Mathematical Physics and Quantum Field Theory, Electronic Journal of Differential Equations, Conf. 04, 2000, pp. 265–288 http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu or ejde.math.unt.edu (login: ftp)

Exponential decay of two-body eigenfunctions: A review ^{∗}

P. D. Hislop

Dedicated to Eyvind H. Wichmann

Abstract

We review various results on the exponential decay of the eigenfunctions of two-body Schrödinger operators. The exponential, isotropic bound results of Slaggie and Wichmann [15] for eigenfunctions of Schrödinger operators corresponding to eigenvalues below the bottom of the essential spectrum are proved. The exponential, isotropic bounds on eigenfunctions for nonthreshold eigenvalues due to Froese and Herbst [5] are reviewed. The exponential, nonisotropic bounds of Agmon [1] for eigenfunctions corresponding to eigenvalues below the bottom of the essential spectrum are developed, beginning with a discussion of the Agmon metric. The analytic method of Combes and Thomas [4], with improvements due to Barbaroux, Combes, and Hislop [2], for proving exponential decay of the resolvent, at energies outside of the spectrum of the operator and localized between two disjoint regions, is presented in detail. The results are applied to prove the exponential decay of eigenfunctions corresponding to isolated eigenvalues of Schrödinger and Dirac operators.

1 Introduction

The decay properties of bound state wave functions of Schrödinger operators have been intensively studied for many years. We are concerned here with the simplest part of the theory: The decay of the wave functions for two-body Schrödinger operators. We consider a Schrödinger operator of the form

H = −Δ + V,  (1.1)

acting on L^{2}(ℝ^{n}). The potential V is a real-valued function assumed to be sufficiently regular so that H is essentially self-adjoint on C_{0}^{∞}(ℝ^{n}). We will assume that

lim_{∥x∥→∞} |V (x)| = 0.  (1.2)

∗Mathematics Subject Classifications: 81Q10.

Key words: Schrodinger operator, eigenfunction, exponential decay, Dirac operator.

\begin{matrix}c \\ ◯ \end{matrix}2000 Southwest Texas State University and University of North Texas.

Published November 3, 2000.

Supported in part by NSF grant DMS 97-07049

We also assume that V is well-enough behaved so that the spectrum of H, denoted by σ(H), has the standard form

σ(H) = σ_{ess}(H) ∪ σ_{d}(H)
= [0, ∞) ∪ {E_{j} | − E_{j} > 0, j = 0, 1, …}  (1.3)

For example, if V is continuous and satisfies (1.2), then the spectrum of H has this form. This is due to the invariance of the essential spectrum under relatively-compact perturbations [9]. In many commonly encountered cases, for example, when V has decaying derivatives, the essential spectrum is purely absolutely continuous. However, it is known that condition (1.2) is not sufficient to guarantee the absolute continuity of the spectrum. There are examples due to Pearson [12] of bounded, decaying potentials for which the Schrödinger operator has purely singular continuous spectrum. We will not need these fine spectral results here.

Suppose that E < 0 is an eigenvalue of H with eigenfunction ψ_{E} ∈ L^{2}(ℝ^{n}) so that Hψ_{E} = Eψ_{E}. We will always assume that the eigenfuncton is normalized so that

∫_{ℝ^{n}} |ψ_{E}(x)|^{2} d^{n}x = 1.

We are interested in the spatial behavior of this function ψ_{E}(x), as ∥x∥ → ∞. The well-known example of the eigenfunctions of the hydrogen atom Hamiltonian provides a guide. The hydrogen atom Hamiltonian on L^{2}(ℝ^{3}) has the form

H = −Δ − \frac{1}{∥x∥}.  (1.4)

It is easy to check that the spherically symmetric function

ψ_{E}(x) = \frac{1}{\sqrt{8π}} e^{−\sqrt{\frac{1}{4}} ∥x∥},  (1.5)

is a normalized eigenfunction of H with eigenvalue E = −1/4, the ground state energy. We note that the eigenfunction decays exponentially with a factor given by the square root of the distance from the eigenvalue E = −1/4 to the bottom of the essential spectrum inf σ_{ess}(H) = 0. We will see that this is a characteristic exponential decay behavior.

In general, we do not know if an eigenfunction is continuous, and so pointwise bounds are not meaningful. For the general case, it is convenient to describe the decay of an eigenfunction in the L^{2}-sense. For a nonnegative function F, we say that ψ decays like e^{−F} in the L^{2}-sense if

∥e^{F} ψ∥ ≤ C_{0},  (1.6)

for some finite constant C_{0} > 0. Of course, if we know more about the regularity of the potential V, we can use a simple argument to conclude the regularity of the eigenfunction ψ_{E} corresponding to an eigenvalue E < 0. From this regularity and an L^{2}-exponential decay estimate, we can prove the pointwise decay of the eigenfunction. Simple regularity results are based on the Sobolev Embedding Theorem, which we now state.

Theorem 1.1 Any function f ∈ H^{s+k}(ℝ^{n}), for s > n/2 and k ≥ 0, can be represented by a function f ∈ C^{k}(ℝ^{n}).

Proposition 1.2 Suppose that H is a self-adjoint Schrödinger operator and the potential V is bounded and satisfies (1.2). If, additionally, V ∈ C^{2k}(ℝ^{n}), with bounded derivatives, then an eigenfunction ψ_{E} ∈ L^{2}(ℝ^{n}), corresponding to an eigenvalue E < 0, satisfies ψ_{E} ∈ H^{2k+2}(ℝ^{n}). If (2k + 2) > n/2 + l, then the eigenfunction satisfies ψ_{E} ∈ C^{l}(ℝ^{n}).

Proof. We note that since E < 0, the resolvent of H_{0} = −Δ exists at energy E. Furthermore, the resolvent R_{0}(E) = (H_{0} − E)^{−1} maps H^{s}(ℝ^{n}) → H^{s+2}(ℝ^{n}), for all s ∈ ℝ. The eigenvalue equation can be written as

ψ_{E} = −R_{0}(E)V ψ_{E}.  (1.7)

Since V is bounded, the potential is a bounded operator on L^{2}(ℝ^{n}). Equation (1.7) then shows that ψ_{E} ∈ H^{2}(ℝ^{n}). We now repeat this argument k-times since V is a bounded operator on H^{s}(ℝ^{n}), provided s ≤ k. From this, we conclude that ψ_{E} ∈ H^{2k+2}(ℝ^{n}). The last statement follows from the Sobolev Embedding Theorem. □

Once regularity of the eigenfunction has been established, the pointwise decay estimate is derived from a local estimate of the following type (cf. [9] for a simple case, or [1], for a general proof). Let B(y, r) denote the ball of radius r > 0 about the point y ∈ ℝ^{n}. There exists a constant C_{E,V}, depending on the potential V, the energy E, and inf σ(H), but independent of x_{0} ∈ ℝ^{n}, so that

max_{x∈B(x_{0},1/2)}|ψ_{E}(x)| ≤ C_{E,V} ∥ψ_{E}∥_{L^{2}(B(x_{0},1))}.  (1.8)

Let us suppose that the exponential weight F in (1.6) is translation invariant and satisifes a triangle inequality: F (x) ≤ F (x − y) + F (y). With this assumption and estimate (1.8), we find

max_{x∈B(x_{0},1/2)} |ψ_{E}(x)e^{F (x)}|

≤ C_{E,V} (max_{x∈B(x_{0},1)}e^{F (x)}) (max_{y∈B(x_{0},1)}e^{−F (y)}) ∥ψ_{E}e^{F} ∥_{L^{2}(B(x_{0},1))}
≤ C_{1}∥ψ_{E}e^{F} ∥_{L^{2}(ℝ^{n})}
≤ C_{2},  (1.9)

where we used the triangle inequality to combine the exponential terms. This proves that |ψ_{E}(x_{0})| ≤ C_{1}e^{−F (x_{0})}, for a constant C_{1} independent of x_{0}. Since x_{0} is arbitrary, the eigenfunction satisfies a pointwise exponential bound.

