STABILITY OF SINGULAR SPECTRAL TYPES UNDER DECAYING

PERTURBATIONS

ALEXANDER KISELEV^{1}, YORAM LAST^{2}, AND BARRY SIMON^{3}

Dedicated to Jean Michel Combes on his 60th birthday

Abstract. We look at invariance of a.e. boundary condition spectral behavior under perturbations, W, of half-line, continuum or discrete Schrödinger operators. We extend the results of del Rio, Simon, Stolz from compactly supported W ’s to suitable shortrange W. We also discuss invariance of the local Hausdorff dimension of spectral measures under such perturbations.

1. Introduction

We want to discuss aspects of the spectral theory of Schrödinger operators on a half-line, both continuous

(Hψ)(x) = −ψ′′(x) + V (x)ψ(x)  (1.1)

on L^{2}(0, ∞; dx) and discrete

(hψ)(n) = ψ(n + 1) + ψ(n − 1) + V (n)ψ(n)  (1.2)

on ℓ^{2}({1, 2, … }) with ψ(0) determined by the boundary condition. These operators have a boundary condition determined by a parameter θ in [0, π):

ψ(0) cos(θ) + ψ′(0) sin(θ) = 0  (1.3)

in the continuum case and

ψ(0) cos θ + ψ(1) sin(θ) = 0  (1.4)

in the discrete case. Thus (1.4) is equivalent to defining

(h_{θ}ψ)(1) = ψ(2) + [V (1) − tan(θ)]ψ(1).

1Supported in part by NSF Grant No. DMS-0102554 and by an Alfred P. Sloan fellowship.

2Supported in part by The Israel Science Foundation Grant No. 447/99 and by an Allon fellowship.

3Supported in part by NSF Grant No. DMS-9707661. Date: June 18, 2002.

In the continuum case, we will suppose that V (x) is locally integrable and sometimes that it is bounded from below for reasons that will become clear. We will need to impose a mild restriction on the growth of the potential in Theorem 1.9 on Lyapunov behavior (both in the continuum and discrete cases).

We will use H_{θ} and h_{θ} to indicate the operators with boundary condition. It is well known (see, e.g., Simon [20]) that there are spectral measures dρ_{θ}(λ) for H_{θ} and h_{θ} (so that H_{θ} or h_{θ} is unitarily equivalent to multiplication by λ on L^{2}(ℝ, dρ_{θ}(λ))) normalized so that

∫_{θ=0}^{π} dρ_{θ}(λ)\frac{dθ}{π} = dλ.  (1.5)

A major theme in this paper (as in many recent papers) is the relation of spectral properties with solutions of the differential/difference equation. Given V and θ, for each λ ∈ ℂ, we will define φ_{1,θ}(λ, x) (or φ_{1,θ}(λ, n)) to be the solution of

Hφ = λφ (or hφ = λφ)  (1.6)

(intended as a differential/difference equation with no L^{2} condition at ∞) obeying the boundary condition (1.3)/(1.4) and normalized by

φ_{1,θ}(λ, 0) = sin(θ) φ′_{1,θ}(λ, 0) = cos(θ)  (1.7)

(or φ_{1,θ}(λ, 1) = cos(θ) in the discrete case). We will also define

φ_{2,θ} ≡ φ_{1,(θ−π/2)}.

While we consider θ ∈ [0, π) in the basic definition of φ_{1,θ}, it makes sense for all θ with φ_{1,θ+nπ} = (−1)^{n}φ_{1,θ}. In particular, in the last equation θ − π/2 lies in [−π/2, π/2). With this definition, the Wronskian obeys

W (φ_{1,θ}, φ_{2,θ}) = 1  (1.8)

with W (f, g) = fg′ − f′g in the continuum case and W (f, g)(n) = f(n)g(n + 1) − f(n + 1)g(n) in the discrete case.

Following Jitomirskaya-Last [9], for L > 0, we define

∥f∥_{L}^{2} = ∫_{0}^{L} |f(x)|^{2} dx

in the continuum case and

∥f∥_{L}^{2} = ∑_{n=1}^{[L]} |f(n)|^{2} + (L − [L])|f([L] + 1)|^{2}

in the discrete case (so ∥f∥_{L}^{2} is the obvious analog at integer L, with linear interpolation in between).

When one looks at the decomposition of dρ_{θ} into spectral types, for example, into a.c., s.c., and pure point pieces (see Reed-Simon [14]), a basic pair of facts says that the a.c. spectrum is stable and the singular spectrum is unstable — explicitly (see Simon [20] for references), the essential support of dρ_{θ}^{ac} is θ independent, while for any pair θ ≠ θ′, dρ_{θ}^{sing} and dρ_{θ′}^{sing} are mutually singular. These facts seem to be at variance with the notion that spectral properties should depend on the behavior of V at infinity since they suggest that dρ^{sing} will be unstable under perturbations of compact support. The resolution of this conundrum is the idea of del Rio, Simon, and Stolz [5] that one should look at the union over θ of spectral supports. Explicitly, we proceed as follows: Definition (Gilbert-Pearson [7]). We say there is a subordinate solution at energy λ ∈ ℝ if and only if there is some θ ∈ [0, π) so lim_{L→∞} ∥φ_{1,θ}∥_{L}/∥φ_{2,θ}∥_{L} = 0. θ is necessarily unique and we call it θ(λ). Definition.

P = {λ | φ_{1,θ(λ)} ∈ L^{2}}
S = {λ | there is a subordinate solution but φ_{1,θ(λ)} ∉ L^{2}}
L = {λ | there is no subordinate solution}

When we need to discuss the V -dependence of these sets, we will write P (V), etc.

Then:

Theorem 1.1. (i) P = ∪_{θ} σ_{pp}(H_{θ}).

(ii) L = essential support of σ_{ac}(H_{θ}) for all θ.

(iii) For any θ, dρ_{θ}^{sc} = dρ_{θ}(S ∩ ·) and if S̃ is any other set with that property, then |S△S̃| = 0 where | · | is Lebesgue measure.

Remarks. 1. This is close to a theorem in [5], although S and L are defined differently there.

2. σ_{pp} in (i) means the set of eigenvalues, not their closure.

3. (i) is obvious since λ ∈ σ_{pp}(H_{θ(λ)}) if and only if φ_{1,θ(λ)}(λ, ·) ∈ L^{2}.

4. (ii) is the main result of Gilbert-Pearson [7].

5. That dρ_{θ}^{sc}(P) = 0 is obvious since dρ_{θ}^{sing} is mutually singular to each dρ_{θ′}^{pp} for θ′ ≠ θ′ and dρ_{θ}^{sc} is obviously mutually singular to dρ_{θ}^{pp}.

6. That dρ_{θ}^{sc}(L) = 0 is a result of Gilbert-Pearson showing that dρ_{θ}^{sc} = dρ_{θ}(S ∩ ·).

7. The |S△S̃| = 0 result follows from (1.5).

Since P, L, S are defined purely in terms of the behavior of solutions at infinity, the following result of del Rio et al. [5] is immediate:

Theorem 1.2. Let V = V_{0} + W where W has compact support. Then P (V) = P (V_{0}), L(V) = L(V_{0}), S(V) = S(V_{0}).

