Semiclassical Dynamics with Exponentially Small Error Estimates

George A. Hagedorn^{∗} Department of Mathematics and Center for Statistical Mechanics and Mathematical Physics Virginia Polytechnic Institute and State University Blacksburg, Virginia 24061-0123, U.S.A.

Alain Joye^{†} Institut Fourier Unité Mixte de Recherche CNRS-UJF 5582 Université de Grenoble I BP 74 F–38402 Saint Martin d’Hères Cedex, France

Abstract

We construct approximate solutions to the time–dependent Schrödinger equation

i h̄ \frac{∂ψ}{∂t} = − \frac{h̄^{2}}{2} Δ ψ + V ψ

for small values of h̄. If V satisfies appropriate analyticity and growth hypotheses and |t| ≤ T, these solutions agree with exact solutions up to errors whose norms are bounded by

C exp { − γ/h̄ } ,

for some C and γ > 0. Under more restrictive hypotheses, we prove that for sufficiently small T′, |t| ≤ T′ | log(h̄)| implies the norms of the errors are bounded by

C′ exp { − γ′/h̄^{σ} } ,

for some C′, γ′ > 0, and σ > 0.

MSC: 81Q20 Keywords: Semiclassical methods, quantum dynamics, exponential asymptotics

∗Partially Supported by National Science Foundation Grant DMS–9703751.

†Partially Supported by Fonds National Suisse de la Recherche Scientifique, Grant 8220-037200.

1 Introduction

In this paper, we construct exponentially accurate semiclassical approximations ψ(x, t, h̄) to certain normalized exact solutions Ψ(x, t, h̄) of the d–dimensional time–dependent Schrödinger equation

i h̄ \frac{∂Ψ}{∂t} = − \frac{h̄^{2}}{2} Δ Ψ + V Ψ. (1.1)

More precisely, our main result is that for |t| ≤ T and small values of h̄, these approximations satisfy error estimates of the form

∥ ψ(x, t, h̄) − Ψ(x, t, h̄) ∥_{L^{2}(IR^{d})} ≤ C exp { − γ/h̄ } , (1.2)

where γ > 0.

Our construction of ψ(x, t, h̄) is technically complicated, but quite explicit. It uses a particular collection of semiclassical wave packets { φ_{j}(A, B, h̄, a, η, ·) } that are defined in [6], [7], and the next section. Here A and B are d × d complex matrices that satisfy certain conditions. The quantities a and η are elements of IR^{d}. For fixed A, B, h̄, a, and η, { φ_{j}(A, B, h̄, a, η, ·) } is an orthonormal basis of L^{2}(IR^{d}) as j ranges over all d-dimensional multi-indices.

The function φ_{j}(A, B, h̄, a, η, ·) is concentrated near position a, and its Fourier transform is concentrated near momentum η. Its position and momentum uncertainties are proportional to \sqrt{h̄}. The position uncertainty is determined by the matrix |A| = \sqrt{A A^{∗}}, and the momentum uncertainty is determined by |B| = \sqrt{B B^{∗}}.

We construct ψ(x, t, h̄) by applying the idea of “optimal truncation” of an asymptotic expansion. For initial conditions of the form

Ψ(x, 0, h̄) = ∑_{|j|≤J} c_{j} φ_{j}(A(0), B(0), h̄, a(0), η(0), x), (1.3)

with ∑_{|j|≤J} |c_{j}|^{2} = 1, there exist ([5], [6]) approximate solutions

ψ_{l}(x, t, h̄) = e^{iS(t)/h̄} ∑_{|j|≤_{J}(l)} c_{j}(l, t, h̄) φ_{j}(A(t), B(t), h̄, a(t), η(t), x), (1.4)

that satisfy

sup_{t∈[−T,T]} ∥ψ_{l}(x, t, h̄) − Ψ(x, t, h̄)∥_{L^{2}(IR^{d})} ≤ C(l) h^{l/2} (1.5)

for some constant C(l). Here J̃(l) = J + 3l − 3, and A(t), B(t), a(t), η(t), and S(t) are solutions to the classical equations of motion

ȧ(t) = η(t),
η̇(t) = − ∇V (a(t)),
Ȧ(t) = i B(t),  (1.6)
Ḃ(t) = i V^{(2)}(a(t)) A(t),
Ṡ(t) = \frac{η(t)^{2}}{2} − V (a(t)),

where V^{(2)} denotes the Hessian matrix for V, and the initial conditions A(0), B(0), a(0), η(0), and S(0) = 0 satisfy

A^{t}(0) B(0) − B^{t}(0) A(0) = 0,  (1.7)
A^{∗}(0) B(0) + B^{∗}(0) A(0) = 2 I.  (1.8)

The c_{j}(l, t, h̄) satisfy a linear system of ordinary differential equations that we describe in the next section.

We carefully estimate the l–dependence of C(l) in (1.5). Then, for each h̄, we choose l(h̄) to minimize the error C(l(h̄)) h̄^{l(h̄)/2} over all choices of l. It turns out that l(h̄) behaves like a constant times 1/h̄. We define ψ(x, t, h̄) = ψ_{l(h̄)}(x, t, h̄) and prove that (1.2) is satisfied.

For t in a fixed compact interval, the precise statement of our results is the following:

Theorem 1.1 Suppose V is a real-valued function on IR^{d} that is bounded below and has an analytic continuation to the set

D = { z ∈ I C^{d} : |Im z_{j}| < δ, j = 1, 2, ⋯, d }. (1.9)

Suppose further that there exist M > 0 and τ > 0, such that

| V (z) | ≤ M exp(τ|z|^{2}),  for all z ∈ D,  (1.10)

where |z|^{2} = ∑_{j=1}^{d} |z_{j}|^{2}. Suppose initial conditions A(0), B(0), a(0), η(0), S(0) = 0, and c_{j} for |j| ≤ J are specified that satisfy (1.7) and ∑_{|j|≤J} |c_{j}|^{2} = 1. Then for any T > 0, there exist C and γ > 0, such that the difference between the semiclassical approximation ψ(x, t, h̄) and the exact solution Ψ(x, t, h̄) to the Schrödinger equation (1.1) with initial condition (1.3) satisfies

∥ ψ(x, t, h̄) − Ψ(x, t, h̄) ∥_{L^{2}(IR^{d})} ≤ C exp { − γ/h̄ } ,

whenever |t| ≤ T.

Remarks 1. Theorem 1.1 can be generalized to allow time–dependent potentials. For example, suppose a potential V (x, t) depends smoothly on t, is bounded below, and satisfies

\frac{1}{m!} |D^{m}V (x, t))| ≤ \frac{M exp(τ|x|^{2})}{δ^{|m|}},

for |t| ≤ T for all multi-indices m. Suppose further that a classical solution (a(t), η(t)) to Newton’s equations with potential V (x, t) is bounded for |t| ≤ T. Then the conclusion to Theorem 1.1 holds.

2. Our results can also be extended to obtain weaker error estimates of the form C exp { − γ/h̄^{σ} } for some σ ∈ (0, 1), when the potential belongs to a Gevrey class.

3. Theorem 1.1 is optimal in the sense that the conclusion fails if the hypotheses are relaxed slightly. For example, consider the one-dimensional potential V (x) = exp(−1/x^{u}), for x > 0, and V (x) = 0 for x ≤ 0, where u > 0. It is shown in [8] that this potential belongs to the Gevrey class of order 1 + 1/u. For initial conditions a(0) = 0, η(0) = 0, A(0) = B(0) = 1, S(0) = 0, and c_{j}(0) = δ_{j,0}, our approximation yields φ_{0}(1 + it, 1, h̄, 0, 0), for all times. This function is very simple, and we can write the error term explicitly. By steepest descent analysis, we can show that there exist δ > 0 and Σ_{1} > Σ_{2} > 0, such that t ∈ (0, h̄^{δ}) implies

exp(−Σ_{1}/h̄^{u/(1+u)}) ≤ ∥ e^{−itH(h̄)/h̄} φ_{0}(1, 1, h̄, 0, 0, ·) − φ_{0}(1 + it, 1, h̄, 0, 0, ·) ∥
≤ exp(−Σ_{2}/h̄^{u/(1+u)}).