There are two basic types of upper bounds on wave functions as ∥x∥ → ∞. We will present these as pointwise estimates on the eigenfunction, although they can be formulated in the L^{2}-sense as described above. We say that a function ϕ satisfies an isotropic decay estimate if there exists a nonnegative function F : ℝ^{+} → ℝ^{+} so that

|ϕ(x)| ≤ C_{α}e^{−F (∥x∥)}.  (1.10)

In many situations, we have F (∥x∥) = α∥x∥, for some 0 < α < ∞. In the situation where |V (x)| → 0 uniformly with respect to ω ∈ S^{n−1}, as ∥x∥ → ∞, isotropic decay estimates are optimal. We have seen this from the example of the hydrogen atom ground state wave function. In the situation that the potential has different limits at infinity, depending on the direction, or, the limit at infinity is not achieved uniformly, as in the case of some nonspherically-symmetric potentials, it is more precise to replace the isotropic exponential factor with an anisotropic function which expresses this variation. These anisotropic exponents are described by a function ρ_{E}(x) : ℝ^{n} → ℝ^{+} that, as we will see, depends on the eigenvalue E and the potential V. An anisotropic upper bound has the form

|ϕ(x)| ≤ C_{0}e^{−ρ_{E}(x)}.  (1.11)

This is the case for N-body Schrödinger operators when the potential is a sum of pair potentials. Agmon [1] has developed an extensive theory of anisotropic decay estimates.

We will generalize the family of Hamiltonians described so far in order to incorporate Schrödinger operators with gaps in the essential spectrum. The situation we envision is the perturbation of a periodic Schrödinger operator H_{per} = −Δ + V_{per} by a compactly supported potential W. It is well-known that the spectrum of H_{per} is the union of intervals B_{j}, called bands, so that

σ(H_{per}) = ∪_{j≥0}B_{j}.  (1.12)

For many periodic potentials, there exist two consecutive bands B_{j} and B_{j+1} that do not overlap. We say that there is an open spectral gap G between the two bands. A local perturbation W, with compact support, preserves the essential spectrum. In the case that W has fixed sign, say W ≥ 0, it is easy to show, using the Birman-Schwinger principle, that for λ > 0 sufficiently large, the perturbed Hamiltonian, H(λ) = H_{per} + λW, has bound states at energies in the gap G. Suppose that E ∈ G. The existence of an eigenvalue for H(λ) = H_{0} +λW at E, for some λ ≠ 0, is equivalent to the existence of an eigenvalue of the compact, self-adjoint operator K(E) ≡ W^{1/2}(H_{0} − E)^{−1}W^{1/2} equal to −1/λ. We simply choose λ ∈ ℝ so that −1/λ is an eigenvalue of the compact, self-adjoint operator K(E). This argument can be generalized to perturbations W with compact support (or, sufficiently rapid decay) and not necessarily fixed sign [7]. We are interested in the exponential decay of the eigenfunctions corresponding to eigenvalues in open spectral gaps.

The methods used to describe eigenfunction decay below cover three main cases encountered in the study of Schrödinger operators:

1. Isolated eigenvalues below the bottom of the essential spectrum;

2. Eigenvalues embedded in the essential spectrum;

3. Eigenvalues lying in the spectral gap of an unperturbed operator.

As we will discuss in section 3, embedded eigenvalues at positive energies do not occur for most Schrödinger operators. The embedded eigenvalues referred to in point 2 occur at negative energies in the N-body case for N ≥ 3. A notable exception is the Wigner-von Neumann potential in one-dimension that has an embedded eigenvalue at positive energy. In general, embedded eigenvalues do occur for Schrödinger operators, but they are rare. The methods presented here, especially the Agmon technique, can be extended so as to apply to resonance eigenfunctions (cf. [9]). The methods can also be applied to the study of eigenfunctions for the Laplace-Beltrami operator on noncompact Riemannian manifolds (cf. [8]). Other references on the exponential decay of eigenfunctions can be found in [13].

2 The Slaggie-Wichmann Results on Two-Body Wave Functions

In 1962, Slaggie and Wichmann [15] published a paper in which they studied the decay properties of the eigenfunctions of three-body Schrödinger operators using integral operator methods. Although we will not discuss the many-body problem here, we are interested in their proof of exponential decay of eigenfunctions corresponding to negative energies E_{j} < 0 in the two-body situation. The proof of Slaggie and Wichmann is very simple and requires minimum regularity on the potential. The method capitalizes on a basic fact that will be used again below: The Green’s function for the unperturbed operator H_{0} = −Δ, in dimensions n ≥ 3 and at negative energies, decays exponentially in space. In three dimensions, the kernel of the resolvent is given by

R_{0}(z)(x, y) = \{4π∥x − y∥\}^{−1}e^{i\sqrt{z}∥x−y∥},  (2.1)

with the branch cut for the square root taken along the positive real axis. In higher dimensions, the kernel of the resolvent is given by a Hankel functions of the first kind that exhibits similar exponential decay. Since the spectrum of H_{0} is the half-axis [0, ∞), the bound state energies of H lie outside the spectrum of H_{0} and, consequently, the free Green’s function exhibits exponential decay at those energies. In section 5, we will discuss the exponential decay of the resolvent in more detail.

The hypotheses of Slaggie and Wichmann on the real-valued potential V are rather general.

Hypothesis 1. There exists a positive, continuous function Q(s), with s ∈ ℝ^{+}, having the properties

lim_{s→∞} Q(s) = 0  (2.2)
lim_{s→0} sQ(s) ≤ C_{0},  (2.3)

and such that

|V (x)| ≤ Q(∥x∥), for x ≠ 0.  (2.4)

We also assume that V ∈ L_{loc}^{1}(ℝ^{3}), and that V is relatively Laplacian bounded with relative bound less than one.

These conditions allow a Coulomb singularity at the origin and slow decay at infinity. It is not difficult to prove, using a Weyl sequence argument, that σ_{ess}(H) = [0, ∞). Alternatively, one can show that |V |^{1/2}(−Δ + 1)^{−1} is a compact operator. It follows that (−Δ + 1)^{−1}V (−Δ + 1)^{−1} is compact so that σ_{ess}(H) = σ_{ess}(−Δ) = [0, ∞).

Theorem 2.1 Let H = −Δ + V be a two-body Schrödinger operator satisfying Hypothesis 1. Let E_{j} < 0 be a negative bound state energy and let ϕ_{j} be any corresponding normalized eigenfunction satisfying Hϕ_{j} = E_{j}ϕ_{j}. For any 0 < θ < \sqrt{|E_{j}|}, there is a constant 0 < C(θ) < ∞, so that

|ϕ_{j}(x)| ≤ C(θ)e^{−θ∥x∥}.  (2.5)

Let us note that this bound is saturated for the Coulomb ground state wave function which is ψ(x) = C_{0}e^{−\sqrt{|E_{0}|} ∥x∥}, with E_{0} = −1/4. The square root in the energy behavior comes from the dependence of the free Green’s function on the energy, as seen in (2.1).

Simon [14] proved a similar result on the exponential decay of eigenfunctions corresponding to negative eigenvalues of two-body Schrödinger operators on L^{2}(ℝ^{3}), with V ∈ L^{2}(ℝ^{3}) (and also for V in the Rollick class), using the integral equation (2.7). Simon used a result, proved in [14], on the solutions of certain integral equations associated with Hilbert-Schmidt kernels. Suppose that K(x, y) is a Hilbert-Schmidt kernel, that is,

∫_{ℝ^{3}} ∫_{ℝ^{3}} |K(x, y)|^{2} d^{3}xd^{3}y < ∞,

and suppose that G(x) is a nonzero, measurable function, and that G(x)^{−1} exists. Consider the kernel M(x, y) = G(x)K(x, y)G(y)^{−1}, and suppose that the kernel M(x, y) is also Hilbert-Schmidt. If ψ ∈ L^{2}(ℝ^{n}) satisfies Kψ = ψ, then G(x)ψ(x) ∈ L^{2}(ℝ^{n}). In the application of this general result, we note that the kernel of the integral equation (2.7) is Hilbert-Schmidt. For the function G(x), we choose G(x) = e^{θ∥x∥}, for 0 ≤ θ < \sqrt{|E_{j}|}, as in Theorem 2.1. It is easy to show that the modified kernel M(x, y) is also Hilbert-Schmidt, so the exponential decay of the eigenfunction follows.