A major theme of this paper will be to examine when this result still holds for W ’s not of compact support. Before discussing our theorems, we will further refine the set S in connection with the breakdown of singular spectrum according to Hausdorff measures and dimensions.

As usual for α ∈ (0, 1), α-dimensional Hausdorff measure is defined on Borel sets, T, by

h^{α}(T) ≡ lim_{δ→0} inf_{δ−covers} ∑_{ν=1}^{∞} |b_{ν}|^{α},

where a δ-cover is a countable collection of intervals each of length at most δ so T ⊂ ∪_{ν=1}^{∞}b_{ν}. h^{1} is Lebesgue measure and h^{0} is counting measure.

Given α ∈ [0, 1] (following Rogers and Taylor [16, 17]; see also Last [12]), we define a measure μ to be α-continuous (αc) if μ(S) = 0 for any set S with h^{α}(S) = 0 and α-singular (αs) if it is supported on a set of S with h^{α}(S) = 0. For every such α and any measure μ, one can uniquely decompose μ = μ^{αc} + μ^{αs} with μ^{αc} α-continuous and μ^{αs}, α-singular.

We call a measure zero-dimensional if it is supported on a set S with h^{α}(S) = 0 for all α > 0. We call it one-dimensional if it is α-continuous for all α < 1.

It will be useful, following Jitomirskaya-Last, to have a pair of inverse functions A, B : [0, 1] to [0, 1] by

B(α) = α/(2 − α)
A(β) = 2β/(1 + β).

Definition. Let λ ∈ S, the set of energies for which there is a non-L^{2} subordinate solution. Define

β(λ) = lim inf_{L→∞} [ ln ∥φ_{1,θ(λ)}∥_{L}/ ln ∥φ_{2,θ(λ)}∥_{L}].

Notice that since φ_{1,θ} ∉ L^{2}, ∥φ_{1,θ(λ)}∥_{L} → ∞ as L → ∞ and since φ_{1} is subordinate, eventually ∥φ_{2,θ}∥_{L} ≥ ∥φ_{1,θ}∥_{L}, and thus

lim_{L→∞} ∥φ_{2,θ}∥_{L} = ∞  (1.9)

and

0 ≤ β(λ) ≤ 1.

When we want to indicate the V -dependence of β, we will write β(λ; V). We note the following elementary:

Proposition 1.3. If β > β(λ), then

\underline{lim} ∥φ_{1}∥_{L}/∥φ_{2}∥_{L}^{β} = 0  (1.10)

and if β < β(λ),

lim ∥φ_{1}∥_{L}/∥φ_{2}∥_{L}^{β} = ∞.  (1.11)

Proof. Write

∥φ_{1}∥_{L}/∥φ_{2}∥_{L}^{β} = exp [ ln ∥φ_{2}∥_{L}\{\frac{ln ∥φ_{1}∥_{L}}{ln ∥φ_{2}∥_{L}} − β\}].

By (1.9), ln ∥φ_{2}∥_{L} → ∞. If β > β(λ), then there is a subsequence where the expression in { } goes to β(λ) − β < 0, so a subsequence where the expression in [ ] goes to −∞ and (1.10) holds. If β < β(λ), then eventually the expression in \{ \} is larger than \frac{1}{2}(β(λ) − β), and so (1.11) holds. □

For each β_{0}, decompose S into four sets:

S_{β_{0}}^{++} = {λ | β_{0} > β(λ)}
S_{β_{0}}^{−−} = {λ | β_{0} < β(λ)}
S_{β_{0}}^{+} = {λ | β_{0} = β(λ) and (1.10) holds for β_{0} = β(λ)}
S_{β_{0}}^{−} = \{λ | β_{0} = β(λ) and \underline{lim} ∥φ_{1}∥_{L}/∥φ_{2}∥^{β_{0}} > 0\}.

Thus (1.10) holds for β = β_{0} if and only if λ ∈ S_{β_{0}}^{++} ∪ S_{β_{0}}^{+}.

It follows from Theorem 1.1 and the discussion following equation (2.2) of Jitomirskaya-Last [9] that

Theorem 1.4. Let β_{0} = B(α_{0}).

(i) dρ_{θ}^{α_{0}c} = dρ_{θ}((S_{β_{0}}^{−} ∪ S_{β_{0}}^{−−} ∪ L) ∩ ·)

(ii) dρ_{θ}^{α_{0}s} = dρ_{θ}((S_{β_{0}}^{+} ∪ S_{β_{0}}^{++} ∪ P) ∩ ·)

(iii) dρ_{θ}^{sc} is one-dimensional for a.e. θ if and only if β = 1 a.e. on

S. (iv) dρ_{θ}^{sc} is zero-dimensional for a.e. θ if and only if β = 0 a.e. on

S.

Remark. More generally, dρ_{θ}^{sc} has exact dimension α_{0} for a.e. θ if β = B(α_{0}) for a.e. λ ∈ S.

Clearly, β only depends on V near infinity, so we extend the result of del Rio et al. [5] to handle dimensional decomposition of dρ via

Theorem 1.5. Let V = V_{0} + W where W has compact support. Then β(λ; V) = β(λ; V_{0}).

The purpose of this paper is to study when invariance results of the genre of Theorems 1.2 and 1.5 extend to cases where W does not have compact support but has “suitable” decay; that is, we want to determine what suitable decay is. For the a.c. spectrum, the standard rate of decay is W ∈ L^{1}:

Theorem 1.6. In the continuum case, suppose V_{0} and V ≡ V_{0} + W are such that H_{0} + V_{0} and H_{0} + V are bounded below by εH_{0} − c. In the discrete case, no hypothesis is needed on V_{0}. Suppose that W ∈ L^{1} (or ℓ^{1}). Then

|L(V)△L(V_{0})| = 0.  (1.12)

Proof. In the discrete case, W is trace class, and in the continuum case, (H_{0} + 1)^{−1/2}W (H_{0} + 1)^{−1/2} is trace class. So (H_{0} + V + c − 1)^{−1} − (H_{0} + V_{0} + c + 1)^{−1} is trace class. The trace class theory of scattering [15] implies that H_{0} on H_{ac}(H_{0}) is unitarily equivalent to H_{0} + V_{0} on H_{ac}(H_{0} + V_{0}) from which (1.12) follows by Theorem 1.1. □

Remark. We conjecture that (1.12) holds if W is merely assumed in L^{2}. In [11], we made this conjecture when V_{0} = 0 and it was proven by Deift-Killip [4]. Killip [10] proved the result when V_{0} is periodic. We conjecture the result for all V_{0}.

We now turn to the substantially new results in this paper. As spectrum moves from the most smooth (a.c.) to the least smooth (point), we need to successively strengthen the conditions on the perturbation W.

We begin with several results we prove in Section 3 concerning point spectrum that all hold in the discrete and continuum case.

Theorem 1.7. For each λ ∈ P (V_{0}), define

f_{+}(λ, x) = (1 + |x|) sup_{|y|≤x} |φ_{2,θ(λ)}(y)|.  (1.13)

Suppose that for all λ ∈ Q ⊆ P (V_{0}), we have that

∫ |W (x)|f_{+}(λ, x) dx < ∞

and that the L^{2} solution is bounded. Then Q ⊆ P (V_{0} + W).

Remarks. 1. In (1.13), one can replace (1 + |x|) by (1 + |x|)^{γ} for any γ > \frac{1}{2}.