Note also that if we choose a(0) = −a < 0 and η(0) = η > 0, it is easy to check that the error term is O(exp(−γ/h̄)), for each t < a/η.

4. For all practical purposes, we can replace the c_{j}’s by the corresponding Dyson expansion up to order l(h̄) (see (2.26)–(2.28)) without spoiling our exponential estimate. The normalization of the approximation, however, will be lost.

5. One cannot expect to get better agreement with the exact solution when constructing an approximation using wave packets associated with a single classical path. Such wave packets do not describe tunneling, which is an exponentially small effect in h̄.

Theorem 1.1 can also be generalized to allow time intervals that grow like | log(h̄)| as h̄ tends to zero. However, we obtain a somewhat weaker conclusion. Our precise results are summarized by the following theorem.

Theorem 1.2 Suppose V is bounded below and analytic in

D = { z ∈ I C^{d} : |Im z_{j}| < δ, j = 1, 2, ⋯, d }. (1.11)

Suppose further that there exist M > 0 and τ > 0, such that

| V (z) | ≤ M exp(τ|z|),  for all z ∈ D.  (1.12)

Suppose initial conditions A(0), B(0), a(0), η(0), S(0) = 0, and c_{j} for |j| ≤ J are specified that satisfy (1.7) and ∑_{|j|≤J} |c_{j}|^{2} = 1, and further assume there exist N > 0 andλ > 0, such that

∥A(t)∥ ≤ N exp(λ|t|). (1.13)

Then for sufficiently small T′ > 0, there exist C′, γ′ > 0, and σ > 0, such that the difference between the semiclassical approximation ψ(x, t, h̄) and the exact solution Ψ(x, t, h̄) to the Schrödinger equation (1.1) with initial condition (1.3) satisfies

∥ ψ(x, t, h̄) − Ψ(x, t, h̄) ∥_{L^{2}(IR^{d})} ≤ C′ exp { − γ′/h̄^{σ} } ,

whenever |t| ≤ T′ | log(h̄)|.

Remark Standard existence and uniqueness theorems for systems of ODE’s show that condition (1.13) is satisfied if the norm of the Hessian V^{(2)}(a(t)) is uniformly bounded. That is the case if V is the sum of a quadratic polynomial plus an analytic function bounded on D. It is also the case if E denotes the energy of the considered trajectory and ∥V^{(2)}(x)∥ is bounded on the connected component of the classically allowed region D_{E} = {x ∈ IR^{d} : V (x) ≤ E} that contains a(t). This is satisfied for all confining potentials.

It is easily deduced from the proof that if τ can be chosen arbitrarily small, then we can take T′ = (\frac{1}{6} − ϵ)\frac{1}{λ} with ϵ arbitrarily small. This yields exponential control over the same time intervals as in [3].

The propagation of coherent states is also considered by Combescure and Robert in [3], see also [9], using an approximation given by a linear combination of squeezed states. (The squeezed states coincide with our semiclassical wave packets, although the notation is quite different.) Their emphasis is on the long time behavior of this approximation. The bound on the error term is of the form C_{l}(t)h̄^{l/2}, with explicit control of the time-dependence of C_{l}(t) in terms of classical quantities. The l behavior is however not investigated.

Results of a flavor similar to ours can be found in the work of Yajima [12]. They are obtained by means of the pseudo-differential techniques developed in the analytic context by Sjöstrand in [10]. These results concern the propagation of wave packets of the form φ(x) = e^{iS(x)/h̄}f(x), where S is analytic and f belongs to the set of compactly supported Gevrey functions of order s > 1. Assuming the potential V is analytic, Yajima constructs approximations to the actual evolution of such wave packets that are valid up to an error, whose L^{2}(IR^{d}) norm is of order e^{−γ/h̄^{1/(2s−1)}}, with γ > 0 (see Theorems 1.2, 1.2 and Lemma 2.5 in [12]). However, it should be possible to make use of the theory [10] to recover our results.

Similar issues have been dealt with by Bambusi, Graffi and Paul in [2]. They focus on the validity for large times of the semiclassical approximation of the Heisenberg evolution of a smooth observable, under analyticity assumptions on the hamiltonian. They prove that the semiclassical approximation remains useful for times up to order | ln(h̄)|, the Ehrensfest time scale. However, the Hamiltonians they can accomodate consist more or less of analytic perturbations of the harmonic oscillator that decay as x and p tend to infinity.

The paper is organized as follows: In the Section 2, we prove Theorem 1.1 under the assumption that two types of error terms satisfy certain bounds. We prove the two required bounds in Sections 3 and 4. In Section 5 we describe the proof of Theorem 1.2.

2 Proof of Theorem 1.1

We begin this section by presenting the definition of the semiclassical wave packets φ_{j}(A, B, h̄, a, η, x) that is given in [7]. A more explicit, but more complicated definition is given in [6]. Since [7] provides a detailed discussion of these wave packets, we do not prove all their properties here.

We adopt the standard multi-index notation. A multi-index j = (j_{1}, j_{2}, …, j_{d}) is a d-tuple of non-negative integers. We define |j| = ∑_{k=1}^{d} j_{k}, j! = (j_{1}!)(j_{2}!) ⋯ (j_{d}!),

x^{j} = x_{1}^{j_{1}}x_{2}^{j_{2}} ⋯ x_{d}^{j_{d}}, and D^{j} = \frac{∂^{|j|}}{(∂x_{1})^{j_{1}}(∂x_{2})^{j_{2}} ⋯ (∂x_{d})^{j_{d}}}.

Throughout the paper we assume a ∈ IR^{d}, η ∈ IR^{d} and h̄ > 0. We also assume that A and B are d × d complex invertible matrices that satisfy

A^{t} B − B^{t} A = 0,  (2.1)

A^{∗} B + B^{∗} A = 2 I.  (2.2)

These conditions guarantee that both the real and imaginary parts of BA^{−1} are symmetric. Furthermore, Re BA^{−1} is strictly positive definite, and ( Re BA^{−1} )^{−1} = A A^{∗}. We note that conditions (2.1) and (2.2) are preserved under the dynamics generated by (1.6).

Our definition of φ_{j}(A, B, h̄, a, η, x) is based on the following raising operators that are defined for m = 1, 2, …, d.

A_{m}(A, B, h̄, a, η)^{∗} = \frac{1}{\sqrt{2h̄}} [ ∑_{n=1}^{d} \overline{B}_{n m} (x_{n} − a_{n}) − ∑_{n=1}^{d} \overline{A}_{n m} (−ih̄\frac{∂}{∂x_{n}} − η_{n}) ] .

Definition For the multi-index j = 0, we define the normalized complex Gaussian wave packet (modulo the sign of a square root) by

φ_{0}(A, B, h̄, a, η, x) = π^{−d/4} h̄^{−d/4} (det(A))^{−1/2}
× exp { − ⟨ (x − a), B A^{−1} (x − a) ⟩/(2h̄) + i ⟨ η, (x − a) ⟩/h̄ } .  (2.3)

Then, for any non-zero multi-index j, we define

φ_{j}(A, B, h̄, a, η, ·) = \frac{1}{\sqrt{j!}} ( A_{1}(A, B, h̄, a, η)^{∗})^{j_{1}} ( A_{2}(A, B, h̄, a, η)^{∗})^{j_{2}} ⋯
× ( A_{d}(A, B, h̄, a, η)^{∗})^{j_{d}} φ_{0}(A , B, h̄, a, η, ·).

Remarks 1. For A = B = I, h̄ = 1, and a = η = 0, the φ_{j}(A, B, h̄, a, η, ·) are just the standard Harmonic oscillator eigenstates with energies |j| + d/2.

2. For each A, B, h̄, a, and η, the set { φ_{j}(A, B, h̄, a, η, ·) } is an orthonormal basis for L^{2}(IR^{d}).

3. The raising operators can also be given by another formula that was omitted from [7] in the multi-dimensional case. If we set

g(A, B, h̄, a, x) = exp { − ⟨ (x − a), (BA^{−1})^{∗} (x − a) ⟩/(2h̄) − i ⟨ η, (x − a) ⟩/h̄ } ,

then we have

( A_{m}(A, B, h̄, a, η)^{∗} ψ) (x) = − \sqrt{\frac{h̄}{2}} \frac{1}{g(A, B, h̄, a, x)} ∑_{n=1}^{d} \overline{A}_{n m} \frac{∂}{∂x_{n}} ( g(A, B, h̄, a, x) ψ(x) ) .