Proof of Theorem 2.1. We will repeated use one basic fact below. An L^{2}- eigenfunction in 3-dimensions is necessarily continuous. This is a consequence of the facts that such an eigenfunction is in the Sobolev space H^{2}(ℝ^{3}), and the Sobolev Embedding Theorem, Theorem 1.1. The eigenvalue equation

(−Δ + V)ψ_{j} = E_{j}ψ_{j},  (2.6)

implies the integral equation for ψ_{j},

ψ_{j}(x) = − ∫_{ℝ^{3}} \frac{e^{−\sqrt{|E_{j}|}∥x−x′∥}}{4π∥x − x′∥} V (x′)ψ_{j}(x′) d^{3}x′.  (2.7)

It follows immediately from Hypothesis 1 and (2.7) that

|ψ_{j}(x)| ≤ ∫_{ℝ^{3}} \frac{e^{−\sqrt{|E_{j}|}∥x−x′∥}}{4π∥x − x′∥} Q(∥x′∥)|ψ_{j}(x′)| d^{3}x′.  (2.8)

Let us define a function m(x) by

m(x) ≡ sup_{x′}{ |ψ_{j}(x′)|e^{−θ∥x−x′∥} },  (2.9)

and, motivated by (2.8), another function h_{θ}(x) by

h_{θ}(x) ≡ ∫_{ℝ^{3}} \frac{e^{−(\sqrt{|E_{j}|}−θ)∥x−x′∥}}{4π∥x − x′∥} Q(∥x′∥) d^{3}x′,  (2.10)

for 0 < θ < |E_{j}|. It is easy to check that h_{θ} is continuous, rotationally invariant, and thus a function of ∥x∥ only. These two definitions and inequality (2.8) imply that

|ψ_{j}(x)| ≤ h_{θ}(∥x∥)m(x).  (2.11)

We next prove that

lim_{∥x∥→∞} h_{θ}(∥x∥) = 0.  (2.12)

We divide the region of integration ℝ^{3} into two regions: ∥x − x′∥ < ϵ and ∥x−x′∥ > ϵ, for some ϵ > 0, to be determined below. In the first region, we easily show that h_{θ}(x) ≤ Cϵ^{2}∥x∥^{−1}. In the second region, we use the boundedness of Q and write

∫_{∥x−x′∥>ϵ} \frac{e^{−λ∥x−x′∥}}{4π∥x − x′∥} Q(∥x∥)d^{3}x′ ≤ \frac{C_{0}}{4πϵ} ∫_{ℝ^{3}} e^{−λ∥u∥}d^{3}u,  (2.13)

where λ = \sqrt{|E_{j}|} − θ > 0. If we choose ϵ = ∥x∥^{1/4}, for example, property (2.12) follows from these two estimates.

This decay of the function h_{θ}(x) as ∥x∥ → ∞ controls the decay of the eigenfunction in the following sense. Because h_{θ} vanishes at infinity, there exists a region R ⊂ ℝ^{3} on which h_{θ}(x) < 1. We can simply take R = ℝ^{3}\B_{R}(0), for a radius R sufficiently large. We denote by R^{c} the complement of this region. We have from (2.11) that |ψ_{j}(x)| < m(x), for x ∈ R. On the otherhand, we have for all x ∈ ℝ^{3} that

m(x) = max ( sup_{x′∈R}{|ψ_{j}(x′)|e^{−θ∥x−x′∥}}, sup_{x′∈R^{c}}{|ψ_{j}(x′)|e^{−θ∥x−x′∥}}) . (2.14)

Our goal is to show that the maximum is obtained by the R^{c}-term. This will immediately imply the result. To this end, we first note that from the definition of m, for any x′′ ∈ ℝ^{3}, we have

m(x) = sup_{x′}{|ψ_{j}(x′)|e^{−θ∥x−x′∥}}
= sup_{x′}{|ψ_{j}(x′)|(sup_{x′′} e^{−θ∥x′−x′′∥}e^{−θ∥x′′−x∥})}
= sup_{x′′}{m(x′′)e^{−θ∥x′′−x∥}}.  (2.15)

We have used the identity

e^{−θ∥x−x′∥} = sup_{x′′} e^{−θ∥x−x′′∥}e^{−θ∥x′′−x′∥},  (2.16)

that is proved by the triangle inequality and the definition of the supremum. Using (2.15), and the fact that |ψ_{j}(x)| < m(x), for x ∈ R, we compute,

sup_{x′∈R}{|ψ_{j}(x′)|e^{−θ∥x−x′∥}}

< sup_{x′∈R}{m(x′)e^{−θ∥x−x′∥}}
≤ sup_{x′∈ℝ^{3}}{m(x′)e^{−θ∥x−x′∥}}
= m(x).  (2.17)

That is, the supremum over x′ ∈ R occurring in (2.14) is strictly less that m(x). Hence, we have that

m(x) = sup_{x′∈R^{c}}{|ψ_{j}(x′)|e^{−θ∥x−x′∥}}.  (2.18)

We can take R to be the exterior of a ball of radius R, for sufficiently large R, due to the vanishing of h_{θ}. It follows immediately from the continuity of the eigenfunction ψ_{j} and (2.18) that

m(x) ≤ C(R, θ)e^{−θ∥x∥},  (2.19)

for all x ∈ ℝ^{n}, and for some constant depending on R > 0 and θ. Inequality (2.11), that |ψ_{j}(x)| ≤ h(∥x∥)m(x), for all x ∈ ℝ^{3}, and the boundedness of h, implies that there exists a constant C_{0} > 0 so that |ψ_{j}(x)| ≤ C_{0}m(x). This, together with (2.19), establishes the upper bound on the eigenfunction. □

As noted by Slaggie and Wichmann, the proof requires less restrictive conditions on the potential V. The potential must satisfy conditions (2.2)–(2.4), and, for each 0 < θ < \sqrt{|E_{j}|}, there must exist an R_{θ} > 0 so that h_{θ}(∥x∥) < 1 for ∥x∥ > R_{θ}.

3 The Froese-Herbst Method

We indicate the basic ideas of the Froese-Herbst method [5] for proving the decay of eigenfunctions. The authors’ main motivation and results concern the exponential decay of the eigenfunctions of N-body Schrödinger operators, and the absence of positive eigenvalues for N-body Schrödinger operators. We will only give the simplest version of the results here. The Froese-Herbst method does not depend upon the explicit properties of the free Green’s function, as in the Slaggie-Wichmann method. Consequently, the Froese-Herbst method can be applied to more general classes of differential operators, such as Laplace-Beltrami operators on noncompact manifolds, and to the study of eigenfunctions corresponding to eigenvalues embedded in the essential spectrum.

The Froese-Herbst method is tied to the theory of positive commutators as developed by E. Mourre [10]. We will briefly review the main points of this theory below. The Froese-Herbst method yields L^{2}-exponential bounds of the form

e^{F} ψ ∈ L^{2}(ℝ^{n}),  (3.1)

for some function F. Under more regularity assumptions on V, this L^{2}-expo-nential bound can be converted to a pointwise exponential bound, as explained in section 1.

The Froese-Herbst method identifies the threshhold energies associated with the Hamiltonian H as controlling the rate of decay of the eigenfunctions. This means the following. Let Σ = inf σ_{ess}(H) be the bottom of the essential spectrum. For many-body systems, this can be strictly negative. The bound state energies of subsystems lie between Σ and 0. These energies are called thresholds of the system. More generally, we define threshold energies as those energies at which the Mourre estimate (3.7) fails to hold. For many two-body Schrödinger operators, the only threshold energy is zero, which is also the bottom of the essential spectrum. We mention that there may be an eigenvalue at zero energy, or at any threshold. There is no general method for obtaining estimates on the decay rate of the corresponding eigenfunctions. There are examples for which the decay rate is only inverse polynomial, rather than exponential.

The general Froese-Herbst result states that any eigenfunction ψ_{E} ∈ L^{2}(ℝ^{n}), with Hψ_{E} = Eψ_{E}, decays exponentially at a rate given by the square root of the distance from the eigenvalue to a threshold above the eigenvalue. That is, for some threshold energy τ > E, we have the bound,

e^{(\sqrt{τ−E}−ϵ) ∥x∥}ψ_{E}(x) ∈ L^{2}(ℝ^{n}),  (3.2)

for any ϵ > 0. Note that when τ = 0, this is basically the Slaggie-Wichmann result.

In the two-body case studied here, the potential V must satisfy the following hypothesis. We write R_{0}(z) for the resolvent of the Laplacian, R_{0}(z) = (−Δ − z)^{−1}.

Hypothesis 2. We assume that the potential V ∈ C^{1}(ℝ^{n}), and is relatively −Δ-bounded, with relative bound less than one. Furthermore, we assume that there exists a constant 0 < C_{0} < ∞, so that V satisfies

∥R_{0}(−1)(x · ∇V)R_{0}(−1)∥ ≤ C_{0}.  (3.3)

We can relax the C^{1}-condition and assume that condition (3.3) holds in the sense of quadratic forms, cf. [6]. The main result of the Froese-Herbst in the two-body case is the following theorem.