2. By a Sobolev estimate if φ, φ′ ∈ L^{2}, then φ ∈ L^{∞}, so, for example, if V_{0} is bounded from below, L^{2} solutions will be bounded.

When V_{0} is bounded, f_{+} does not grow faster than exponentially for any λ.

Corollary 1.8. Let V_{0} be bounded from below and suppose that

∫ |W (x)| e^{A|x|} dx < ∞

for all A > 0. Then

P (V_{0}) = P (V_{0} + W).

Finally, we have a result on preservation of Lyapunov behavior. Recall that we say there is Lyapunov behavior at energy λ if the transfer matrix

T_{λ}(0, x) = (\begin{matrix}φ′_{1,θ}(x) & φ′_{2,θ}(x) \\ φ_{1,θ}(x) & φ_{2,θ}(x) \end{matrix})

obeys

lim_{x→∞} \frac{1}{|x|} ln ∥T_{λ}(0, x)∥ ≡ γ(λ).  (1.14)

Assume in addition that

lim_{x→∞} \frac{1}{|x|} ln(1 + |V (x)|) = 0.  (1.15)

Theorem 1.9. Suppose V_{0} satisfies (1.15), has Lyapunov behavior at energy λ and that for some ε > 0,

∫ |W (x)| e^{ε|x|} dx < ∞.

Then V_{0} + W has Lyapunov behavior at λ with the same value of γ.

Remarks. 1. If γ > 0, we have much more than merely the same Lyapunov behavior.

2. Theorem 1.9 isn’t new. It is essentially a special case of Theorem 4.I of [18].

3. There is an obvious discrete analog of this theorem.

In Section 4, we will discuss stability of singular spectrum and its components. Our results will hold only for energies with an extra condition. Definition. An energy λ is called regular if and only if for some θ (= θ(λ) if there is a subordinate solution) we have for all ε > 0,

∥φ_{1,θ}∥_{L} ≤ C_{ε}L^{1/2+ε}.  (1.16)

By the general theory of eigenfunction expansions [1, 19], a.e. λ is regular both with respect to each dρ_{θ}, and so by (1.5) for a.e. λ with respect to Lebesgue measure dλ. Indeed, we could replace L^{1/2+ε} by L^{1/2}(ln L)^{κ} for any κ > \frac{1}{2}.

Remark. If V (x) = −\frac{3}{16} x^{−2} for large x, then the subordinate solution at λ = 0 is ∼ x^{1/4} at infinity. So ∥φ_{1}∥_{L} ∼ L^{3/4} and λ = 0 is not a regular energy, so not all energies need to be regular.

In the discrete case, constancy of the Wronskian implies

∥φ_{1,θ}∥_{L} ∥φ_{2,θ}∥_{L} ≥ \frac{1}{2}(L − 1),  (1.17)

but in the continuum case, this is not automatic since the Wronskian involves φ′. But, by a Sobolev estimate, if V is bounded from below (uniform locally L^{1} will do!), then

∥φ_{1,θ}∥_{L} ∥φ_{2,θ}∥_{L} ≥ c(L − 1)  (1.18)

for some c, dependent on V and λ (see e.g. [19]), and so we will need to suppose that V is bounded from below in the continuum case. Remark. The case V (x) = −x where ∥φ_{1,θ}∥_{L} ∼ ∥φ_{2,θ}∥_{L} ∼ L^{1/4} shows (1.18) really can fail if V is unbounded from below.

Here are the theorems we will prove in Section 4.

Theorem 1.10. In the continuum case, suppose V_{0} is bounded from below. Let λ ∈ S(V_{0}) be a regular energy with β(λ, V_{0}) = 1. Suppose that

|W (x)| ≤ C(1 + |x|)^{−1−ε}  (1.19)

for some ε > 0. Then λ ∈ S(V_{0} + W) with β(λ, V_{0} + W) = 1. In particular, if, for V_{0}, H_{θ} has one-dimensional spectrum for a.e. θ, the same is true for V_{0} + W.

Theorem 1.11. In the continuum case, suppose V_{0} is bounded from below. Let λ ∈ S(V_{0}) be a regular energy. Suppose that for all η > 0,

|W (x)| ≤ C_{η}(1 + |x|)^{−η}.  (1.20)

Suppose that β(λ, V_{0}) ≠ 0. Then λ ∈ S(V_{0} + W) and β(λ, V_{0} + W) = β(λ, V_{0}). Suppose β(λ, V_{0}) = 0. Then either λ ∈ S(V_{0} + W) with β(λ, V_{0} + W) = 0 or λ ∈ P (V_{0} + W).

Remarks. 1. The latter shows that having zero-dimensional spectrum is preserved under perturbations obeying (1.20), although to preserve point spectrum, we need a stronger exponential bound.

2. In fact, our proof shows that for a given β(λ, V_{0}) = β_{0}, we only need (1.20) for some

η > \frac{1}{β_{0}} .

In terms of the case of Hausdorff dimension α, one needs

η > \frac{2}{α} − 1.  (1.21)

We will prove our new results, Theorem 1.7, Corollary 1.8, and Theorems 1.9–1.11, by proving stability of the asymptotics of solutions of the Schrödinger differential/difference equation. We use φ_{−} for φ_{1,θ(λ)}, the subordinate solution with potential V_{0}, and φ_{+} for φ_{2,θ(λ)}. The basic construction we will use is variation of parameters. That is, we will write (in the continuum case):

ψ(x) = u_{1}(x)φ_{−}(x) + u_{2}(x)φ_{+}(x)  (1.22)
ψ′(x) = u_{1}(x)φ′_{−}(x) + u_{2}(x)φ′_{+}(x).  (1.23)

With u(x) = \binom{u_{1}(x)}{u_{2}(x)}, the differential equation for ψ is equivalent, given the normalization (1.7), to

u′(x) = A(x)u(x)  (1.24)

with

A(x) = −W (x) (\begin{matrix}φ_{+}(x)φ_{−}(x) & φ_{+}(x)^{2} \\ −φ_{−}(x)^{2} & −φ_{+}(x)φ_{−}(x) \end{matrix}) .  (1.25)

(1.25) is sometimes written (e.g., in [9]) in the integral form:

\begin{aligned}ψ(x) & = u_{1}(x_{0})φ_{−}(x) + u_{2}(x_{0})φ_{+}(x) \\ & + ∫_{x_{0}}^{x} W (y)[φ_{+}(x)φ_{−}(y) − φ_{−}(x)φ_{+}(y)]ψ(y) dy. \end{aligned} (1.26)

In the discrete case, the result is similar. One writes

ψ(n) = u_{1}(n)φ_{−}(n) + u_{2}(n)φ_{+}(n)  (1.27)
ψ(n − 1) = u_{1}(n)φ_{−}(n − 1) + u_{2}(n)φ_{+}(n − 1).  (1.28)

(1.24) becomes

u(n + 1) − u(n) = A(n)u(n),  (1.29)

where

A(n) = −W (n) (\begin{matrix}φ_{+}(n)φ_{−}(n) & φ_{+}(n)^{2} \\ −φ_{−}(n)^{2} & −φ_{+}(n)φ_{−}(n) \end{matrix})  (1.30)

or its integral form

\begin{aligned}ψ(n) & = u_{1}(n_{0})φ_{−}(n) + u_{2}(n_{0})φ_{+}(n) \\ & + ∑_{j=n_{0}}^{n} W (j)[φ_{+}(n)φ_{−}(j) − φ_{−}(n)φ_{+}(j)]ψ(j). \end{aligned}  (1.31)

The standard control for perturbing solutions at infinity is to require ∫_{x_{0}}^{∞} ∥A(x)∥ dx < ∞. For the diagonal matrix elements of A, that cannot be improved without detailed oscillation estimates, but it is well known that one can try to trade off the growth of one offdiagonal matrix element by the decay of the other. In Section 2, we present a version of this fact made for our applications. These ideas are not new; for example, our method of proof is patterned after problem XI.97 of Reed-Simon [15]. In Section 3, we present the results of stability of a solution L^{2} at ∞ and in Section 4, the results on stability of polynomially bounded solutions. The Appendix discusses some results concerning the preservation of WKB asymptotic behavior of solutions.