4. In [6], the state φ_{j}(A, B, h̄, a, η, x) is defined as a normalization factor times

H_{j}(A; h̄^{−1/2} |A|^{−1} (x − a)) φ_{0}(A, B, h̄, a, η, x).

Here H_{j}(A; y) is a |j|^{th} order polynomial in y that depends on A only through U_{A}, where A = |A| U_{A} is the polar decomposition of A.

5. By the argument on page 370 of [6] or by scaling out the |A| and h̄ dependence and using Remark 3 above, one can show that H_{j}(A; y) e^{−y^{2}/2} is an (unnormalized) eigenstate of the usual Harmonic oscillator with energy |j| + d/2.

6. When the dimension d is 1, the position and momentum uncertainties of the φ_{j}(A, B, h̄, a, η, ·) are \sqrt{(j + 1/2)h̄} |A| and \sqrt{(j + 1/2)h̄} |B|, respectively. In higher dimen-

sions, they are bounded by \sqrt{(|j| + d/2)h̄} ∥A∥ and \sqrt{(|j| + d/2)h̄} ∥B∥, respectively.

7. When we approximately solve the Schrödinger equation, the choice of the sign of the square root in the definition of φ_{0}(A, B, h̄, a, η, ·) is determined by continuity in t after an arbitrary initial choice.

The proof of the theorem depends on the following abstract lemma, whose proof is an easy application of Duhamel’s formula (see e.g. [7]).

Lemma 2.1 Suppose H(h̄) is a family of self-adjoint operators for h̄ > 0. Suppose ψ(t, h̄) belongs to the domain of H(h̄), is continuously differentiable in t, and approximately solves the Schrödinger equation in the sense that

i h̄ \frac{∂ψ}{∂t} (t, h̄) = H(h̄) ψ(t, h̄) + ξ(t, h̄), (2.4)

where ξ(t, h̄) satisfies

∥ ξ(t, h̄) ∥ ≤ μ(t, h̄). (2.5)

Then, for t > 0,

∥ e^{−itH(h̄)/h̄} ψ(0, h̄) − ψ(t, h̄) ∥ ≤ h̄^{−1} ∫_{0}^{t} μ(s, h̄) ds. (2.6)

The analogous statement holds for t < 0.

Because V is smooth and bounded below, there exist global solutions to the first two equations of the system (1.6) for any initial condition. It then follows immediately that the remaining three equations of the system (1.6) have global solutions. Furthermore, it is not difficult ([5], [6]) to prove that (2.1) and (2.2) are preserved by the flow.

As mentioned in the introduction, it is proved in [5] and [6] that initial conditions of the form (1.3) give rise to approximate solutions of the form

ψ_{l}(x, t, h̄) = e^{iS(t)/h̄} ∑_{|j|≤_{J}(l)} c_{j}(l, t, h̄) φ_{j}(A(t), B(t), h̄, a(t), η(t), x),

with errors whose norms are of order h̄^{l/2}. Here J̃(l) = J + 3l − 3, and A(t), B(t), a(t), η(t), and S(t) satisfy (1.6). The coefficients c_{j}(l, t, h̄) satisfy the linear system

i h̄ ċ_{k}(l, t, h̄) = ∑_{|j|≤_{J}(l)} K_{k j}(l, t, h̄) c_{j}(l, t, h̄),  |k| = 0, 1, …, J̃(l),  (2.7)

with initial conditions c_{j}(l, 0, h̄) = c_{j}, for |j| ≤ J in accordance with (1.3) and c_{j}(l, 0, h̄) = 0 for |j| > J.

To specify the (J + 3l − 2) × (J + 3l − 2) matrix K(l, t, h̄) that appears in (2.7), we first decompose the potential as

V (x) = W_{a}(x) + Z_{a}(x) ≡ W_{a}(x) + (V (x) − W_{a}(x)), (2.8)

where W_{a}(x) denotes the second order Taylor approximation (with the obvious abuse of notation)

W_{a}(x) ≡ V (a) + V^{(1)}(a) (x − a) + V^{(2)}(a) (x − a)^{2}/2, (2.9)

Next (reverting to multi-index notation), we approximate Z_{a}(x) by its Taylor approximation of order l + 1,

Z_{a}^{[l]}(x) = ∑_{3≤|m|≤l+1} \frac{(D^{m}V)(a)}{m!} (x − a)^{m}, (2.10)

and define the infinite matrix

K̃_{k j}(l, t, h̄) =
⟨ φ_{k}(A(t), B(t), h̄, a(t), η(t), x), Z_{a(t)}^{[l]}(x) φ_{j}(A(t), B(t), h̄, a(t), η(t), x) ⟩.  (2.11)

Then, we obtain the matrix K(l, t, h̄) from K̃(l, t, h̄) by restricting the indices to |j| ≤ J̃(l) and |k| ≤ J̃(l).

The general strategy [5], [6], [7] to show that ψ_{l}(x, t, h̄) is an approximation to the actual solution Ψ(x, t, h̄) of (1.1) and (1.3) up to order h̄^{l/2} is as follows: From [7], we know that for all multi-indices j,

i h̄ \frac{∂}{∂t} [ e^{iS(t)/h̄} φ_{j}(A(t), B(t), h̄, a(t), η(t), x) ]

= ( − \frac{h̄^{2}}{2} Δ + W_{a(t)}(x) ) [ e^{iS(t)/h̄} φ_{j}(A(t), B(t), h̄, a(t), η(t), x) ] .  (2.12)

Thus, the φ_{j} take into account the kinetic energy and W_{a(t)}(x) parts of the Hamiltonian. Next, we expand the exact solution as

Ψ(x, t, h̄) = ∑_{j} b_{j}(h̄, t) e^{iS(t)/h̄} φ_{j}(A(t), B(t), h̄, a(t), η(t), x),

where the b_{j}(h̄, t) satisfy an infinite linear system of ordinary differential equations whose matrix is obtained from the Z_{a(t)}(x) term in the Hamiltonian. In that system, we make a first approximation by replacing the function Z_{a(t)}(x) by its Taylor approximation Z_{a(t)}^{[l]}(x). This yields an infinite linear system whose matrix is K̃(l, t, h̄). Its entries are time dependent polynomials in h̄^{1/2} of order l − 1. We make a second approximation by truncating the infinite system to obtain (2.7) that is satisfied by the c_{j}(l, t, h̄). The result (1.5) is proved by using Lemma 2.1 to show that the errors generated by the Taylor approximation and the truncation approximation are of order h̄^{l/2}.

As described in the introduction, we construct the exponentially accurate approximate solution ψ(x, t, h̄) = ψ_{l(h̄)}(x, t, h̄), by keeping track of the l–dependence of C(l) in (1.5) and then choosing l(h̄) in order to minimize the error.

In the remainder of this section, we prove Theorem 1.1 under the assumption that the following three technical lemmas are true. We prove the first lemma in Section 3. It estimates errors that arise from our replacement of Z_{a(t)}(x) by Z_{a(t)}^{[l]}(x). The second and third lemmas are proved in Section 4. They bound certain matrix elements and combinatorial quantities that arise from the truncation approximation discussed above.

Lemma 2.2 Suppose V satisfies the hypotheses of Theorem 1.1, |m| = l+2, J̃(l) = J+3l−3, and

ψ_{l}(x, t, h̄) = e^{iS(t)/h̄} ∑_{|j|≤_{J}(l)} c_{j}(l, t, h̄) φ_{j}(A(t), B(t), h̄, a(t), η(t), x),

with ∑_{|j|≤_{J}(l)} |c_{j}(l, t, h̄)|^{2} = 1.