Theorem 3.1 Let H = −Δ+V be a two-body Schrödinger operator with potential V satisfying Hypothesis 2. Suppose that for E < 0, is a bound state energy with eigenfunction ψ_{E} ∈ L^{2}(ℝ^{n}), satisfying Hψ_{E} = Eψ_{E}. Then, we have for any ϵ > 0,

e^{(\sqrt{|E|}−ϵ) ∥x∥}ψ_{E}(x) ∈ L^{2}(ℝ^{n}).  (3.4)

If, in addition, we assume that x · ∇V is relatively Laplacian bounded with relative bound less than 2, then H has no positive eigenvalues.

One of the main applications of the method of proof of this theorem (and its counterpart in the N-body case) is to prove the nonexistence of positive eigenvalues of Schrödinger operators. The idea is to prove that any L^{2}-eigenfunction ψ_{E}, corresponding to a positive energy eigenvalue E, must decay faster than any exponential. That is, for all θ > 0, we have

e^{θ∥x∥}ψ_{E}(x) ∈ L^{2}(ℝ^{n}).  (3.5)

Since the decay of an eigenfunction is controlled by the distance to a threshold larger than the eigenvalue, one must prove that an N-body Schrödinger operator has no positive thresholds. For the two-body case, we will show that the Mourre estimate holds at all positive energies, so there are no positive thresholds. Consequently, we see that the eigenfunction for a positive eigenvalue must decay faster than any exponential. A variant of a unique continuation argument then shows that such a function ψ_{E} = 0.

We now give an outline of the proof of the exponential decay part of the Froese-Herbst Theorem. A complete textbook presentation is given in [6]. We will work with the specific case of two-body operators. We begin with the Mourre theory of positive commutators [10]. Let A = \frac{1}{2}(x · ∇ + ∇ · x) be the skew-adjoint operator so that −iA is the self-adjoint generator of the dilation group on L^{2}(ℝ^{n}). We assume that the potential V satisfies Hypothesis 2. A simple computation shows that, formally, the commutator [H, A] ≡ HA − AH, is

[H, A] = 2H − 2V − 2x · ∇V = 2H + K.  (3.6)

The operator K is relatively-Laplacian compact. Let I = [I_{0}, I_{1}] ⊂ ℝ be a finite, closed interval, and let E_{H}(I) be the spectral projection for H and the interval I. Conjugating the commutator relation (3.6) by the spectral projectors E_{H}(I), we obtain the Mourre estimate,

E_{H}(I)[H, A]E_{H}(I) = 2E_{H}(I)H + E_{H}(I)KE_{H}(I)
≥ 2I_{0}E_{H}(I) + E_{H}(I)KE_{H}(I).  (3.7)

This estimate implies a Virial Theorem of the following type. Suppose that the Mourre estimate (3.7) holds in a neighborhood I with compact operator K = 0. Then, the operator H cannot have an eigenvalue E ∈ I. Since, if E ∈ I is an eigenvalue with an eigenfunction ψ_{E}, we have (neglecting domain considerations),

⟨ψ_{E}, [H, A]ψ_{E}⟩ = ⟨ψ_{E}, [H − E, A]ψ_{E}⟩
= 0.  (3.8)

On the otherhand, the Mourre estimate (3.7) with K = 0 implies

⟨ψ_{E}, [H, A]ψ_{E}⟩ ≥ 2I_{0} > 0.  (3.9)

This inequality clearly contradicts (3.8). Consequently, the energy E cannot be an eigenvalue for H.

This simple idea lies behind the proof of the Froese-Herbst Theorem. Suppose E is an eigenvalue of H, and define ⟨x⟩ ≡ (1 + ∥x∥^{2})^{1/2}. We define τ ≡ sup {E + α^{2} | α > 0, and e^{α⟨x⟩}ψ_{E} ∈ L^{2}(ℝ^{n})}. If τ = α_{0}^{2} + E is not a threshold of H, then there exist α_{1} ≥ 0 and γ > 0, with α_{1} < α_{0} < α_{1} + γ, so that e^{α_{1}⟨x⟩}ψ_{E} ∈ L^{2}(ℝ^{n}), but e^{(α_{1}+γ)⟨x⟩}ψ_{E} is not in L^{2}(ℝ^{n}). Because τ is not a threshold of H, the Mourre estimate holds in a neighborhood of τ. In particular, it holds in a neighborhood of E + α_{1}^{2}, for some α_{1} sufficiently close to α_{0}, since the set of thresholds is closed. We will construct a sequence of approximate eigenfunctions Ψ_{s} for H and the eigenvalue E + α_{1}^{2} in the sense that ∥(H − E − α_{1}^{2})Ψ_{s}∥ ≤ C_{0}γ, and Ψ_{s} converges weakly to zero as s → 0. The Virial Theorem then implies that the matrix element of the [H, A] in the state Ψ_{s}, which is approximately an eigenfunction with eigenvalue E + α_{1}^{2}, is very small with respect to γ. On the other hand, the Mourre estimate holds in a small neighborhood E + α_{1}^{2}, and, since Ψ_{s} converges weakly to zero, the matrix element ⟨Ψ_{s}, KΨ_{s}⟩ → 0. This implies that the matrix element of [H, A] in the state Ψ_{s} is strictly positive. This gives a contradiction for small γ.

In the first step of the proof, we construct states with shifted energy. For motivation, recall that a translation in momentum space has the effect of shifting the classical energy. Let ψ_{E} be an L^{2}-eigenfunction of H, and assume that F is a differentiable function such that ψ_{F} = e^{F} ψ_{E} ∈ L^{2}(ℝ^{n}). We want to compute the conjugated operator e^{F} He^{−F}. To do this, we note that for any u ∈ C_{0}^{∞}(ℝ^{n}), we have

e^{F} (−i∇)e^{−F} u = (−i∇ + i∇F)u,

so that

−e^{F} Δe^{−F} = (−i∇ + i∇F)^{2} = −Δ + (∇ · ∇F + ∇F · ∇) − |∇F |^{2}.

It then follows that

H_{F} ≡ e^{F} He^{−F} = H + (∇F · ∇ + ∇ · ∇F) − |∇F |^{2}.  (3.10)

It follows, after a short calculation, that the expected value of the Hamiltonian H in the state ψ_{F} is

⟨ψ_{F} , Hψ_{F} ⟩ = ⟨ψ_{F} , [E + |∇F |^{2}]ψ_{F} ⟩,  (3.11)

so the state ψ_{F} appears as a state with energy E + |∇F |^{2}.

Next, we choose a family of functions F_{s} so that |∇F_{s}|^{2} ∼ α_{1}^{2}, and so that the sequence ψ_{F_{s}} ≡ ψ_{s} converges weakly to zero. Let χ_{s}(t) be a smooth function of compact support satisfying lim_{s→0} χ_{s}(t) = t. We now define a weight F_{s}(x) ≡ α_{1} + γχ_{s}(⟨x⟩) having the property that lim_{s→0} F_{s}(x) = α_{1} + γ. It then follows that ψ_{s} = e^{F_{s}}ψ_{E} ∈ L^{2}(ℝ^{n}), provided s > 0, but that ∥ψ_{s}∥ → ∞, as s → 0. Furthermore, a calculation reveals that |∇F_{s}|^{2} ∼ α_{1}^{2}, for small γ. We define Ψ_{s} ≡ ψ_{s} ∥ψ_{s}∥^{−1}, so that Ψ_{s} converges weakly to zero. It is not too difficult to show that Ψ_{s} is the sequence of approximate eigenfunctions we desire, in the sense that

∥(H − E − α_{1}^{2})Ψ_{s}∥ ∼ 0,  (3.12)

for small γ. Finally, it follows from the Virial Theorem (3.8) that the matrix element ⟨Ψ_{s}, [H, A]Ψ_{s}⟩ ∼ 0. On the otherhand, since the Mourre estimate holds in a neighborhood of E + α_{1}^{2}, by the assumption that E + α_{0}^{2} is not a threshold, and the sequence Ψ_{s} converges weakly to zero, we know that this matrix element is bounded from below by a strictly positive constant. This gives a contradiction, so that τ = E + α_{0}^{2} must be a threshold.

Some final comments are in order. The Froese-Herbst technique depends upon the existence of a conjugate operator A for a given Hamiltonian H. It is not always easy to construct a conjugate operator, but this has now been done in a variety of situations. Secondly, if the energy E itself is a threshold, the method gives no information about the rate of decay of a corresponding eigenfunction. Thirdly, the proof indicates that the rate of decay of the eigenfunction is controlled by the square root of the distance to some threshold above E, similar to the Slaggie-Wichmann result. The proof, however, does not indicate that it is always the nearest threshold above E that controls the exponential decay.