2. A Perturbation Lemma

In this preliminary section, we will be interested in solutions of

u′(x) = A(x)u(x),  (2.1)

where

A(x) = (\begin{matrix}a_{11}(x) & a_{12}(x) \\ a_{21}(x) & a_{22}(x) \end{matrix})  (2.2)

is in L_{loc}^{1}[0, ∞) and

u(x) = \binom{u_{1}(x)}{u_{2}(x)}  (2.3)

is a two-component vector. By a solution of (2.1), we mean an absolutely continuous function so that (2.1) holds for a.e. x. As usual, given any x_{0} and ω ∈ ℂ^{2}, there is a unique solution of (2.1) with u(x_{0}) = ω.

We will use a pair of non-negative functions f_{±}(x) with

f_{+}(x)f_{−}(x) ≥ 1  (2.4)

and f_{+} monotone increasing and f_{−} monotone decreasing (in some applications, we will take f_{±} = e^{2(±γ+ε)|x|} so you can have this example in mind). Define

G(x) = max(|a_{11}(x)| + |a_{12}(x)|f_{−}(x), |a_{21}(x)|f_{+}(x) + |a_{22}(x)|). (2.5)

Lemma 2.1. Define ∥ · ∥_{x}^{±} as norms on ℂ^{2} by

∥ω∥_{x}^{+} = max(|ω_{1}|, |ω_{2}|f_{+}(x))
∥ω∥_{x}^{−} = max(f_{−}(x)|ω_{1}|, |ω_{2}|).

Then

∥A(x)ω∥_{x}^{±} ≤ G(x)∥ω∥_{x}^{±}.  (2.6)

Proof. We will prove the ∥ · ∥^{+} result. The ∥ · ∥^{−} is similar. Note that

|(A(x)ω)_{1}| ≤ |a_{11}(x)| |ω_{1}| + |a_{12}|f_{+}(x)^{−1}f_{+}(x)|ω_{2}|
≤ [|a_{11}(x)| + |a_{12}|f_{+}^{−1}(x)] ∥ω∥_{x}^{+} ≤ G(x)∥ω∥_{x}^{+}

since f_{+}^{−1} ≤ f_{−} by (2.4) and

f_{+}(x)|(A(x)ω)_{2}| ≤ [|a_{21}(x)f_{+}(x)|ω_{1}| + |a_{22}(x)|f_{+}(x)|ω_{2}|]

≤ (|a_{21}(x)|f_{+}(x) + |a_{22}(x)|)∥ω∥_{x}^{+}
≤ G(x)∥ω∥_{x}^{+}.

□

Theorem 2.2. Suppose f_{+} is monotone increasing, f_{−} is monotone decreasing, (2.4) holds, and

∫_{x}^{∞} G(y) dy < ∞.

Then there exist solutions u^{±} of (2.1) so that as x → ∞ (i) u_{1}^{−}(x) → 1, f_{+}(x)u_{2}^{−}(x) → 0 (ii) u_{1}^{+}(x)f_{−}(x) → 0, u_{2}^{+}(x) → 1.

Proof. Define u^{−(n)} by

u^{−(0)} = \binom{1}{0}
u^{−(n+1)}(x) = − ∫_{x}^{∞} A(y)u^{−(n)}(y) dy,

where we will deal with the convergence of the integral below. Since f_{+} is increasing, if y > x, then ∥ω∥_{x}^{+} ≤ ∥ω∥_{y}^{+}. Thus

∥u^{−(n+1)}(x)∥_{x}^{+} ≤ ∫_{x}^{∞} ∥A(y)u^{−(n)}(y)∥_{x}^{+} dy
≤ ∫_{x}^{∞} ∥A(y)u^{−(n)}(y)∥_{y}^{+} dy
≤ ∫_{x}^{∞} G(y) ∥u^{−(n)}(y)∥_{y}^{+} dy

by (2.6). Thus

sup_{y≥x} ∥u^{−(n+1)}(y)∥_{y}^{+} ≤ sup_{y≥x} ∥u^{−(n)}(y)∥_{y}^{+} ∫_{x}^{∞} G(y) dy

proving convergence of the integral and

sup_{y≥x} ∥u^{−(n)}(y)∥_{y}^{+} ≤ [ ∫_{x}^{∞} G(y) dy]^{n}

inductively.

It follows that

u^{−}(y) ≡ ∑_{n=0}^{∞} u^{−(n)}(y)

converges for y ≥ x_{0} where ∫_{x_{0}}^{∞} G(y) dy < 1 and that for such y, ∫_{y}^{∞} A(w)u^{−}(w) dw converges and

u^{−}(y) = \binom{1}{0} + ∫_{y}^{∞} A(w)u^{−}(w) dw

so u^{−} solves (2.1). Since ∥u^{−}(y) − \binom{1}{0}∥_{y}^{+} → 0 as y → ∞, we obtain (i). Define ũ^{+(n)} by

ũ^{+(0)} = \binom{0}{1}
ũ^{+(n+1)}(x) = ∫_{x_{0}}^{x} A(y)ũ^{+(n)}(y) dy

for x_{0} chosen so that

∫_{x_{0}}^{∞} G(y) dy ≤ \frac{1}{3}.  (2.7)

As above, using the fact that if y < x, then ∥ω∥_{x}^{−} ≤ ∥ω∥_{y}^{−} since f_{−} is decreasing, we have

sup_{x≥x_{0}} ∥ũ^{+(n)}(x)∥_{x}^{−} ≤ ( ∫_{x}^{∞} G(y) dy)^{n} ≤ (\frac{1}{3})^{n}.

As in the ∥ · ∥^{+} case, we see that ∑_{n=0}^{∞} ũ^{+(n)} = ũ^{+} converges for y > x_{0} and ũ^{+} solves (2.1) and obeys

ũ^{+}(x) = \binom{0}{1} + ∫_{x_{0}}^{x} A(y)ũ^{+}(y) dy.

In particular,

ũ_{2}^{+}(∞) = 1 + ∫_{x_{0}}^{∞} A(y)ũ^{+}(y) dy

exists and |ũ_{2}^{+}(∞) − 1| ≤ \frac{1}{2} so ũ_{2}^{+}(∞) ≡ α > 0. Define

u^{+} = α^{−1}ũ^{+}

and so obtain a vector-valued function u^{+} with u_{2}^{+} → 1 and |u_{1}^{+}f_{−}| bounded. We will show that if f_{−} → 0, then u_{1}^{+}f_{−} → 0. When f_{−} does not go to zero, we will provide an alternative construction of u^{+}.