Let ζ(x, a(t)) = (a(t) + θ_{x,a(t)}(x − a(t))) ∈ IR^{d}, with θ_{x,a(t)} ∈ (0, 1). There exist constants g_{0} and g_{1}, that depend on d and J only, such that for sufficiently small h̄,

∥ \frac{D^{m}V (ζ(x, a(t)))}{m!} (x − a(t))^{m} ψ_{l}(·, t, h̄) ∥

≤ g_{0} M exp(4τ(δ^{2}d + a(t)^{2})) ( g_{1} \sqrt{h̄(l + 2)} ∥A(t)∥/δ )^{l+2} .  (2.13)

Lemma 2.3 We define the infinite matrix

X̃_{j k}^{m}(t, h̄) =
⟨ φ_{j}(A(t), B(t), h̄, a(t), η(t), ·), (x − a(t))^{m} φ_{k}(A(t), B(t), h̄, a(t), η(t), ·) ⟩, (2.14)

and then define the finite matrix X^{m}(l, t, h̄) from X̃^{m}(t, h̄) by restricting its indices to

|j| ≤ J̃(l) and |k| ≤ J̃(l). Then, X_{j k}^{m}(l, s, h̄) = 0 and X̃_{j k}^{m}(s, h̄) = 0 if ||j| − |k|| > |m| and,

for each N > 0, there exists D(N) < ∞, such that

sup_{\begin{matrix}|k| ≤ _{J}(l) \\ _{J}(l) + 1 ≤ |j| ≤ _{J}(l) + l + 1 \end{matrix}} | (X̃^{m_{0}}(t_{0}, h̄)X^{m_{1}}(l, t_{1}, h̄)X^{m_{2}}(l, t_{2}, h̄) ⋯ X^{m_{q}}(l, t_{q}, h̄))_{j k} |

≤ ( D(N) \sqrt{h̄ J̃(l)} sup_{t∈\{t_{0},t_{1},t_{2},···t_{q}\}} ∥A(t)∥ )^{|m_{0}|+|m_{1}|+|m_{2}|···+|m_{q}|} ,  (2.15)

for any collection m_{0}, m_{1}, …, m_{q} of multi-indices that satisfy |m_{j}|/J̃(l) ≤ N.

Lemma 2.4 We define

F_{p}(n, q) = ∑_{\begin{matrix}1 ≤ |m_{1}|, |m_{2}|, ⋯, |m_{q}| ≤ p \\ |m_{1}| + |m_{2}| + ⋯ + |m_{q}| = n \end{matrix}} 1  (2.16)

to be the number of distinct sets { m_{1}, m_{2}, …, m_{q} }, where each m_{j} is a d–dimensional multi-index with 1 ≤ |m_{j}| ≤ p and |m_{1}| + |m_{2}| + ⋯ + |m_{q}| = n. We note that F_{p}(n, q) is zero unless q ≤ n ≤ qp. Suppose that a function L_{q}(l) satisfies

L_{q}(l) ≤ \frac{C_{1}^{q}}{h̄^{q} q!} ∑_{n=3l−2}^{q(l+1)} (C_{2}h̄l)^{n/2} F_{l+1}(n, q), (2.17)

where C_{1} and C_{2} are constants. Let [[ α]] denote the greatest integer less than or equal to α. Then there exists g^{∗} > 0, such that for any g ∈ (0, g^{∗}), there exist positive constants C_{3}, γ_{1}, and h̄^{∗}, that depend only on g, C_{1}, and C_{2}, such that

0 < h̄ < h̄^{∗}, l(h̄) = [[ g/h̄]], and 2 ≤ q ≤ l(h̄) + 2 (2.18)

imply

L_{q}(l(h̄)) ≤ C_{3} exp { − γ_{1}/h̄ } . (2.19)

Proof of Theorem 1.1: We define ψ_{l}(x, t, h̄) by (1.4), where the c_{j}(l, t, h̄) are determined by the system (2.7) and initial conditions described above.

To apply Lemma 2.1 we define

ξ_{l}(x, t, h̄) = i h̄ \frac{∂}{∂t}ψ_{l}(x, t, h̄) − ( − \frac{h̄^{2}}{2} Δ + V (x) ) ψ_{l}(x, t, h̄). (2.20)

By using (2.12), we see that this can be decomposed as a sum of two terms,

ξ_{l}^{(1)}(x, t, h̄) = ( Z_{a(t)}^{[l]}(x) − Z_{a(t)}(x) ) ψ_{l}(x, t, h̄) (2.21)

and

ξ_{l}^{(2)}(x, t, h̄) = − P_{{|j|≥_{J}(l)}} Z_{a(t)}^{[l]}(x) ψ_{l}(x, t, h̄), (2.22)

where P_{{|j|≥l+2}} is the orthogonal projection onto the span of the set { φ_{j}(A(t), B(t), h̄, a(t), η(t), ·) : |j| ≥ l + 2 }.

By the standard Taylor series error formula,

Z_{a(t)}(x) − Z_{a(t)}^{[l]}(x) = ∑_{|m|=l+2} \frac{D^{m}V (ζ(x, a(t)))}{m!} (x − a(t))^{m},

for some ζ(x, a) = (a + θ_{x,a}(x − a)), with θ_{x,a} ∈ (0, 1). Thus, by the crude estimate

∑_{|m|=n} 1 ≤ ∑_{|m|≤n} 1 ≤ ∑_{j=1}^{d} ∑_{m_{j}≤n} 1

= (n + 1)^{d} = e^{d ln(n+1)} ≤ (e^{d})^{n} ,  (2.23)

and Lemma 2.2, we obtain

∥ ξ_{l}^{(1)}(·, t, h̄) ∥
≤ ∑_{|m|=l+2} ∥ \frac{D^{m}V (ζ(x, a(t)))}{m!} (x − a(t))^{m} ψ_{l}(·, t, h̄) ∥

≤ g_{0} M exp(4τ(δ^{2}d + a(t)^{2})) ( g_{1} \sqrt{h̄(l + 2)} ∥A(t)∥/δ )^{l+2} (e^{d})^{l+2} .

Thus, there exist constants C_{4} and C_{5}, such that |t| ≤ T implies

∥ ξ_{l}^{(1)}(·, t, h̄) ∥ ≤ C_{4} ( C_{5} \sqrt{h̄(l + 2)} )^{l+2} . (2.24)

The quantity ξ_{l}^{(2)}(·, t, h̄) satisfies

⟨ φ_{j}(A(t), B(t), h̄, a(t), η(t), x), ξ_{l}^{(2)}(x, t, h̄) ⟩ = 0, if \{ \begin{matrix}0 ≤ |j| ≤ J̃(l) or \\ |j| > J̃(l) + l + 1, \end{matrix}

and

⟨ φ_{j}(A(t), B(t), h̄, a(t), η(t), x), ξ_{l}^{(2)}(x, t, h̄) ⟩ =

∑_{|k|≤_{J}(l)} ⟨ φ_{j}(A(t), B(t), h̄, a(t), η(t), x), Z_{a(t)}^{[l]}(x) φ_{k}(A(t), B(t), h̄, a(t), η(t), x) ⟩ c_{k}(t)

= (K̃(t) c(t))_{j}, if J̃(l) < |j| ≤ J̃(l) + l + 1,  (2.25)

where we denote the c_{j}(l, t, h̄) collectively by the vector c(l, t, h̄). We easily verify these facts by using (2.7), (2.11), (2.12) and Lemma 2.3. To estimate the norm of ξ_{l}^{(2)}(·, t, h̄), we use the Dyson expansion with remainder to decompose

c(l, t, h̄) = ∑_{q=0}^{l} c^{q}(l, t, h̄) + r(l, t, h̄), (2.26)

where (dropping some arguments)

c^{q}(t) = ∫_{0}^{t} ds_{1} ∫_{0}^{s_{1}} ds_{2} ⋯ ∫_{0}^{s_{q−1}} ds_{q}(ih̄)^{−q}K(s_{1})K(s_{2}) ⋯ K(s_{q}) c(0), (2.27)

and

r(t) = ∫_{0}^{t} ds_{1} ∫_{0}^{s_{1}} ds_{2} ⋯ ∫_{0}^{s_{l}} ds_{l+1}(ih̄)^{−(l+1)}K(s_{1})K(s_{2}) ⋯ K(s_{l+1}) c(s_{l+1}). (2.28)

Using (2.11), (2.10), and (2.14), we see that each c^{q}(l, t, h̄) is of order h̄^{q/2} and that r(l, t, h̄) is of order h̄^{(l+1)/2}.