4 Nonisotropic Agmon Decay Estimates

The results that we have discussed so far are exponential decay estimates of the form e^{F} ψ ∈ L^{2}(ℝ^{n}), with F a function of ∥x∥ alone. Hence, the resulting bounds are spatially isotropic. For the case of a two-body potential, these are optimal when the potential is spherically symmetric. In general, isotropic bounds do not reflect the variation of the potential with direction. In the many-body case, when the total potential is the sum of two-body potentials, the behavior of V at infinity depends crucially on the direction. Hence, one is led to develop nonisotropic bounds on the decay of the wave function that more closely reflect the behavior of the potential in each direction. Such bounds and techniques are also crucial for the estimation of the lifetime of quantum resonances in terms of the potential barrier generating the resonance.

A systematic study of the decay of eigenfunctions of second-order partial differential operators, corresponding to eigenvalues below the bottom of the essential spectrum, was performed by Agmon [1]. A key role is played by the Agmon metric on ℝ^{n}. This pseudo-Riemannian metric is constructed directly from the potential and the energy, and thus reflects the variation of the potential with direction. The distance function corresponding to the Agmon metric measures how the potential controls the decay. Explicitly solvable models in one-dimension, and the W KB approximation give some clue as to the form of this metric.

Definition 4.1 Let V be a bounded, real-valued function on ℝ^{n}, and let E ∈ ℝ. For any x ∈ ℝ^{n}, and ξ, η ∈ T_{x}(ℝ^{n}) = ℝ^{n}, the tangent space to ℝ^{n} at x, we define a (degenerate) inner product on T_{x}(ℝ^{n}) by

⟨ξ, η⟩_{x} ≡ (V (x) − E)_{+}⟨ξ, η⟩_{E},  (4.1)

where ⟨·, ·⟩_{E} is the usual Euclidean inner product and f(x)_{+} ≡ max{f(x), 0}. The corresponding pseudo-metric on ℝ^{n} is called the Agmon metric induced by the potential V at energy E.

It is important to note that the Agmon metric depends on both the potential and the energy E. The Agmon metric on ℝ^{n} is degenerate because there may exist nonempty turning surfaces {x ∈ ℝ^{n} | V (x) = E}, and classically forbidden regions {x ∈ ℝ^{n} | V (x) < E}. These sets play an important role in the theory. The turning surface marks the limits of classical motion for a particle with energy E moving under the influence of the potential V, and such a particle cannot penetrate into the classically forbidden region. Consequently, it is expected that the quantum mechanical wave function is small in the classically forbidden region.

We use the structure given in Definition 4.1 to construct a distance function (or, metric) on ℝ^{n}. Let γ : [0, 1] → ℝ^{n} be a differentiable path in ℝ^{n}. The derivative γ̇(t) belongs to the tangent space at the point γ(t). For any Riemannian metric g on a manifold ℝ^{n}, the length of γ is given by the integral

L(γ) = ∫_{0}^{1} ∥ γ̇(t) ∥_{γ(t)} dt,  (4.2)

where ∥ ξ ∥_{x}= ⟨ξ, ξ⟩_{x}^{1/2}, for ξ ∈ T_{x}(ℝ^{n}). In the Agmon structure (4.1), the length of the curve γ (4.2) is:

L_{A}(γ) = ∫_{0}^{1}(V (γ(t)) − E)_{+}^{1/2} ∥ γ̇(t) ∥_{E} dt,  (4.3)

where ∥ · ∥_{E} denotes the usual Euclidean norm. A path γ is a geodesic if it minimizes the energy functional E(γ) ≡ \frac{1}{2} ∫_{0}^{1} ∥ γ̇(t) ∥_{γ(t)}^{2} dt.

Definition 4.2 Given a bounded, real-valued potential V and energy E, the distance between x, y ∈ ℝ^{n} in the Agmon metric is

ρ_{E}(x, y) ≡ inf_{γ∈P_{x,y}}L_{A}(γ),  (4.4)

where P_{x,y} ≡ {γ : [0, 1] → ℝ^{n} | γ(0) = x, γ(1) = y, γ ∈ AC[0, 1]}. Here, the set AC[0, 1] is the space of all absolutely continuous functions on [0, 1].

The distance between x, y ∈ ℝ^{n} with the Agmon metric is the length of the shortest geodesic connecting x to y. The distance function ρ_{E} in (4.4) reduces to the usual WKB factor for the 1-dimensional case,

ρ_{E}(x, y) = ∫_{x}^{y}(V (s) − E)_{+}^{1/2} ds.  (4.5)

The Agmon metric has several nice properties: It satisfies the triangle inequality, and is Lipschitz continuous. The main result of this section is that the Agmon metric at energy E controls the decay of an eigenfunction at energy E, provided E < inf σ_{ess}(H).

Theorem 4.1 Let H = −Δ + V, with V real and continuous, be a closed operator bounded below with σ(H) ⊂ ℝ. Suppose E is an eigenvalue of H, and that the support of the function (E − V (x))_{+} ≡ max (0, E − V (x)) is a compact subset of ℝ^{n}. Let ψ ∈ L^{2}(ℝ^{n}) be an eigenfunction of H such that Hψ = Eψ. Then, for any ϵ > 0, there exists a constant C_{ϵ}, with 0 < C_{ϵ} < ∞, such that

∫ e^{2(1−ϵ)ρ_{E}(x)}|ψ(x)|^{2}dx ≤ C_{ϵ},  (4.6)

where ρ_{E}(x) ≡ ρ_{E}(x, 0).

We note that if V satisfies (1.2) uniformly in the sense that |V (x)| < ϵ for ∥x∥ large enough, and if E < 0, then the support of the positive part of (E −V) is compact.

Sketch of the Proof. We will sketch the proof here. A textbook treatment is given in [9], and the general cases are treated in [1]. The main idea of the Agmon approach is to use the strict positivity of (V − E), outside a compact set, in order to bound the quadratic form ⟨Φ, (H−E)Φ⟩ from below, for suitably chosen vectors Φ. Note that the set on which (V − E) > 0 is the classically forbidden region. A classical particle with energy E cannot penetrate this region. One expects that the corresponding quantum wave function is small in this region. The vector Φ has the form Φ = ηe^{F} ϕ, where F is a distance function built from the Agmon metric, the function η localizes the eigenfunction to the classically forbidden region, and we will take ϕ = e^{F} ψ_{E} after some initial calculations. Because Φ is built from an eigenfunction for H, the quadratic form ⟨Φ, (H−E)Φ⟩ is bounded above by the norm of ψ_{E}, localized near the support of ∇η. Hence, we arrive at an inequality roughly of the form,

∥ηe^{F} ψ_{E}∥^{2} ≤ C_{1}∥g(∇η)e^{F} ψ_{E}∥^{2},  (4.7)

where g(∇η) represents some combination of ∇η. Since e^{F} will be bounded on the support of ∇η, this, in turn, implies the L^{2}-exponential bound of ψ_{E}.

We now illustrate how to implement this strategy. Let F = (1 − ϵ)ρ_{E} and note that almost everywhere,

|∇F |^{2} ≤ (1 − ϵ)^{2}(V − E)_{+} ≤ (1 − ϵ)(V − E)_{+}.  (4.8)

We first compute a lower bound for the quadratic form ⟨Φ, (H − E)Φ⟩, for Φ = e^{F} ηϕ. We take η ∈ C^{2}(ℝ^{n}) to be a nonnegative function supported in the region where (V − E) ≥ δ, and η = 1, except near the boundary of this region. A key computation involves the gauge transformation, as in (3.10), given by H → H_{F} ≡ e^{F} He^{−F}. We recall the result that

e^{F} He^{−F} = H + (∇ · ∇F + ∇F · ∇) − |∇F |^{2}.  (4.9)

Consequently, for any reasonable function ϕ, we compute a lower bound on the quadratic form,

Re⟨e^{F} ηϕ, (H − E)e^{−F} ηϕ⟩
= Re⟨ηϕ, (H + (∇ · ∇F + ∇F · ∇) − |∇F |^{2} − E)ηϕ⟩
≥ ⟨ηϕ, (V − E − |∇F |^{2})ηϕ⟩
≥ ϵδ∥ηϕ∥^{2}.  (4.10)

We made use of the fact that Re⟨ϕ, (∇g · ∇ + ∇ · ∇g)ϕ⟩ = 0, for any real-valued function g. We use this lower bound by setting ϕ = e^{F} ψ_{E}. After some standard computations, the final formula is

Re⟨e^{2F} ηψ_{E}, (H − E)ηψ_{E}⟩ ≥ δϵ∥e^{F} ηψ_{E}∥^{2}.  (4.11)