To prove that u_{1}^{+}f_{−} → 0 if f_{−} → 0, write for x_{0} < y < x:

\begin{aligned}u_{1}^{+}(x)f_{−}(x) & = f_{−}(x) ∫_{x_{0}}^{y}(A(w)u^{+}(w))_{1} dw \\ & + ∫_{y}^{x} f_{−}(x)f_{−}(w)^{−1}f_{−}(w)(A(w)u^{+}(w))_{1} dw \end{aligned}

so, since f_{−} is monotone decreasing,

|u_{1}^{+}(x)f_{−}(x)| ≤ f_{−}(x) ∫_{x_{0}}^{y} |(A(w)u^{+}(w))_{1}| dw +∫_{y}^{∞} G(w)||u^{+}(w)∥_{w}^{−} dw.

(2.8)

Given ε, pick y so the second integral in (2.8) is less than ε/2 and then, since f_{−} → 0, x so that the first term is less than ε/2. Thus u_{1}^{+}f_{−} → 0.

If f_{−}(x) has a non-zero limit as x → ∞, then since f_{−} is monotone, f_{−}(x) ≥ c. Thus, A(x) ∈ L^{1}, and by the same construction as used for u_{−} (i.e., integrating from infinity), one can construct u^{+}(x) → \binom{1}{0} as x → ∞. □

The situation in the discrete case is similar. (2.1) becomes

u(n + 1) − u(n) = A(n)u(n).  (2.9)

f_{±} obey (2.4), although they are only defined (and monotone) on n = 1, 2, . . .. G(n) is defined as in (2.5). The analog of Theorem 2.2 holds with ∫_{x}^{∞} G(y) dy < ∞ replaced by

∑_{n_{0}}^{∞} G(n) < ∞

and x going through discrete values. The proof is identical with obvious changes — for example, the formula for u^{−(n+1)} becomes

u^{−(n+1)}(j) = − ∑_{k=j}^{∞} A(k)u^{−(n)}(k).

We owe to F. Gesztesy an illuminating remark about our result, Theorem 2.2, namely the special case f_{+}f_{−} = 1 (which is true in some of the applications we will make) follows quickly from Levinson’s theorem [6, 13]. One variant of Levinson’s theorem says:

Proposition 2.3. Let A be a 2 × 2 matrix of the form A_{1} + A_{2} where ∫_{t_{0}}^{∞} ∥A_{1}(s)∥ ds < ∞, A_{2} is diagonal with

A_{2}(s) = (\begin{matrix}α_{1}(s) & 0 \\ 0 & α_{2}(s) \end{matrix})

so that

∫_{t_{0}}^{t} Re[α_{1}(s) − α_{2}(s)]  (2.10)

is either bounded below or bounded above. Then, there exist solutions φ_{1,2} of

φ̇ = Aφ

so that

φ_{1}(t) e^{− ∫_{t_{0}}^{t} α_{1}(s) ds} → \binom{1}{0}

and

φ_{2} e^{− ∫_{t_{0}}^{t} α_{2}(s) ds} → \binom{0}{1}

as t → ∞.

Remark. This is essentially equivalent to the general 2 × 2 case.

To apply this to the situation of Theorem 2.2, given a solution, u, of (2.2), let φ be defined by φ_{1} = u_{1}, φ_{2} = u_{2}f_{+}. Then

φ′ = (A_{1} + A_{2})φ,

where

A_{1} = (\begin{matrix}a_{11}(x) & a_{12}(x)f_{+}^{−1}(x) \\ a_{21}(x)f_{+}(x) & a_{22}(x) \end{matrix})

and

A_{2} = (\begin{matrix}0 & 0 \\ 0 & f′_{+}/f_{+}(x) \end{matrix}) .

By hypothesis (∫_{x_{0}}^{∞} G(x) < ∞), A_{1} ∈ L^{1} and the function (2.10) is log(f_{+}(t_{0})/f_{+}(t)) which is bounded by 0 (since f_{+} is monotone). The two Levinson’s theorem solutions obey (i) and (ii) of Theorem 2.2.

3. Stability of Point Spectra

In this section, we will prove Theorems 1.7 and 1.9. We will only consider the continuum case; the discrete case is similar.

Proof of Theorem 1.7. Since λ ∈ P (V_{0}), φ_{−} ≡ φ_{1,θ(λ)} is in L^{2} and so by hypothesis, also in L^{∞} [19]. Pick φ_{+} ≡ φ_{2,θ(λ)} and use variation of parameters (1.22)/(1.23). A has the form (1.25). Let f_{+} be given by (1.13). Since φ_{−} is bounded,

|W (x)φ_{+}(x)φ_{−}(x)| ≤ Cf_{+}(x)|W (x)|
|W (x)| |φ_{+}(x)|^{2}f_{+}(x)^{−1} ≤ f_{+}(x)|W (x)|
|W (x)| |φ_{−}(x)|^{2}f_{+}(x) ≤ Cf_{+}(x)|W (x)|.

So if f_{−}(x) ≡ f_{+}(x)^{−1}, we have that G given by (2.5) obeys

|G(x)| ≤ Cf_{+}(x)W (x).

Thus, Theorem 2.2 is applicable, so there is a solution, ψ, of the perturbed Schrödinger equation of the form:

ψ(x) = u_{1}^{−}(x)φ_{−}(x) + u_{2}^{−}(x)φ_{+}(x)  (3.1)

with u_{1}^{−} bounded and with u_{2}^{−}(x)f_{+}(x) bounded. Since φ_{−} ∈ L^{2}, u_{1}^{−}φ_{−} ∈ L^{2}. Moreover, since u_{2}^{−}f_{+} is bounded, (1.13) says that

|u_{2}^{−}(x)φ_{+}(x)| ≤ C(1 + |x|)^{−1}

which is also in L^{2}. Thus ψ ∈ L^{2}. □

Corollary 1.8 follows immediately since f_{+} is exponentially bounded. Results of this genre are well known; see, for example, Hartman [8]. We proceed to prove Theorem 1.9:

Proof of Theorem 1.9. If γ = 0, any solution, φ, of the unperturbed equation has

|φ(x)| + |φ′(x)| ≤ C_{ε} e^{ε|x|}  (3.2)

so, by hypothesis, the A of (1.25) is in L^{1} for any choice of φ_{±}. Thus, by standard theory (or Theorem 2.2 with f_{+} = f_{−} = 1), any solution ψ also obeys (3.2) which implies that γ = 0.

Now suppose that γ > 0. By the Ruelle-Osceledec Theorem [18], if V satisfies (1.15), there is a solution φ_{−}(x) (≡ φ_{1,θ(λ)}) for the V_{0} equation with

lim_{x→∞} [\frac{1}{2|x|} ln[|φ_{−}(x)|^{2} + |φ′_{−}(x)|^{2}]] = −γ.

Any linearly independent solution and, in particular, φ_{+} = φ_{2,θ(λ)} obeys

lim_{x→∞} [\frac{1}{2|x|} ln[|φ_{+}(x)|^{2} + |φ′_{+}(x)|^{2}]] = γ.