To estimate the norm of ξ_{l}^{(2)}(·, t, h̄) we study the j^{th} component of K̃(t)c(l, t, h̄), with |j| > J̃(l). Because of (2.26), this coefficient is a sum of two types of terms: those that arise from c^{q}(l, t, h̄) and those that arise from r(l, t, h̄). Using (2.10), we expand K(l, t, h̄) in (2.27) and (2.28) and K̃(l, t, h̄) in terms of X^{m}(l, t, h̄) and X̃^{m}(t, h̄), to obtain

K̃(t)c^{q}(t) = (ih̄)^{−q} ∑_{n=3(q+1)}^{(l+1)(q+1)} ∑_{\begin{matrix}m_{0}, m_{1}, m_{2}, ⋯, m_{q} \\ |m_{0}| + |m_{1}| + ⋯ + |m_{q}| = n \\ 3 ≤ |m_{j}| ≤ l + 1 \end{matrix}}

∫_{0}^{t} ds_{1} ∫_{0}^{s_{1}} ds_{2} ⋯ ∫_{0}^{s_{q−1}} ds_{q} \frac{D^{m_{0}}V (a(t))D^{m_{1}}V (a(s_{1})) ⋯ D^{m_{q}}V (a(s_{q}))}{m_{0}!m_{1}!m_{2}! ⋯ m_{q}!}

× X̃^{m_{0}}(t)X^{m_{1}}(s_{1})X^{m_{2}}(s_{2}) ⋯ X^{m_{q}}(s_{q}) c(0),  (2.29)

and

K̃(t)r(t) = (ih̄)^{−(l+1)} ∑_{n=3(l+2)}^{(l+1)(l+2)} ∑_{\begin{matrix}m_{0}, m_{1}, m_{2}, ⋯, m_{l+1} \\ |m_{0}| + |m_{1}| + ⋯ + |m_{l+1}| = n \\ 3 ≤ |m_{j}| ≤ l + 1 \end{matrix}}

∫_{0}^{t} ds_{1} ∫_{0}^{s_{1}} ds_{2} ⋯ ∫_{0}^{s_{l}} ds_{l+1} \frac{D^{m_{0}}V (a(t))D^{m_{1}}V (a(s_{1})) ⋯ D^{m_{l+1}}V (a(s_{l+1}))}{m_{0}!m_{1}!m_{2}! ⋯ m_{l+1}!}

× X̃^{m_{0}}(t)X^{m_{1}}(s_{1})X^{m_{2}}(s_{2}) ⋯ X^{m_{l+1}}(s_{l+1}) c(s_{l+1}). (2.30)

The values of m_{j} that occur in both (2.29) and (2.30) satisfy

|m_{j}|/J̃(l) ≤ (l + 1)/(3l − 3) ≤ 1, (2.31)

as long as l ≥ 3. So, we can apply Lemma 2.3 with N = 1.

Recall that X_{j k}^{m}(l, s, h̄) = 0 and X̃_{j k}^{m}(s, h̄) = 0 if ||j| − |k|| > |m|. Since c_{k}(0) is non-zero

only for |k| ≤ J, and we need only consider the j^{th} coefficient of K̃(t)c(t) for |j| > J̃(l), the only relevant values of n in (2.29) must satisfy

n ≥ 3l − 2. (2.32)

This condition is also satisfied for all values of n in (2.30), since the sum begins with n = 3l+6. To use the analyticity assumptions to get estimates on the derivatives of V, we define

C_{δ}(x) = {z ∈ I C^{d} : z_{j} = x_{j} + δe^{iθ_{j}}, θ_{j} ∈ [0, 2π), j = 1, 2, ⋯, d}.

If z ∈ C_{δ}(a(t)), then, for all j = 1, 2, ⋯, d,

|z_{j}| ≤ δ + |a_{j}(t)|.

Hence, writing \frac{1}{m!} D^{m}V (a(t)) as a d–dimensional Cauchy integral, we get the bound

\frac{1}{m!} |D^{m}V (a(t))| ≤ \frac{sup_{z∈C_{δ}(a(t))} |V (z)|}{δ^{|m|}} ≡ \frac{v(t)}{δ^{|m|}} (2.33)

where

v(t) ≤ M exp(4τ(δ^{2}d + a^{2}(t))).

Furthermore, since ∥A(t)∥ depends continuously on t, there exists w(T), such that

sup_{t∈[−T,T]} ∥A(t)∥ ≤ w(T).

The norm of the vector c(t) is 1 since K(l, t, h̄) is self-adjoint. Thus, the non-zero entries of c(0) are each bounded by 1, and by a crude estimate, there are at most (J + 1)^{d} of them. Similarly, c(t) has at most (J̃(l)+1)^{d} non-zero entries, each of which is bounded by 1. Thus, for l ≥ 3, (2.11), (2.10), (2.14), (2.29), (2.30), (2.32), (2.31), and Lemma 2.3 imply the following two estimates when j satisfies J̃(l) + 1 ≤ |j| ≤ J̃(l) + l + 1:

| (K̃(t) c^{q}(t))_{j} |

≤ v(t)\frac{(∫_{0}^{T} v(s) ds)^{q}}{h̄^{q}q!} ∑_{n=3l−2}^{(l+1)(q+1)} (D(1) w(T)/δ)^{n} h̄^{n/2} J̃(l)^{n/2} F_{l+1}(n, q + 1) (J + 1)^{d}

= \frac{v(t)h̄(q + 1)(J + 1)^{d}}{∫_{0}^{T} v(s)ds} \frac{(∫_{0}^{T} v(s)ds)^{q+1}}{h̄^{q+1}(q + 1)!} ∑_{n=3l−2}^{(l+1)(q+1)}(D(1)w(T)\sqrt{h̄J̃(l)}/δ)^{n}F_{l+1}(n, q + 1)

(2.34)

and

| (K̃(t)r(t))_{j} |

≤ \frac{v(t)h̄(l + 2)(J̃(l) + 1)^{d}}{∫_{0}^{T} v(s)ds} \frac{(∫_{0}^{T} v(s)ds)^{l+2}}{h̄^{l+2}(l + 2)!} ∑_{n=3l−2}^{(l+1)(l+2)}(D(1)w(T)\sqrt{h̄J̃(l)}/δ)^{n}F_{l+1}(n, l + 2),

(2.35)

where F_{p}(n, q) is defined by (2.16).

By Lemma 2.4, (2.34), and (2.35), both | (K̃(t) c^{q}(t))_{j} | and | (K̃(t)r(t))_{j} | are bounded by C_{3} exp { −γ_{1}/h̄ } for an appropriate choice of l(h̄) = [[ g/h̄]] and sufficiently small h̄. Each of the l + 2 terms in (2.26) contributes a term of this type to ξ_{l}^{(2)}(x, t, h̄), so

h̄^{−1} ∥ ξ_{l}^{(2)}(x, t, h̄) ∥ ≤ C_{3} h̄^{−1} (l(h̄) + 2) exp { − γ_{1}/h̄ }
≤ C_{3} exp { − γ_{2}/h̄ } ,

for any γ_{2} < γ_{1} when h̄ is sufficiently small.

We shrink g if necessary to make

C_{5}^{2} g < 1 (2.36)

and set l = l(h̄) in (2.24). This yields a similar estimate

h̄^{−1} ∥ ξ_{l}^{(1)}(·, t, h̄) ∥ ≤ C_{6} exp { − γ_{3}/h̄ } ,

for some γ_{3} > 0.

We combine these two estimates and apply Lemma 2.1 to obtain (1.2) with γ = min {γ_{2}, γ_{3}}. This proves the theorem.

3 Proof of Lemma 2.2

For simplicity, we drop the t dependence in the notation throughout this section.

To prove Lemma 2.2, we use Hölder’s inequality to see that ∑_{|j|≤_{J}(l)} |c_{j}(l, h̄)|^{2} = 1 implies

∑_{|j|≤_{J}(l)} |c_{j}(l, h̄)| ≤ ( ∑_{|j|≤_{J}(l)} |c_{j}(l, h̄)|^{2} )^{1/2} ( ∑_{|j|≤_{J}(l)} 1 )^{1/2}

≤ ( J̃(l) + 1 )^{d/2} .