We now turn to the upper bound. We control the exponentially growing term on the left in (4.12) by the compactness of the support of the gradient of η. Using the fact that ψ_{E} is an eigenfunction, we have

(H − E)ηψ_{E} = [−Δ, η]ψ_{E}
= (−Δη − 2∇η · ∇)ψ_{E}.  (4.12)

Since ∥∇ψ_{E}∥ ≤ C_{E}∥∇(H + M)^{−1}∥, for some M large enough, we obtain an upper bound of the form,

|⟨e^{2F} ηψ_{E}, (H − E)ηψ_{E}⟩| ≤ C_{E} ( sup_{x∈supp|∇η|} e^{2F (x)}) .  (4.13)

By arranging the diameter of the support of |∇η| small enough, we combine (4.12) and (4.14) to obtain

∥e^{(1−ϵ)ρ_{E}}ηψ_{E}∥^{2} ≤ C(E, η).  (4.14)

A simple estimate on ∥e^{(1−ϵ)ρ_{E}}(1−η)ψ_{E}∥^{2}, using the compactness of the support of (1 − η), completes the proof of the result. □

5 Resolvent Decay Estimates and the Combes-Thomas Method

In this final section, we examine another manner of proving exponential decay bounds on eigenfunctions based on the exponential decay of the resolvent at energies outside of the spectrum of the Hamiltonian. The Slaggie-Wichmann proof discussed in section 2 used the decay of the free Green’s function at energies below the bottom of the essential spectrum. We will first prove a result, due to Combes and Thomas [4], on the exponential decay of the resolvent of a self-adjoint Schrödinger operator, at energies outside of the spectrum, when localized between two disjoint regions. We will then use this estimate to prove the decay of eigenfunctions corrsponding to eigenvalues outside of the essential spectrum in certain cases. We conclude this section with an application to the exponential decay of eigenfunctions of the Dirac operator corresponding to eigenvalues in the spectral gap (−m, m).

We begin with a form of the Combes-Thomas method [4], due to Barbaroux, Combes, and Hislop [2], which allows an improvement on the rate of decay of the resolvent. Combes and Thomas, motivated by the work of O’Connor [11] and dilation analyticity, emphasized the use of analytic methods in the study of the decay of eigenfunctions. Their method is more flexible than the Slaggie-Wichmann or O’Connor method in that it can be applied, for example, to Schrödinger operators with nonlocal potentials, to N-body Schrödinger operators, and to other forms of differential operators. As a consequence, one obtains exponential decay for eigenfunctions corresponding to isolated eigenvalues in gaps of the essential spectrum, not just to those below the bottom of the essential spectrum.

The Combes-Thomas method also applies to N-body Schrödinger operators with dilation analytic two-body potentials. For such Schrödinger operators, the Combes-Thomas method allows one to prove the exponential decay on nonthreshold eigenfunctions. The result is similar to that obtained by the Froese-Herbst method. If ψ_{E} is an eigenfunction of H corresponding to a nonthreshold eigenvalue E, then e^{a∥x∥}ψ_{E} ∈ L^{2}(ℝ^{n}), for any a satisfying a^{2} < 2 inf{|E − ReE_{α}| + ImE_{α}}. The infimum is taken over all thresholds of the nonself-adjoint operator H(iπ/4), obtained from H by dilation analyticity. Since we will not discuss dilation analytic operators here, we refer the reader to [13]. As with the works mentioned above, Combes and Thomas are mainly concerned with the N-body problem. We will discuss the method as applied to two-body problems only.

A Simple Proof of Resolvent Decay Estimates

The idea of Combes and Thomas is to study the deformation of the Hamiltonian by a unitary representation of an abelian Lie group, and then to analytically continue in the group parameters. Typically, one uses the group of dilations in coordinate space, or boost transformations in momentum space. As an example, let the Hamiltonian H = −Δ + V be self-adjoint. It follows as in previous sections, that for a constant vector λ ∈ ℝ^{n}, we have

H(λ) ≡ e^{ix·λ}He^{−ix·λ} = H + 2iλ · ∇ + |λ|^{2}.  (5.1)

Provided that the operator ∇ is relatively H-bounded, the operator H(λ) extends to an analytic family of type A on ℂ^{n}.

We next study the resolvent of this analytic type A family of operators. Let us suppose that E ∈ ρ(H), the resolvent set of H. Then, the operator (H − E) is boundedly invertible, and we can write,

(H(λ) − E) = (1 + 2iλ · ∇(H − E)^{−1} + |λ|^{2}(H − E)^{−1})(H − E).  (5.2)

Let us choose λ so that

∥2iλ · ∇(H − E)^{−1}∥ < 1/2.  (5.3)

We define a constant C_{V,E} ≡ ∥∇(H − E)^{−1}∥, that we assume is finite. We require that |λ| satisfies

|λ| < \frac{1}{4C_{V,E}} .  (5.4)

Let us write B ≡ i∇(H − E)^{−1}. Assuming the condition (5.4) for the moment, and returning to (5.2), we see that

(H(λ) − E) = (1 + 2λB)(1 + (1 + 2λB)^{−1}|λ|^{2}(H − E)^{−1})(H − E). (5.5)

Once again, in order to invert the second factor, we demand |λ| also satisfies

∥(1 + 2λB)^{−1}|λ|^{2}(H − E)^{−1}∥ < 1/2.  (5.6)

Since H is self-adjoint, we have ∥(H − E)^{−1}∥ ≤ {dist(σ(H), E)}^{−1}. Let us define η by η ≡ dist(σ(H), E). To satisfy the bound (5.6), we require

|λ| ≤ \frac{\sqrt{η}}{2} .  (5.7)

Consequently, the inverse of (H(λ) − E) satisfies the bound ,

∥(H(λ) − E)^{−1}∥
≤ ∥(H − E)^{−1}∥ ∥(1 + B)^{−1}∥ ∥(1 + |λ|^{2}(1 + B)^{−1}(H − E)^{−1}∥
≤ 4 {dist (σ(H), E)}^{−1},  (5.8)

for λ ∈ ℂ^{n} with

|λ| ≤ min (\frac{1}{4C_{V,E}} , \frac{\sqrt{η}}{2} ) .  (5.9)

Let ν denote the minimum of the right side of (5.9). Thus, we have proved that for all λ ∈ ℂ^{n} with |λ| < ν, the dilated operator (H(λ) − E) is invertible, and we have the bound

∥e^{iλ·x}(H − E)^{−1}e^{−iλ·x}∥ ≤ \frac{4}{η} .  (5.10)

This bound is the key to proving exponential decay of the resolvent. For any u ∈ ℝ^{n}, let χ_{u} be a function with compact support near u ∈ ℝ^{n}. We consider a fixed vector y ∈ ℝ^{n}, and the origin. First, for any fixed, nonzero unit vector ê ∈ ℝ^{n}, we set λ = κê ∈ ℝ^{n} and find,

χ_{y}(H − E)^{−1}χ_{0} = e^{−iκê·x}χ_{y}(e^{iκê·x}(H − E)^{−1}e^{−iκê·x})e^{iκê·x}χ_{0}
= e^{−iκê·x}χ_{y}(H(κê) − E)^{−1}e^{iκê·x}χ_{0}.  (5.11)

Second, we analytically continue the last term in (5.11) to κ = −iν. This is possible because of the type A analyticity proved above, and because the localization functions have compact support. We obtain from (5.10)–(5.11),

∥χ_{y}(H − E)^{−1}χ_{0}∥ ≤ ∥χ_{y}e^{−νê·x}(H(−iν) − E)^{−1}e^{νê·x}χ_{0}∥
≤ \frac{C_{0}}{η} e^{−νê·y}.  (5.12)

Taking ê = y∥y∥^{−1}, it follows that

∥χ_{y}(H − E)^{−1}χ_{0}∥ ≤ \frac{C_{1}}{η} e^{−ν∥y∥}.  (5.13)

This is our first resolvent decay estimate. Notice that the estimate holds for any energy that is separated from the spectrum of H. The exponential rate of decay ν is given in (5.9). It is the minimum of C_{2}η and C_{3}\sqrt{η}. We will see that this can be improved.

We note an improvement of the above technique when the eigenvalue E satisfies E < Σ ≡ inf σ_{ess}(H). In this case, the operator (H − E) is positive in the sense that for all ϕ ∈ D(H),

⟨ϕ, (H − E)ϕ⟩ ≥ (Σ − E)∥ϕ∥^{2}.  (5.14)

We are back in the case considered by Agmon. We see that in this case the Combes-Thomas method is the same as an isotropic Agmon estimate with an exponential factor \sqrt{Σ − E}.