In particular, for any ε_{1} > 0,

|φ_{+}(x)| ≤ C_{ε_{1}}e^{(γ+ε_{1})|x|},  |φ_{−}(x)| ≤ C_{ε_{1}} e^{−(γ−ε_{1})|x|}.  (3.3)

Pick f_{±}(x) = e^{(±2γ+2ε_{1})|x|} where ε_{1} is chosen so that ε_{1} < γ (so f_{−} is decreasing) and ε_{1} < \frac{1}{4}ε where ε is given in the hypothesis of the theorem. By the estimates of (3.3),

|G(x)| ≤ e^{4ε_{1}|x|}|W (x)|

so G ∈ L^{1} since 4ε_{1} < ε. Theorem 2.2 applies and we get solutions ψ_{±} of the perturbed equation with

|ψ_{+} − u_{2}^{+}φ_{+}| ≤ |u_{1}^{+}||φ_{−}| ≤ C_{ε_{1}}|u_{1}^{+}|f_{−} e^{(γ−ε_{1})|x|}

and a similar estimate for ψ′_{+}. It follows that

\frac{∥ψ_{+}∥_{x}}{∥φ_{+}∥_{x}} − 1 → 0

as x → ∞ where ∥g∥_{x} = (|g(x)|^{2}+|g′(x)|^{2})^{1/2}. Similarly ∥ψ_{−}∥_{x}/∥φ_{−}∥_{x} → 1. Thus not only is the Lyapunov exponent the same, but even the subexponential corrections are unchanged. □

4. Power Law Theorems

In this section we will prove the following result that essentially includes Theorems 1.10 and 1.11 as corollaries. (We will need to make an additional argument for β = 0.)

Theorem 4.1. In the continuum case, suppose V_{0} is bounded from below. Let λ ∈ S_{0}(V_{0}) be a regular energy with β(λ, V_{0}) > 0. Suppose that

|W (x)| ≤ C(1 + |x|)^{−η}

for some η > β(λ, V_{0})^{−1}. Then λ ∈ S(V_{0} + W) and

β(λ, V_{0} + W) = β(λ, V_{0}).

Interestingly enough, we will apply Theorem 2.2 in a situation where f_{+}f_{−} ≠ 1 but is strictly bigger. Essentially, we will not want to take f_{−} as small as f_{+}^{−1} because we will need the error estimate u_{1}^{+}f_{−} → 0 to be stronger than u_{1}^{+}f_{+}^{−1} → 0.

To employ the ideas of Jitomirskaya-Last, we need to relate estimates involving an integral of a product of φ_{+}, φ_{−}, W and 1, f_{+} or f_{−} to ∥ · ∥_{L}. The following is useful:

Lemma 4.2. If

|Q(x)| ≤ C_{1}(1 + |x|)^{−a}  (4.1)

and

∥φ_{+}∥_{L}∥φ_{−}∥_{L} ≤ C_{2}(1 + L)^{b}  (4.2)

and a > b, then

∫_{0}^{∞}|Q(x)φ_{+}(x)φ_{−}(x)| dx < ∞.

Proof. Let g(x) = ∫_{0}^{x}|φ_{+}(y)φ_{−}(y)| dy. By the Schwarz inequality and (4.2),

|g(x)| ≤ C_{2}(1 + |x|)^{b}  (4.3)

and, of course,

g(0) = 0.  (4.4)

Use (4.1) to write

∫_{0}^{L} |Q(x)φ_{+}(x)φ_{−}(x)| dx

≤ C_{1} ∫_{0}^{L}(1 + |x|)^{−a} \frac{dg}{dx} dx

= C_{1}a ∫_{0}^{L}(1 + |x|)^{−a−1}g(x) dx + C_{1}(1 + |L|)^{−a}g(L).

There is no boundary term at x = 0 by (4.4).

Now use (4.3) and b < a to see

lim_{L→∞} ∫_{0}^{L}|Q(x)φ_{+}(x)φ_{−}(x)| dx ≤ C_{1}C_{2} a ∫_{0}^{∞}(1 + |x|)^{b−a−1} dx
< ∞.
□

The next step is obtaining power-law upper and lower bounds on ∥φ_{±}∥_{L}. In principle, the upper and lower powers could be different with oscillation between the two powers of growth.

Lemma 4.3. Let λ be a regular energy with λ ∈ S and β(λ) > 0. Let φ_{−} = φ_{1,θ(λ)} and φ_{+} = φ_{2,θ(λ)}. Then for any ε > 0, there are constants C_{1}, C_{2}, C_{3}, C_{4} (ε-dependent) so that for L large,

C_{2}L^{1−1/2β−ε} ≤ ∥φ_{−}∥_{L} ≤ C_{1}L^{1/2+ε}  (4.5)
C_{4}L^{1/2−ε} ≤ ∥φ_{+}∥_{L} ≤ C_{3}L^{1/2β+ε}.  (4.6)

Proof. The definition of regularity says (1.16) which is the C_{1} estimate in (4.5). (1.17) then implies the C_{4} estimate in (4.6).

By (1.11), if β̃ < β, then ∥φ_{−}∥_{L} ≥ ∥φ_{+}∥β̃ ; L for L large which, given the C_{1} estimate, implies the C_{3} estimate in (4.6). Using (1.17) again, we get the C_{2} estimate in (4.5). □

At first sight, it might appear that all one needs on ∥φ_{±}∥_{L} are upper bounds because they are all that enter in proving the applicability of Theorem 2.2. But one wants to apply Theorem 2.2 to show that

\frac{∥ψ_{+}∥_{L}}{∥φ_{+}∥_{L}} → 1,  \frac{∥ψ_{−}∥_{L}}{∥φ_{−}∥_{L}} → 1  (4.7)

as L → ∞. Consider the second part of (4.7). We have

ψ_{−} = u_{1}^{−}φ_{−} + u_{2}^{−}φ_{+}.

Since u_{1}^{−} → 1, we have that

\frac{|∥ψ_{−}∥_{L} − ∥φ_{−}∥_{L}|}{∥φ_{−}∥_{L}} ≤ \frac{∥ψ_{−} − φ_{−}∥_{L}}{∥φ_{−}∥_{L}} ≤ \frac{∥ψ_{−} − u_{1}^{−}φ_{−}∥_{L}}{∥φ_{−}∥_{L}} + o(1)

and so it is natural to prove the desired relation by showing

\frac{∥u_{2}^{−}φ_{+}∥_{L}}{∥φ_{−}∥_{L}} → 0.

All we basically know about u_{2}^{−} is f_{+}u_{2}^{−} → 0. Thus

Lemma 4.4. Suppose G(x) ∈ L^{1}. In order for (4.7) to hold, it suffices that for large L,

\frac{∥φ_{+}∥_{L}}{∥φ_{−}∥_{L}} ≤ Cf_{+}(L)  (4.8)

and

\frac{∥φ_{−}∥_{L}}{∥φ_{+}∥_{L}} ≤ Cf_{−}(L).  (4.9)

By (4.5)/(4.6), we have (4.8) if f_{+}(L) = L^{μ_{+}} with

\frac{1}{2β} − (1 − \frac{1}{2β} ) < μ_{+}.