Thus, it is sufficient to prove

∥ \frac{D^{m}V (ζ(x, a))}{m!} (x − a)^{m} φ_{j}(A, B, h̄, a, η, x) ∥

≤ g_{3} M exp(4τ(δ^{2}d + a^{2})) ( g_{4} \sqrt{h̄(l + 2)} ∥A∥/δ )^{l+2} ,  (3.1)

for some g_{3} and g_{4}, whenever |j| ≤ J̃(l).

We mimic the proof of 2.33 to obtain a bound on D^{m}V (ζ(x, a))/m!. If z ∈ C_{δ}(ζ(x, a)), then, for all j = 1, 2, ⋯, d,

|z_{j}| ≤ δ + |ζ_{j}(x, a)| ≤ δ + |a_{j}| + |x_{j} − a_{j}|.

Using this and applying (b+c)^{2} ≤ 2(b^{2}+c^{2}) several times, we see that z ∈ C_{δ}(ζ(x, a)) implies

| V (z) | ≤ M exp(2τ(x − a)^{2}) exp(4τ(δ^{2}d + a^{2})).

Hence, writing \frac{1}{m!} D^{m}V (ζ(x, a)) as a d–dimensional Cauchy integral, we easily obtain the bound

\frac{1}{m!} |D^{m}V (ζ(x, a))| ≤ M \frac{exp(4τ(δ^{2}d + a^{2}))}{δ^{|m|}} exp(2τ(x − a)^{2}). (3.2)

Thus, estimate (3.1) follows from the corresponding estimate on the integral

I = ∫_{IR_{d}} (x − a)^{2m} exp(4τ(x − a)^{2}) |φ_{j}(A, B, h̄, a, η, x)|^{2} dx. (3.3)

Performing the change of variables x ↦ y = |A|^{−1}(x−a)/h̄^{1/2}, and using the explicit formula for φ_{j}, we see that

I = \frac{h̄^{|m|}}{2^{|j|} j! π^{d/2}} ∫_{IR_{d}} (|A|y)^{2m} exp(−y^{2} + 4τh̄(|A|y)^{2}) |H_{j}(A; y)|^{2} dy, (3.4)

where H_{l}(A; y) is the polynomial described in Remarks 4 and 5 that immediately follow the definition of φ_{j}(A, B, h̄, a, η, x) in Section 2.

We assume henceforth that h̄ is sufficiently small that

4 τ h̄ ∥A∥^{2} ≤ 1/2. (3.5)

The estimate (|A|y)^{2} ≤ ∥A∥^{2} y^{2} implies (|A|y)_{k}^{2} ≤ ∥A∥^{2} y^{2}, for k = 1, 2, ⋯, d. From this, we conclude

I ≤ \frac{(h̄ ∥A∥^{2})^{|m|}}{2^{|j|} j! π^{d/2}} ∫ y^{2|m|} exp(−y^{2}/2) |H_{j}(A; y)|^{2} dy. (3.6)

We next need an estimate on |H_{j}(A; y)|. By Remark 5 after the definition in Section 2,

(−Δ + y^{2}) H_{j}(A; y) e^{−y^{2}/2} = (2|j| + d) H_{j}(A; y) e^{−y^{2}/2}. (3.7)

This equation states that H_{j}(A; y) e^{−y^{2}/2} is a eigenfunction of −Δ + y^{2} corresponding to the eigenvalue 2|j| + d. Introducing normalization factors, we conclude that

2^{−|j|/2} (j!)^{−1/2} π^{−d/4} H_{j}(A; y) e^{−y^{2}/2} = ∑_{|k|=|j|} b_{k} φ_{k_{1}}(y_{1})φ_{k_{2}}(y_{2}) ⋯ φ_{k_{d}}(y_{d}), (3.8)

where φ_{k}(y) = 2^{−k/2} (k!)^{−1/2} π^{−1/4} H_{k}(y) e^{−y^{2}/2} is the normalized eigenfunction of −\frac{∂^{2}}{∂y^{2}} +y^{2}

corresponding to the eigenvalue 2k + 1, and the coefficients b_{k} satisfy ∑_{|k|=|j|} |b_{k}|^{2} = 1. We can thus deduce an estimate of H_{j}(A; y) from an estimate for usual Hermite polynomials H_{k}(y):

|H_{j}(A; y)|^{2} = | ∑_{|k|=|j|} b_{k} H_{k_{1}}(y_{1})H_{k_{2}}(y_{2}) ⋯ H_{k_{d}}(y_{d})|^{2}
≤ ∑_{|k|=|j|} |H_{k_{1}}(y_{1})H_{k_{2}}(y_{2}) ⋯ H_{k_{d}}(y_{d})|^{2}.  (3.9)

The Hermite polynomials, in turn, satisfy the following bounds.

Lemma 3.1 If |y| > \sqrt{2k + 1}, then

|H_{k}(y)| ≤ 2^{k} |y|^{k}, (3.10)

and there exists a numerical constant κ > 0, such that for all y ∈ IR,

|H_{k}(y)| ≤ κ 2^{k/2} \sqrt{k!} e^{y^{2}/2}. (3.11)

Proof: The second statement is well known. See [1], formula 22.14.17 or [4], formula 8.954.2.

By symmetry, it is sufficient to prove (3.10) for y > 0. It is well known that the k^{th} eigenfuction of the harmonic oscillator is non-zero in the classically forbidden region |y| > \sqrt{2k + 1}. It is also well known that H_{k}(y) = 2^{k}y^{k} + P_{k}(y) where P_{k}(y) is a polynomial of degree k − 2. Thus, we have

H_{k}(y)/(2^{k}y^{k}) = 1 + B_{k}(y),

where B_{k} satisfies

B_{k}(y) = O(1/y^{2}) for large y,

and

B_{k}(y) > −1 if y > \sqrt{2k + 1}.

Using standard properties of the Hermite polynomials, we see that

B′_{k}(y) = (y H′_{k}(y) − k H_{k}(y))/(2^{k} y^{k+1})
= (2 k y H_{k−1}(y) − k H_{k}(y))/(2^{k} y^{k+1})
= k (2 y H_{k−1}(y) − (2 y H_{k−1}(y) − 2 (k − 1) H_{k−2}(y)))/(2^{k} y^{k+1})
= k (k − 1) H_{k−2}(y)/(2^{k−1} y^{k+1}).

So, if y > \sqrt{2k + 1}, we see that B′_{k}(y) > 0. Thus, y > \sqrt{2k + 1} implies −1 < B_{k}(y) < 0, which implies (3.10).

In (3.6), we use the multinomial expansion

(y^{2})^{|m|} = ∑_{|n|=|m|} \binom{|m|}{n_{1} ⋯ n_{d}} y_{1}^{2n_{1}} y_{2}^{2n_{2}} ⋯ y_{d}^{2n_{d}}

to obtain

I ≤ \frac{(h̄ ∥A∥^{2})^{|m|}}{2^{|j|} j! π^{d/2}} ∑_{|n|=|m|} ∑_{|k|=|j|} \binom{|m|}{n_{1} ⋯ n_{d}} ∏_{r=1}^{d} ∫ y_{r}^{2n_{r}}|H_{k_{r}}(y_{r})|^{2} exp(−y_{r}^{2}/2)dy_{r}.

Using Lemma 3.1, the one dimensional integrals in the final factor here can then be estimated as follows (without indices):

∫ y^{2n} |H_{k}(y)|^{2} exp(−y^{2}/2) dy
≤ ∫_{y^{2}≤2k+1} κ^{2} 2^{k} k! y^{2n} exp(y^{2}/2) dy + ∫ 4^{k} y^{2(n+k)} exp(−y^{2}/2) dy

≤ κ^{2}2^{k}k!\frac{2}{2n + 1}(2k + 1)^{n+1/2} exp(k + 1/2) + 4^{k}2^{(k+n+1/2)} ∫ z^{2(k+n)}e^{−z^{2}}dz

= \frac{2^{k+1}}{2n + 1}κ^{2}e^{k+1/2}k!(2k + 1)^{n+1/2} + 4^{k}2^{(k+n+1/2)}π^{1/2} \frac{(2(k + n) − 1)!}{2^{2(n+k)−1}(k + n − 1)!}

= \frac{2^{k+1}}{2n + 1}κ^{2}e^{k+1/2}k!(2k + 1)^{n+1/2} + 2^{k−n+3/2}π^{1/2} \frac{(2(k + n) − 1)!}{(k + n − 1)!}.  (3.12)

Here we have used

∫ z^{2(k+n)} e^{−z^{2}} dz = π^{1/2} \frac{1 · 3 · 5 ⋯ (2(k + n) − 1)}{2^{k+n}}.