The Combes-Thomas Method

We now present an optimal version of the Combes-Thomas method [2] improving the presentation in section 5.1. The basic technical result is the following.

Lemma 5.1 Let A and B be two self-adjoint operators such that d_{±} ≡ dist(σ(A) ∩ ℝ^{±}, 0) > 0, and ∥B∥ ≤ 1. Then,

(i) For β ∈ ℝ such that |β| < \frac{1}{2} \sqrt{d_{+}d_{−}}, one has 0 ∈ ρ(A + iβB),

(ii) For β ∈ ℝ as in (i),

∥(A + iβB)^{−1}∥ ≤ 2 sup(d_{+}^{−1}, d_{−}^{−1}).

Proof. Let P_{±} be the spectral projectors for A corresponding to the sets σ(A)∩ ℝ^{±}, respectively and define u_{±} ≡ P_{±}u. By the Schwarz inequality one has

\begin{aligned}∥u∥ ∥(A + iβB)u∥ & ≥ & Re ⟨(u_{+} − u_{−}), (A + iβB)(u_{+} + u_{−})⟩ \\ & ≥ & d_{+}∥u_{+}∥^{2} + d_{−}∥u_{−}∥^{2} − 2βIm ⟨u_{+}, Bu_{−}⟩ \\ & ≥ & \frac{1}{2}(d_{+}∥u_{+}∥^{2} + d_{−}∥u_{−}∥^{2}), \end{aligned} (5.15)

where we again used the Schwarz inequality to estimate the inner product. It follows that

∥(A + iβB)u∥ ≥ \frac{1}{2} min (d_{+}, d_{−})∥u∥,

and since this is independent of the sign of β, the lemma follows. □

Proposition 5.2 Let H̃ be a semibounded self-adjoint operator with a spectral gap G ≡ (E_{−}, E_{+}) ⊂ ρ(H̃). Let W be a symmetric operator such that D(W) ⊃ D((H̃ + C_{0})^{\frac{1}{2}} ) and ∥(H̃ + C_{0})^{− \frac{1}{2}} W (H̃ + C_{0})^{− \frac{1}{2}} ∥ < 1, for some C_{0} such that H̃ + C_{0} > 1. For any E ∈ G, let Δ_{±} ≡ dist (E_{±}, E). Then, we have

(i) The energy E ∈ ρ(H̃ + iβW) for all real β satisfying

|β| < \frac{1}{2} \{\frac{Δ_{+}Δ_{−}}{(E_{+} + C_{0})(E_{−} + C_{0})}\}^{\frac{1}{2}} ;

(ii) for any real β and energy E as in (i),

∥(H̃ + iβW − E)^{−1}∥ ≤ 2 sup (\frac{E_{+} + C_{0}}{Δ_{+}} , \frac{E_{−} + C_{0}}{Δ_{−}} ) .

Proof. Let E ∈ G and C_{0} be as above. Define a self-adjoint operator A ≡ (H̃ + C_{0})^{−1}(H̃ − E) and B ≡ (H̃ + C_{0})^{− \frac{1}{2}} W (H̃ + C_{0})^{− \frac{1}{2}} . By hypothesis, the operator B is self-adjoint and satisfies ∥B∥ < 1. Note that 0 ∈ ρ(A) and

d_{±} ≡ dist (σ(A) ∩ ℝ^{±}, 0) = Δ_{±}(E_{±} + C_{0})^{−1} > 0  (5.16)

Applying Lemma 5.2 to these operators A and B, we see that for β as in (i), 0 ∈ ρ(A + iβB) and that

∥(A + iβB)^{−1}∥ ≤ 2 sup (\frac{E_{+} + C_{0}}{Δ_{+}} , \frac{E_{−} + C_{0}}{Δ_{−}} ) .

Let P_{±} be as in the proof of Lemma 5.2. For any w ∈ D(H̃),

\begin{aligned}∥(H̃ + iβW − E)w∥ & = & ∥(H̃ + C_{0})^{\frac{1}{2}} (A + iβB)(H̃ + C_{0})^{\frac{1}{2}} w∥ \\ & ≥ & ∥(A + iβB)(H̃ + C_{0})^{\frac{1}{2}} w∥ , \end{aligned}

since (H̃ + C_{0}) ≥ 1. In order to estimate the lower bound, we now repeat estimate (5.20) taking u ≡ (H̃ + C_{0})^{\frac{1}{2}} w. This gives

\begin{aligned}∥(H̃ + iβW − E)w∥ & ≥ & \frac{1}{2}∥(H̃ + C_{0})^{\frac{1}{2}} u∥^{−1} (d_{+}∥P_{+}(H̃ + C_{0})^{\frac{1}{2}} w∥^{2} \\ & & +(d_{−}∥P_{−}(H̃ + C_{0})^{\frac{1}{2}} w∥^{2}) \\ & ≥ & \frac{1}{2}min (d_{+}, d_{−})∥(H̃ + C_{0})^{\frac{1}{2}} w∥ . \end{aligned} (5.17)

Since ∥(H̃ + C_{0})^{\frac{1}{2}} w∥ ≥ ∥w∥ and d_{±} are defined in (5.21), result (ii) follows from (5.22) and Lemma 5.2. □

We now apply these results first to the exponential decay of the localized resolvent, and second, to the decay of eigenfunctions. We let ρ(x) ≡ \sqrt{1 + ∥x∥^{2}} be the regularized distance function. We assume that the unperturbed operator H_{0} has the form H_{0} = H_{A} + V_{0}, where H_{A} ≡ (−i∇ − A)^{2}, and V_{0} is relatively H_{A} bounded. The electric potential V_{0} and the vector potential A are assumed to be sufficiently well-behaved so that H_{0} is essentially self-adjoint on C_{0}^{∞}(ℝ^{n}). We also assume that the spectrum of H_{0} is semibounded from below. Most importantly, we suppose that the spectrum has an open spectral gap in the sense that there exist constants −∞ ≤ −C_{0} ≤ B_{−} < B_{+} ≤ ∞ so that

σ(H_{0}) ⊂ [−C_{0}, B_{−}] ∪ [B_{+}, ∞).  (5.18)

Of course, this gap might be the half line (−∞, Σ_{0}), where Σ_{0} = inf σ(H_{0}). For a less trivial example, we can take H_{0} = −Δ + V_{per}, where V_{per} is a periodic potential.

Finally, we assume that the perturbation potential V satisfies the following hypothesis.

Hypothesis 3. The potential V is relatively H_{0}-compact with relative bound less than one. For each ϵ > 0, the potential V admits a decomposition V = V_{c} + V_{ϵ}, where V_{c} has compact support and ∥V_{ϵ}∥ < ϵ.

Since the essential spectrum of H_{0} is stable under relatively compact perturbations, the effect of the potential V is to create isolated eigenvalues for H = H_{0} + V in the spectral gap G = (B_{−}, B_{+}). We are interested in the decay of the corresponding eigenfunctions. We first prove that the resolvent of H_{0} at energies E ∈ G decays exponentially when locialized between two disjoint regions.

Theorem 5.3 Let the unperturbed operator H_{0} satisfy the conditions above. Then, the dilated operator H(α) ≡ e^{iαρ}He^{−iαρ}, α ∈ ℝ, admits an analytic continuation as a type A family on the strip S(α_{0}), for any α_{0} > 0. For any E ∈ G = (B_{−}, B_{+}), define Δ_{±} ≡ dist (B_{±}, E). Then there exist finite constants C_{1}, C_{2} > 0, depending only on H_{0} and E, such that

(i) for any real β satisfying |β| < min (α_{0}, C_{1}\sqrt{Δ_{+}Δ_{−}}, \sqrt{Δ_{+}/2}), the energy E ∈ ρ(H_{0}(iβ));

(ii) for any real β as in (i),

∥(H_{0}(iβ) − E)^{−1}∥ ≤ C_{2} max(Δ_{+}^{−1}, Δ_{−}^{−1}) ;  (5.19)

(iii) let χ_{u} be a function of compact support localized near u ∈ ℝ^{n}, then for

β ∈ ℝ as in (i),

∥χ_{y}(H_{0} − E)^{−1}χ_{0}∥ ≤ C_{3}e^{βρ(y)}.  (5.20)

Proof. By a calculation similar to that done in section 5.1, we have, for α ∈ ℝ,

H_{0}(α) = e^{iαρ}H_{0}e^{−iαρ}
= H_{0} + α^{2}|∇ρ|^{2} + αW,  (5.21)

where W = −(∇ρ · (−i∇ − A) + (−i∇ − A) · ∇ρ) is symmetric. Note that ∥∇ρ∥_{∞} = 1 and ∥Δρ∥_{∞} = 1. Under the assumption that V_{0} is relatively H_{A}- bounded, it suffices to show that for some z ∈ ρ(H_{A}), the operator

{α^{2}|∇ρ|^{2} + W }(H_{A} − z)^{−1},  (5.22)

is bounded with norm less than one. Let us take z = −iη, for η > 0 and sufficiently large. It is then easy to show that the operator in (5.22) is bounded above by C_{0}η^{−1/2}, for some constant C_{0} depending on |α|. Since this bound can be made as small as desired for any fixed α_{0}, it folows that H_{0}(α) is an analytic type A family on any strip S(α_{0}).

