that β > 1 − p0 γ, where p0 = p/(p − 1).
We may assume that 0 < β < 1, γ < 1 and p > 1; otherwise there is nothing to
prove, or the result is already known [1]. Let Hβ denote β–dimensional Hausdorff
measure. Fix a compact subinterval K of (0, ∞). Throughout this section, V, F, h
are functions of x ≥ 0.
By its construction, the exponent h(x, λ) defined in (2.5) satisfies |∂ k h(x, λ)/∂λk | ≤
C + Cx, uniformly in λ ∈ K, for every k ≥ 0. Indeed, ∂ k+1 h/∂λk ∂x is bounded.
Likewise, (1 + x)γ times any partial derivative of F(x, λ) with respect to λ belongs to
`p (L1 )(R+ ). This follows from natural analogues of Lemmas 3.1 and 3.2; in particular,
(1 + x)γk/n ∂ k W (x, λ)/∂xk ∈ L∞ ∩ Lpk/n for 0 ≤ k ≤ n, and the same holds for each
of its partial derivatives with respect to λ.
By subtracting a constant from V , we can also assume that esslimsupx→+∞ V (x)
equals 0. As in §7, consider the formal series solution (7.1) of u0 = Du. Define
Z
0
Tm (F1 , . . . , Fm )(x, x , λ) =
m
Y
e±ih(tk ,λ) Fk (tk ) dtk .
x≤t1 ≤t2 ≤···≤tm ≤x0 k=1
Define intervals Ejm ⊂ R+ by the same construction used in §7, but applied to
(1 + x)γ · F(·, λ). To any function F (x, λ) associate the doubly indexed sequence of
numbers
¾
½Z
ih(t,λ)
e
F (t, λ) dt
.
g(F )(λ) =
Ejm
m≥1, 1≤j≤2m
Recall
P§7 that the B–norm of a doubly indexed sequence a(m, j) is defined to
P from
be m m[ j |a(m, j)|2 ]1/2 .
Define F (N ) (x, λ) = F(x, λ) for x ≥ N , and = 0 for x < N . A direct consequence
of the definitions is that
(8.1)
Tm (F, . . . , F)(x, x0 , λ) ≡ Tm (F (N ) , . . . , F (N ) )(x, x0 , λ) for all x, x0 ≥ N.
Throughout this discussion, the exponent h and the sets Ejm appearing in the definitions of Tm and g are defined in terms of the original potential V ; they are independent
of N .
Define
Λc = {λ ∈ K : kg(F (N ) )(λ)kB ≥ c for every N ≥ 0} .
16
MICHAEL CHRIST AND ALEXANDER KISELEV
We will prove that for any c > 0, Hβ (Λc ) = 0. Since by (7.5) and (8.1), whenever
N ≥ M,
∞
X
sup |Tm (F (N ) , . . . , F (N ) )(x, x0 , λ)| ≤ C exp(Ckg(F (M ) )(λ)k2B ),
0
m=1 x,x ≥N
this implies that for any c > 0,
(8.2)
Hβ {λ ∈ K : lim sup
x,x0 →∞
∞
X
|Tm (F(·, λ), . . . , F(·, λ))(x, x0 , λ)| ≥ c} = 0.
m=1
As in the proofs of Proposition 7.1 of this paper, and Proposition 4.1 of [6], that
suffices to establish convergence of the series defining u, and validity of the WKB
asymptotics, for all λ outside a set whose Hβ measure equals zero.