Now, Stirling’s formula guarantees the existence of a > 0, such that

a^{n} n^{n} ≤ n! ≤ n^{n}, (3.13) for all integers n ≥ 1. Using this in (3.12), we obtain

∫ y^{2n} |H_{k}(y)|^{2} exp(−y^{2}/2) dy ≤ f_{0} f_{1}^{k+n} (k + n)!,

for some constants f_{0} and f_{1}. From this we conclude (restoring the indices)

I ≤ \frac{(h̄ ∥A∥^{2})^{|m|}}{2^{|j|} j! π^{d/2}} ∑_{|n|=|m|} ∑_{|k|=|j|} f_{0}^{d} f_{1}^{|m|+|j|} \binom{|m|}{n_{1} ⋯ n_{d}} (k + n)! (3.14)

In this expression, we have

(k + n)! = (|k| + |n|)! \binom{|k| + |n|}{k_{1} + n_{1}  ⋯  k_{d} + n_{d}}^{−1}

= (|m| + |j|)! \binom{|m| + |j|}{k_{1} + n_{1}  ⋯  k_{d} + n_{d}}^{−1}
≤ (|m| + |j|)!  (3.15)

We further use ∑_{|n|=|m|} \binom{|m|}{n_{1} ⋯ n_{d}} = d^{|m|} and (2.23) in (3.14) to obtain

I ≤ (h̄ ∥A∥^{2})^{|m|} ( \frac{f_{0}}{\sqrt{π}} )^{d} (f_{1} d)^{|m|} ( \frac{e^{d} f_{1}}{2} )^{|j|} \frac{(|m| + |j|)!}{j!}

= (h̄ ∥A∥^{2})^{|m|} ( \frac{f_{0}}{\sqrt{π}} )^{d} (f_{1} d)^{|m|} ( \frac{e^{d} f_{1}}{2} )^{|j|} \binom{|j|}{j_{1} ⋯ j_{d}} ( \begin{matrix}|m| + |j| \\ |j| \end{matrix} ) |m|!

≤ (h̄ ∥A∥^{2})^{|m|} ( \frac{f_{0}}{\sqrt{π}} )^{d} (2 f_{1} d)^{|m|} (e^{d} f_{1})^{|j|} \binom{|j|}{j_{1} ⋯ j_{d}} |m|!

≡ (h̄ ∥A∥^{2}d)^{|m|} f_{2}^{d} f_{3}^{|m|} f_{4}^{|j|} \binom{|j|}{j_{1} ⋯ j_{d}} |m|!,  (3.16)

where f_{2}, f_{3}, f_{4} are numerical constants.

Estimates (3.2) and (3.16) imply

∥ \frac{D^{m}V (ζ(x, a))}{m!} (x − a)^{m} φ_{j}(A, B, h̄, a, η, x) ∥

≤ M exp(4τ(δ^{2}d + a^{2}))(h̄∥A∥^{2}(d/δ^{2})f_{3})^{|m|/2}f_{2}^{d/2}f_{4}^{|j|/2}\sqrt{|m|!}\sqrt{\binom{|j|}{j_{1} ⋯ j_{d}}}.

So, by the Schwarz inequality, (1.4), (3.9), and (3.18),

∥ \frac{D^{m}V (ζ(x, a))}{m!} (x − a)^{m} ψ_{l}(x, t, h̄) ∥^{2}

≤ ∑_{n=0}^{_{J}(l)} ∑_{|j|=n} ∥ \frac{D^{m}V (ζ(x, a))}{m!} (x − a)^{m} φ_{j}(A, B, h̄, a, η, x) ∥^{2}

≤ M^{2} exp(8τ(δ^{2}d + a^{2}))(h̄∥A∥^{2}(d/δ^{2})f_{3})^{|m|}f_{2}^{d}|m|! ∑_{n=0}^{_{J}(l)} ∑_{|j|=n} f_{4}^{|j|} \binom{|j|}{j_{1} ⋯ j_{d}}

= M^{2} exp(8τ(δ^{2}d + a^{2})) (h̄∥A∥^{2}(d/δ^{2})f_{3})^{|m|} f_{2}^{d} |m|! ∑_{n=0}^{_{J}(l)} f_{4}^{n} d^{n}

= M^{2} exp(8τ(δ^{2}d + a^{2})) (h̄∥A∥^{2}(d/δ^{2})f_{3})^{|m|} f_{2}^{d} |m|! ( \frac{(f_{4}d)^{_{J}(l)+1} − 1}{f_{4}d − 1} )

≤ M^{2} exp(8τ(δ^{2}d + a^{2})) (h̄∥A∥^{2}(d/δ^{2})f_{3})^{|m|} f_{2}^{d} |m|! 2 (f_{4}d)^{_{J}(l)}.  (3.17)

The last step depends on the following fact: We can assume without loss that q = f_{4}d ≥ 2, so that for any positive integer p, we have −1 ≤ q^{p+1}−2q^{p}, and hence, q^{p+1}−1 ≤ 2q^{p+1}−2q^{p}. Thus,

(q^{p+1} − 1)/(q − 1) ≤ 2q^{p}. (3.18)

We use the hypothesis |m| = l + 2 and note that |m|! ≤ (l + 2)^{l+2}. We then conclude that there exist constants g_{0} and g_{1}, that depend only on d and J, such that (2.13) holds. This implies the lemma.

4 Proofs of Lemmas 2.3 and 2.4

The proof of Lemma 2.3 relies on two preliminary lemmas.

Lemma 4.1 The matrix elements of (x − a)^{m} satisfy

| ⟨ φ_{j}(A, B, h̄, a, η, x), (x − a)^{m} φ_{k}(A, B, h̄, a, η, x)⟩ |
≤ h̄^{|m|/2} (\sqrt{2}d)^{|m|} ∥A∥^{|m|} \sqrt{(|k| + 1)(|k| + 2) ⋯ (|k| + |m|)},  (4.1)

and

⟨ φ_{j}(A, B, h̄, a, η, x), (x − a)^{m} φ_{k}(A, B, h̄, a, η, x) ⟩ = 0, if ||j| − |k|| > |m|.

Proof: For i = 1, 2, ⋯, d, we can use equation (3.28) of [7] to express (x_{i} − a_{i}) in terms of raising and lowering operators. Doing so, we obtain

(x_{i} − a_{i}) φ_{k}(A, B, h̄, a, η, x) = \sqrt{h̄/2} ∑_{p=1}^{d} A_{ip} \sqrt{k_{p} + 1} φ_{k′(p)}(A, B, h̄, a, η, x)

+ \sqrt{h̄/2} ∑_{p=1}^{d} Ā_{ip} \sqrt{k_{p}} φ_{k′′(p)}(A, B, h̄, a, η, x),

where k′(p) and k′′(p) are constructed from k by replacing the component k_{p} by k_{p} + 1 and k_{p} − 1, respectively. Thus, (x_{i} − a_{i}) φ_{k}(A, B, h̄, a, η, x) may be written as a sum of 2d terms, each of which has the form b_{q} φ_{q}(A, B, h̄, a, η, x), where |q| ≤ |k| + 1 and |b_{q}| ≤ \sqrt{h̄/2} ∥A∥ \sqrt{|k| + 1}.

From this and a simple induction, (x − a)^{m} φ_{k}(A, B, h̄, a, η, x) may be written as a sum of (2d)^{|m|} terms, each of which has the form b_{q} φ_{q}(A, B, h̄, a, η, x), where |k| − |m| ≤ |q| ≤ |k| + |m| and |b_{q}| ≤ (h̄/2)^{|m|} ∥A∥^{|m|} \sqrt{(|k| + 1)(|k| + 2) ⋯ (|k| + |m|)}.

This implies the lemma.

In the next lemma we use the shorthand X^{m}(t) to denote X^{m}(l, t, h̄) of Lemma 2.3.