If we only apply the similar bound for (4.9), we see that we need f_{−}(L) ≥ L^{2ε} which is incompatible with f_{−} decreasing. We therefore do not gain from (4.5)/(4.6) and instead define f_{−}(x) ≡ 1 so that (4.9) holds since ∥φ_{−}∥_{L} is subordinate. Thus we will take

f_{+}(x) = x^{μ_{+}},  f_{−}(x) = 1  (4.10)

with

μ_{+} > \frac{1}{β} − 1.  (4.11)

Proof of Theorem 4.1. By the above analysis, if we take f_{+}, f_{−} to obey (4.10)/(4.11), we have (4.7) so long as Theorem 2.2 is applicable. But (4.7) implies that [∥ψ_{−}∥_{L}/∥ψ_{+}∥β̃ ; L]/[∥φ_{−}∥_{L}/∥φ_{+}∥β̃ ; L] → 1 and thus by Proposition 1.3, β(λ, V_{0} + W) = β(λ, V_{0}).

To apply Theorem 2.2, we need G to be in L^{1}. By Lemma 4.2 and the upper bounds in (4.5)/(4.6), this is true if the following three inequalities hold

\frac{1}{2} + \frac{1}{2β} < η ⟵ (a_{11}, a_{22} terms)  (4.12)
1 + μ_{+} < η ⟵ (a_{21} terms)  (4.13)
\frac{1}{β} < η ⟵ (a_{12} terms).  (4.14)

By the basic hypothesis of the theorem, η > β^{−1} and, of course, β^{−1} ≥ 1. Thus (4.12) and (4.14) hold, and to get (4.13) and (4.11), we need only choose μ_{+} > 0 so that

\frac{1}{β} < 1 + μ_{+} < η.

This can be done since β^{−1} ≥ 1. □

Theorem 1.10 is an immediate corollary of Theorem 4.1 as is Theorem 1.11 if β(λ, V_{0}) ≠ 0. In case β(λ, V_{0}) = 0, then we claim β(λ, V_{0} + W) = 0 for if not, we can turn this argument around (think of V_{0} = (V_{0} +W)−W) and find that β(λ, V_{0}) = β(λ, V_{0} +W) ≠ 0. That means β(λ, V_{0} + W) = 0 which implies λ ∈ P (V_{0} + W) or λ ∈ S(V_{0} + W) with β = 0.

The condition η > β^{−1} of Theorem 4.1 is needed because we assume no extra information about the behavior of ∥u_{1}∥_{L} and ∥u_{2}∥_{L} other than the value of β. If one has additional information, one can often do better. Here is an extreme example, but one that holds in some explicit examples. Definition. We say there is power Lyapunov-Osceledec behavior with exponent γ > 0 at energy λ if and only if there exist solutions φ_{1,θ(λ)} and φ_{2,θ(λ)} with

lim_{x→∞} \frac{ln[|φ_{1,θ(λ)}(x)|^{2} + |φ′_{1,θ(λ)}(x)|^{2}]}{ln|x|} = −γ

lim_{x→∞} \frac{ln[|φ_{2,θ(λ)}(x)|^{2} + |φ′_{2,θ(λ)}(x)|^{2}]}{ln|x|} = γ.

Notes. 1. In the discrete case, replace φ′(x) by φ(n + 1).

2. Under these circumstances, if γ < 1/2 and V is bounded, we have ∥φ_{1}∥_{L} ∼ L^{−γ+1/2}, ∥φ_{2}∥_{L} ∼ L^{γ+1/2} (where ∼ means up to factors of L^{ε}) so λ ∈ S(V_{0}) and β(λ) = (1/2 − γ)/(1/2 + γ) and α(λ) = 1 − 2γ.

3. One example where it is known [11] there is power Lyapunov-Osceledec behavior is the discrete n^{−1/2} decaying Anderson model where V_{ω}(n) = λn^{−1/2}X_{ω}(n) where the X_{ω} are bounded i.i.d.’s with E(X_{ω}) = 0, E(X_{ω}^{2}) = 1. Then there is power Lyapunov-Osceledec behavior with γ = λ^{2}/(8 − 2E^{2}) in the region |E| < 2.

Theorem 4.5. Suppose V_{0} has power Lyapunov-Osceledec behavior with γ > 0 at energy λ and that

∫ (1 + |x|)^{ε}|W (x)| dx < ∞

for some ε > 0. Then V_{0} + W has power Lyapunov-Osceledec behavior at energy λ with the same value of γ.

The proof is essentially identical to the proof of Theorem 1.9 (in Section 3) with f_{±}(x) = (1 + |x|)^{2(±γ+ε)}.

This shows the improvement over the power in Theorem 4.1. Instead of η > β^{−1}, we only need η > 1.

5. Appendix: WKB Asymptotic Behavior

In this appendix, we illustrate with an example how Theorem 2.2 can be used to obtain precise asymptotic behavior of solutions in some concrete situations (where the perturbation does not even have to be decaying). Namely, we show how to use Theorem 2.2 to prove the existence of WKB solutions at +∞ for

−ψ′′ + V ψ = λψ,  (5.1)

when V = V_{1} + V_{2} with

V_{1} ∈ L^{1}, V′_{2} ∈ L^{1}, V_{2}(x) → 0 as x → ∞,  (5.2)

when λ ≠ 0. For λ > 0, it is well known that such solutions exist (see, e.g., [21]). For λ < 0, one can also apply Levinson’s theorem (see [13] or [3], Theorem 8.1) to prove this result, but it is nice to get it from Theorem 2.2. As a preliminary, we note one can try an Ansatz, (1.22)/(1.21) for solving (5.1) even if φ_{±} do not solve a related Schrödinger equation. The result is that u still obeys (1.24) but A is now given by

A(x) = w(x)^{−1} (\begin{matrix}−φ_{+}(x)(H_{λ}φ_{−})(x) & −φ_{+}(x)(H_{λ}φ_{+})(x) \\ φ_{−}(x)(H_{λ}φ_{−})(x) & φ_{−}(x)(H_{λ}φ_{+})(x) \end{matrix}) , (5.3)

where

w(x) = φ_{−}(x)φ′_{+}(x) − φ_{+}(x)φ′_{−}(x)  (5.4)

and H_{λ} is the differential expression

H_{λ} = −\frac{d^{2}}{dx^{2}} + V − λ.  (5.5)

We can now prove

Theorem 5.1. Let V obey (5.2) and λ ≠ 0. If λ < 0, let

φ_{±}(x) = exp(±η(x)),

where

η(x) = ∫_{s_{0}}^{x} \sqrt{−λ + V_{2}(s)} ds,

and s_{0} is chosen so that |V_{2}(s)| ≤ |λ| for s > s_{0}. If λ > 0, let

φ_{±} = exp(±η(x)),

where

η = i ∫_{s_{0}}^{x} \sqrt{λ − V_{2}(s)} ds.

Then there exist solutions ψ_{±} of (5.1) so

ψ_{±}(x) = φ_{±}(x) (1 + o(1))
ψ′_{±}(x) = φ′_{±}(x) (1 + o(1))

as x → ∞.

Proof. Consider first the case λ < 0. Then φ′_{±} = ±η′e^{±η} and φ′′_{±} = (±η′′ + (η′)^{2})φ_{±} and thus, since (η′)^{2} = −λ + V_{2},

H_{λ}φ_{±} = [±η′′ + V_{1}]φ_{±}

= [\frac{±V′_{2}}{2η′} + V_{1}]φ_{±}

so we define

Q_{±} = \frac{±V′_{2}}{2η′} + V_{1}.