Let q = p0 . We claim that g(F) belongs to the Sobolev space Lqγ of all B–valued
functions having γ derivatives in Lq , in a fixed neighborhood of K. To prove this,
consider the analytic family of functions Fz (x, λ) = (1 + x)z F(x, λ). For Re (z) = γ,
g(Fz ) ∈ Lq , by Lemma 6.1 and Proposition
7.1. For Re (z) = γ − 1, we have
R
±ih(x,λ)
q
Fz (x, λ) dx is differentiated with
∂λ g(Fz ) ∈ L (K, dλ). Indeed, when E m e
j
respect to λ, the derivative falls either on F, or on the exponent h. In the former
case, no harm is done, because each partial derivative of F with respect to λ satisfies
the same bounds as does F itself; moreover, matters are improved by the factor
of (1 + x)z in the definition of Fz , since Re (γ − 1) ≤ 0. In the latter case, F is
replaced by ±i∂λ h · F . Since ∂ k h/∂λk = O(x) for every k ≥ 1, this results in an extra
O(1 + x) factor; when combined with the factor of (1 + x)γ−1 in the definition of Fz ,
this means that we are applying g to a function all of whose λ–derivatives belong
to `p (L1 ) for each λ. Thus Lemma 6.1, in its λ–dependent version developed in the
proof of Proposition 7.1, applies once more.
Moreover, the Sobolev norm of g(F (N ) ) tends to zero as N → ∞. This follows from
three facts. Firstly, in a fixed neighborhood of K, by the discussion
in the preceding
R
paragraph, for any fixed m, j, the scalar-valued function E m eih(t,λ) F (t, λ) dt has
j
P
Lqγ norm bounded by C 1r=0 supλ∈K k(1 + |t|)γ ∂λr F (t, λ)k`p (L1 )(R+ , dt) . Secondly, for
F = F (N ) , the two `p (L1 ) norms in this last expression tend to zero as N → ∞.
Thirdly, the claim of the preceding paragraph remains valid if the norm on B is
£P
¤
P
2
2 1/2
changed so that the norm of a sequence a(m, j) is
(the
mm
j |a(m, j)|
2
weight m has been changed to m ). Indeed, this remains true with any power of m,
as follows from the proof of Proposition 7.1, the argument two paragraphs above,
and [7].
Suppose now that for some c > 0, Hβ (Λc ) > 0. Then by Theorem II.1 of [4], there
exists a finite positive measure µ with µ(Λc ) > 0, satisfying µ(I) ≤ |I|β for every
interval I.
Let N be large. By a potential-theoretic characterization
of Sobolev spaces [24],
R
we conclude that kg(F (N ) )(λ)kB ≤ J ∗ fN (λ) = R J(λ − ρ)fN (ρ) dρ, where fN , J are
nonnegative, fN ∈ Lq (R), kfN kLq → 0 as N → ∞, J(ρ) ≤ C|ρ|γ−1 for all |ρ| ≤ 1,
and J(ρ) ≤ C exp(−c|ρ|) for |ρ| ≥ 1.
ONE-DIMENSIONAL SCHRÖDINGER OPERATORS
17
A simple calculation using the hypothesis β > 1−p0 γ in conjunction with the upper
0
bound on µ(I)Pdemonstrates that J ∗ µ ∈ Lq = Lp . (Decompose J as a Schwartz
∞
−r
function plus
r=0 Jr where Jr (x) is supported where |x| ≤ 2 , and kJr kL∞ ≤
C2r(1−γ) . Estimate the Lp norm of µ ∗ Jr by interpolating between simple L1 and L∞
bounds, then sum over r.) Thus
Z
Z
(J ∗ fN ) dµ = fN · (J ∗ µ) dλ ≤ kfN kLq · kJ ∗ µkLq0 ,
which tends to zero as N → ∞. But by hypothesis and the definition of Λc ,
kg(F (N ) )(λ)kB ≥ c for every λ ∈ Λc and every N . Therefore
Z
Z
(J ∗ fN )dµ ≥ kg(F (N ) )kB dµ ≥ cµ(Λc ) > 0,
a contradiction.
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Michael Christ, Department of Mathematics, University of California, Berkeley,
CA 94720-3840, USA
E-mail address: [email protected]
Alexander Kiselev, Department of Mathematics, University of Chicago, Chicago,
Ill. 60637
E-mail address: [email protected]