Next, we take α = iβ, β real and |β| < α_{0}, so from (5.21), we have

H_{0}(iβ) = H_{0} − β^{2}|∇ρ|^{2} + iβW.  (5.23)

We apply Proposition 5.2 to this operator taking H̃ ≡ H_{0} − β^{2}|∇ρ|^{2}. This operator has a spectral gap which contains (B̃_{−}, B̃_{+}), where B̃_{−} = B_{−} and B̃_{+} = B_{+} − β^{2}. In order that Δ̃_{+} ≡ dist (B̃_{+}, E) > (Δ_{+}/2), we require |β| < \sqrt{Δ_{+}/2}. (Note that Δ̃_{−} = Δ_{−}). It follows from Proposition 5.2 that E ∈ ρ(H_{0}(iβ)) for |β| < min \{α_{0}, C_{1}\sqrt{Δ_{+}Δ_{−}}, \sqrt{Δ_{+}/2}\}, and that (ii) holds. Result(iii) follows from (ii) as in section 5.1. □

We now consider the perturbation of H_{0} by V. Assuming Hypothesis 3 on V, the operator H ≡ H_{0} + V is self adjoint on the same domain as H_{0}. The perturbation may introduce isolated eigenvalues of finite multiplicity in the spectral gap G. We apply the resolvent bound (5.20) to prove the exponential decay of the corresponding eigenfunctions.

Theorem 5.4 We assume that H_{0} satisfies the hypotheses given above, and that the potential V satisfies Hypothesis 3. Suppose that H = H_{0} + V has an eigenvalue E ∈ G, the gap in the spectrum of H_{0}, with an eigenfunction ψ_{E}. We assume that ∥ψ_{E}∥ = 1. For any α ∈ ℝ, with α < ν ≡ min (C_{1}\sqrt{Δ_{+}Δ_{−}}, \sqrt{Δ_{+}/2}), we have

e^{αρ}ψ_{E} ∈ L^{2}(ℝ^{n}).  (5.24)

Proof. Let us first suppose that V has compact support and that suppV ⊂ K, for some compact K ⊂ ℝ^{n}. We write R_{0}(E) ≡ (H_{0}−E)^{−1}. From the eigenvalue equation, we write, for λ ∈ ℝ,

e^{iλρ}ψ_{E} = −(e^{iλρ}R_{0}(E)e^{−iλρ}) (e^{iλρ}V ψ_{E})
= (H_{0}(λ) − E)^{−1} (e^{iλρ}V ψ_{E}).  (5.25)

Because of the type A analyticity of H_{0}(λ), and the compactness of the support of V, each term on the right in (5.25) admits an analytic continuation onto any the strip S(α_{0}). We set λ = −iα, for α ∈ ℝ with |α| < ν. Taking the norm of of both sides of the equation, and using the bound in (ii) of Proposition 5.3, we find that there exists a constant C_{E,V} > 0 so that

∥e^{αρ}ψ_{E}∥ ≤ C_{E,V} .  (5.26)

When the support of V is not compact, we consider the operator H_{0} + V_{ϵ}, instead of H_{0}. This operator also extends to an analytic type A family on the same strip as H_{0} since V_{ϵ} is a bounded operator. We choose ϵ < η ≡ dist(E, σ(H_{0})), so that H_{0} + V_{ϵ} has a spectral gap around E of size at least ϵ. It follows that (H_{0} + V_{ϵ} − E) is invertible, and its inverse can be computed from the equation

(H_{0} + V_{ϵ} − E) = (1 + V_{ϵ}(H_{0} − E)^{−1})(H_{0} − E),  (5.27)

since ∥V_{ϵ}R_{0}(E)∥ < 1/2. The eigenvalue equation is now written as

ψ_{E} = −(H_{0} + V_{ϵ} − e)^{−1}V_{c}ψ_{E}.  (5.28)

The exponential decay bound now follows from Proposition 5.2 applied to H_{0} + V_{ϵ}, and the argument given above. □

Application: Eigenfunction Decay for the Dirac Operator

As an application of the method of Combes-Thomas, we prove the decay of eigenfunctions corresponding to discrete, isolated eigenvalues of the Dirac operator. The free Dirac operator for a particle of mass m > 0 is constructed as follows. The Dirac matrices γ_{μ}, with μ = 0, 1, 2, 3 are four 4 × 4 matrices that form a representation of the canonical anticommutation relations,

γ_{μ}γ_{ν} + γ_{ν}γ_{μ} = 2δ_{μν},  (5.29)

where δ_{μν} is the Kronicker delta function. We write γ = (γ_{1}, γ_{2}, γ_{3}) for the three-vector, and β = γ_{0}. The free Dirac Hamiltonian is

H_{0} = −iγ · ∇ + βmc,  (5.30)

where c > 0 is the speed of light. We set c = 1. The unperturbed operator H_{0} is a first-order, matrix-valued linear operator. It is self-adjoint on its natural domain in the Hilbert space H = L^{2}(ℝ^{3}, ℂ^{4}). A simple calculation, based on the relations (5.29), shows that H_{0}^{2} = −Δ + m^{2}. It follows that the spectrum of H_{0} consists of two branches: σ(H_{0}) = (−∞, −m] ∪ [m, ∞). The interval G = (−m, m) is a gap in the spectrum of the free Dirac operator.

We now consider local perturbations V of H_{0}. Let us suppose that V > 0 and that V has compact support. Let I_{4} denote the 4 × 4 identity matrix. An application of the Birman-Schwinger principle shows that H_{λ} = H_{0} + λV · I_{4} has an eigenvalue in the gap G provided λ is suitably chosen. In general, relatively H_{0}-compact perturbations V will create eigenvalues in the spectral gap of H_{0}. We are interested in the isotropic exponential decay of the corresponding eigenfunctions.

The simplified argument presented at the beginning of section 5 applies direcly to this situation. Let α ∈ ℝ and take ρ(x) ≡ \sqrt{1 + ∥x∥^{2}}, as above. We define d_{±}(E) = dist(E, ±m). We apply a standard boost transformation to H_{0} = −iγ · ∇ + βm to obtain

H_{0}(α) ≡ e^{iαρ}H_{0}e^{−iαρ}
= −iγ · (∇ − iα∇ρ) + βm
= H_{0} − αγ · ∇ρ.  (5.31)

We now apply Lemma 5.1 with A = H_{0} and B = ∇ρ · γ. Since |∇ρ| ≤ 1, we have ∥B∥ ≤ 1. Hence, with α = iη, for η ∈ ℝ, we require

|η| ≤ (1/2)\sqrt{d_{+}(E)d_{−}(E)}.  (5.32)

Under this condition, we obtain

∥(H_{0}(iη) − E)^{−1}∥ ≤ 2 max (\frac{1}{d_{+}(E)}, \frac{1}{d_{−}(E)}) .  (5.33)

We now proceed as in Theorem 5.4. First, we suppose that the potential V has compact support, and that H = H_{0} +V has an eigenvalue E ∈ G. We write the eigenvalue equation as

ψ = −R_{0}(E)V ψ.  (5.34)

By analytic continuation from α ∈ ℝ to α = −iη, with η ∈ ℝ and satisfying the bound (5.32), we have

∥e^{ηρ}ψ∥ ≤ ∥H_{0}(−iη) − E)^{−1}∥ ∥e^{ηρ}V ∥ ≤ C_{E,V} ,  (5.35)

proving the L^{2}-exponential decay of the eigenfunction for any η ∈ ℝ satisfying (5.32). When V does not have compact support, but vanishes uniformly at infinity, we use the same argument as in the proof of Theorem 5.4. The same exponential decay results hold in the presence of a magnetic field for which H_{0} = γ · (−i∇ − A) + βm, for reasonable magnetic vector potentials A. (A similar result was recently obtained by Breit and Cornean [3]).

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P. D. Hislop Mathematics Department University of Kentucky Lexington, KY 490506-0027 USA e-mail: hislop@ms.uky.edu