Since η′ → \sqrt{−λ} as x → ∞, we see that Q_{±} ∈ L^{1}. Moreover,

w(x) = 2η′ → 2\sqrt{−λ} as x → ∞.  (5.6)

It follows with f_{±} = φ_{±}^{2} (so f_{+} = f_{−}^{−1}) and A given by (5.3) that G(x) ∈ L^{1} since Q_{±}(x) ∈ L^{1}.

Applying Theorem 2.2, there are solutions ψ_{±} with

ψ_{±} = φ_{±}(1 + o(1)) + φ_{∓}(f_{∓}^{−1})o(1)
= φ_{±}(1 + o(1))

and similarly for ψ′_{±}.

The calculation for λ > 0 is similar, except we use |φ_{+}| = |φ_{−}| = 1 in that case to pick f_{+} = f_{−} = 1. □

In this paper, we considered only perturbations which are absolutely integrable. It is reasonable to ask what one can expect for stronger perturbations, for example, in situations where there is Lyapunov behavior. While in general the picture is not complete, we provide a sample result which gives L^{2} stability under additional assumptions on the behavior of solutions of the unperturbed equation. As a bonus, we also obtain a stronger version of Theorem 5.1 in the case λ < 0.

Proposition 5.2. Assume that there exist functions φ_{±}(λ, x) such that (−\frac{d^{2}}{dx^{2}} + V − λ)φ_{±} = U_{±}φ_{±}, with U_{±} ∈ L^{2}, and that the inverse of the Wronskian W [φ_{−}, φ_{+}]^{−1} is bounded. Define functions

η_{±}(λ, x) = φ_{±}e^{± ∫_{0}^{x} \frac{U_{±}φ_{+}φ_{−}}{W [φ_{−},φ_{+}]} dt}  (5.7)

and the kernel

K(x, y) = φ_{+}^{2}(x)φ_{−}^{2}(y)e^{− ∫_{x}^{y} \frac{(U_{−}+U_{+})φ_{−}φ_{+}}{W [φ_{−},φ_{+}]} dt}.  (5.8)

Assume in addition that

inf_{x}|φ_{+}(x)φ_{−}(x)| ≥ c > 0  (5.9)

and

∫_{0}^{∞}sup_{x}|K(x, x + y)| dy < ∞, sup_{y≥x}|K(x, y)| ≤ C.  (5.10)

Then there exist solutions ψ_{±} of the equation (H + V − λ)ψ_{±} = 0 with the asymptotic behavior

ψ_{±}(λ, x) = η_{±}(λ, x)(1 + o(1)).  (5.11)

Remarks. 1. In order for (5.10) to hold, one needs, roughly speaking, Lyapunov behavior at λ and moreover ϕ_{+}(x, λ)ϕ_{−}(x, λ) ∼ const (or grows very slowly) for large x.

2. In the case where V_{2} = 0, the result follows from the Hartman-Wintner theorem (see, e.g., [6]).

3. Notice that the asymptotic behavior of solutions of the perturbed equation differs from φ_{±} by an additional factor.

Before sketching the proof, let us illustrate the result with the following generalization of Theorem 5.1 for λ < 0.

Corollary 5.3. Assume that V = V_{1}+V_{2}, V_{1} ∈ L^{2}, V′_{2} ∈ L^{2}, V_{2}(x) → 0 as x → ∞. Then for λ < 0 there exist solutions ψ_{±} of the equation −ψ′′ + V ψ = λψ such that

ψ_{±}(x, λ) = η_{±}(x, λ)(1 + o(1)).

Here η(x, λ) is given by (5.7) with

φ_{±}(x, λ) = exp (∫_{s_{0}}^{x} \sqrt{−λ + V_{2}(s)} ds) ,

and s_{0} is such that |V_{2}(s)| < |λ| for s > s_{0}.

Remark. For λ > 0, the result is generally not true. It holds for a.e. λ > 0 for V_{1} ∈ L_{p}, V′_{2} ∈ L^{p} with p < 2 [2]. It is not known if the result remains true for p = 2 and a.e. λ > 0.

Proof. Choosing φ_{±} as in the statement of the corollary, one directly verifies that all conditions of Proposition 5.2 hold. Notice that the Lyapunov behavior is preserved, since

W [ϕ_{−}, ϕ_{+}] = −2\sqrt{−λ + V_{2}},

U_{±} = V_{1} ∓ \frac{V′_{2}}{2\sqrt{−λ + V_{2}}}

and therefore the additional factor in (5.7) is bounded by e^{Cx^{1/2}}. □

We now sketch the proof of Proposition 5.2. Seeking solution ψ(x) of the equation −ψ′′ + V ψ = λψ, apply variation of parameters-type transformation

( \begin{matrix}ψ \\ ψ′ \end{matrix} ) = ( \begin{matrix}φ_{−} & φ_{+} \\ φ′_{−} & φ′_{+} \end{matrix} ) u(x),

obtaining a system

u′(x) = \frac{1}{W [φ_{−}, φ_{+}]} ( \begin{matrix}−U_{−}φ_{−}φ_{+} & −U_{+}φ_{+}^{2} \\ U_{−}φ_{−}^{2} & U_{+}φ_{−}φ_{+} \end{matrix} ) u(x).

Do one more transformation to bring this system to a simpler form:

u(x) = ( \begin{matrix}e^{− ∫_{0}^{x} \frac{U_{−}φ_{−}φ_{+}}{W [φ_{−},φ_{+}]} dt} & 0 \\ 0 & e^{∫_{0}^{x} \frac{U_{+}φ_{−}φ_{+}}{W [φ_{−},φ_{+}]} dt} \end{matrix} ) z(x),

then

z′(x) = ( \begin{matrix}0 & −\frac{U_{+}φ_{+}^{2}}{W [φ_{−},φ_{+}]}e^{∫_{0}^{x} \frac{(U_{−}+U_{+})φ_{−}φ_{+}}{W [φ_{−},φ_{+}]} dt} \\ −\frac{U_{−}φ_{−}^{2}}{W [φ_{−},φ_{+}]}e^{− ∫_{0}^{x} \frac{(U_{−}+U_{+})φ_{−}φ_{+}}{W [φ_{−},φ_{+}]} dt} & 0 \end{matrix} ) z(x).

(5.12)

One can obtain the formal series for solutions of (5.12) by iteration; starting with the vector (1, 0)^{T} will lead to the solution ψ_{−}(x). Properties (5.9) and (5.10) allow one to prove the convergence of this series and (5.11) using elementary estimates. We leave the details to the interested reader.

Acknowledgments. A. Kiselev and Y. Last would like to thank T. Tombrello for the hospitality of Caltech. We would like to thank F. Gesztesy and E. Harrell for useful comments.

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A. Kiselev, Department of Mathematics, University of Chicago, Chicago, IL 60637, USA

E-mail address: kiselev@math.uchicago.edu

Y. Last, Institute of Mathematics, The Hebrew University, 91904 Jerusalem, Israel

E-mail address: ylast@math.huji.ac.il

B. Simon, Division of Physics, Mathematics, and Astronomy, 253-37, California Institute of Technology, Pasadena, CA 91125, USA

E-mail address: bsimon@caltech.edu