Lemma 4.2 For each N > 0, there exists D(N) < ∞, such that

sup_{{j,k: |j|, |k|≤_{J}(l)}} | (X^{m_{1}}(t_{1}) X^{m_{2}}(t_{2}) ⋯ X^{m_{q}}(t_{q}))_{j k} |

≤ (D(N) \sqrt{h̄} sup_{t∈\{t_{1},t_{2},···t_{q}\}} ∥A(t)∥)^{|m_{1}|+|m_{2}|···+|m_{q}|} J̃(l)^{(|m_{1}|+|m_{2}|+···+|m_{q}|)/2},  (4.2)

for any collection m_{1}, m_{2}, …, m_{q} of multi-indices that satisfy |m_{j}|/J̃(l) ≤ N.

Proof: The matrix X^{m} is obtained from X̃^{m} by restricting its indices to |j|, |k| ≤ J̃(l). Using Lemma 4.1, we see that for such values of the indices,

| X_{j k}^{m}(t) | ≤ (\sqrt{2h̄} d ∥A(t)∥)^{|m|} J̃(l)^{|m|/2} \sqrt{(1 + 1/J̃(l))(1 + 2/J̃(l))) ⋯ (1 + |m|/J̃(l))}.

Thus, |m|/J̃(l) ≤ N implies

| X_{j k}^{m}(t) | ≤ (\sqrt{2h̄} d ∥A(t)∥)^{|m|} (1 + N)^{|m|/2} J̃(l)^{|m|/2}.

We estimate the absolute value of the j, k matrix element of the product

| (X^{m_{1}}(t_{1}) X^{m_{2}}(t_{2}) ⋯ X^{m_{q}}(t_{q}))_{j k} |
≤ ∑_{\begin{matrix}|p_{i}| ≤ _{J}(l), i = 1, ⋯, q − 1 \\ |p_{1} − j| ≤ |m_{1}|, |p_{2} − p_{1}| ≤ |m_{2}|, ⋯, |k − p_{q−1}| ≤ |m_{q}| \end{matrix}} | X_{j p_{1}}^{m_{1}}(t_{1}) X_{p_{1} p_{2}}^{m_{2}}(t_{2}) ⋯ X_{p_{q−1} k}^{m_{q}}(t_{q}) |

(4.3)

by bounding the absolute values of each of the matrix elements X_{p_{i−1} p_{i}}^{m_{i}}(t_{i}) by

| X_{p_{i−1} p_{i}}^{m_{i}}(t_{i}) | ≤ (\sqrt{2h̄} d ∥A(t_{i})∥)^{|m_{i}|} (1 + N)^{|m_{i}|/2} J̃(l)^{|m_{i}|/2}, (4.4)

and multiplying by a bound on the number of terms.

To estimate the number of terms, we first note that for any multi-index r, the number of multi-indices indices p that satisfy ||p| − |r|| ≤ |m| is equal to the number of vectors v with integer components, such that | ∑_{i=1}^{d} v_{i}| ≤ |m|. This is bounded by the number of vectors with integer components, such that |v_{i}| ≤ |m| for i = 1, 2, ⋯, d, which is (2|m| + 1)^{d}.

Thus, the number of terms is bounded by

∑_{\begin{matrix}|p_{i}| ≤ _{J}(l), i = 1, ⋯, q − 1 \\ |p_{1} − j| ≤ |m_{1}|, |p_{2} − p_{1}| ≤ |m_{2}|, ⋯, |k − p_{q−1}| ≤ |m_{q}| \end{matrix}} 1

≤ (2|m_{1}| + 1)^{d}(2|m_{2}| + 1)^{d} ⋯ (2|m_{q}| + 1)^{d}
≤ (e^{2})^{d(|m_{1}|+|m_{2}|+···+|m_{q}|)}.  (4.5)

The final inequality follows because 1 + 2|m_{j}| ≤ e^{2|m_{j}|}.

We obtain the lemma by using (4.4) and (4.5) to bound (4.3).

Proof of Lemma 2.3 We mimic the proof of Lemma 4.2. Let m̃ = |m_{0}| + |m_{1}| + |m_{2}| + … + |m_{q}|. By Lemmas 4.1 and 4.2,

(\sqrt{h̄} sup_{t∈\{t_{0},t_{1},···,t_{q}\}} ∥A(t)∥)^{−_{m}} | (X̃^{m_{0}}(t_{0}) X^{m_{1}}(t_{1}) X^{m_{2}}(t_{2}) ⋯ X^{m_{q}}(t_{q}))_{j k} |

= (\sqrt{h̄} sup_{t∈\{t_{0},t_{1},···,t_{q}\}} ∥A(t)∥)^{−_{m}} ∑_{|r|≤_{J}(l)} | X̃_{j r}^{m_{0}}(t_{0}) | | (X^{m_{1}}(t_{1}) X^{m_{2}}(t_{2}) ⋯ X^{m_{q}}(t_{q}))_{r k} |

≤ ∑_{\begin{matrix}|r| ≤ _{J}(l) \\ |j − r| ≤ |m_{0}| \end{matrix}} 2^{|m_{0}|/2} \sqrt{(|r| + 1)(|r| + 2) ⋯ (|r| + |m_{0}|)}

× D(N)^{|m_{1}|+|m_{2}|+...+|m_{q}|} J̃(l)^{(|m_{1}|+|m_{2}|+...+|m_{q}|)/2}

≤ ∑_{\begin{matrix}|r| ≤ _{J}(l) \\ |j − r| ≤ |m_{0}| \end{matrix}} 2^{|m_{0}|/2} J̃(l)^{|m_{0}|/2} \sqrt{(1 + 1/J̃(l))(1 + 2/J̃(l)) … (1 + |m_{0}|/J̃(l))}

× D(N)^{|m_{1}|+|m_{2}|+...+|m_{q}|} J̃(l)^{(|m_{1}|+|m_{2}|+...+|m_{q}|)/2}
≤ ∑_{\begin{matrix}|r| ≤ _{J}(l) \\ |j − r| ≤ |m_{0}| \end{matrix}} 2^{|m_{0}|/2} (1 + N)^{|m_{0}|/2} D(N)^{|m_{1}|+|m_{2}|+...+|m_{q}|} J̃(l)^{(|m_{1}|+|m_{2}|+...+|m_{q}|)/2}

≤ 2^{|m_{0}|/2} (1 + N)^{|m_{0}|/2} D(N)^{|m_{1}|+|m_{2}|+...+|m_{q}|} J̃(l)^{(|m_{1}|+|m_{2}|+...+|m_{q}|)/2} ∑_{|j−r|≤|m_{0}|} 1.

Since the sum in the last line is (1+2|m_{0}|)^{d} ≤ e^{2d|m_{0}|}, we obtain the desired estimate.

We now turn to the proof of Lemma 2.4. Our first step is to study the combinatorial factor F_{p}(n, q) in the one dimensional case.

Lemma 4.3 Let m_{j} denote positive integers, and let

G_{p}(n, q) = ∑_{\begin{matrix}1 ≤ m_{1}, m_{2}, ⋯, m_{q} ≤ p \\ m_{1} + m_{2} + ⋯ + m_{q} = n \end{matrix}} 1.  (4.6)

This quantity is zero if n < q or n > qp. Otherwise, it satisfies

G_{p}(n, q) ≤ \{ \begin{matrix}( \begin{matrix}n − 1 \\ q − 1 \end{matrix} ) & if & q ≤ n ≤ [q(p + 1)/2] \\ ( \begin{matrix}q(p + 1) − n − 1 \\ q − 1 \end{matrix} ) & if & [q(p + 1)/2] + 1 ≤ n ≤ qp. \end{matrix} (4.7)

Proof: If n < q or n > qp, the sum in (4.6) contains no terms, so G_{p}(n, q) = 0.

The quantity G_{p}(n, q) is the number of ways that the number n can be decomposed as n = m_{1} + m_{2} + … + m_{q} with each m_{j} satisfying 1 ≤ m_{j} ≤ p. We uniquely associate to each such decomposition of n, a corresponding decomposition of n′ = q(p+1)−n = m′_{1} +m′_{2} +…+m′_{q} by setting m′_{j} = p + 1 − m_{j}. This association is a one-to-one correspondence, so we see that G_{p}(n, q) satisfies the symmetry relation

G_{p}(n, q) = G_{p}(q(p + 1) − n, q). (4.8)

As a consequence, the second inequality in (4.7) (with [q(p + 1)/2] + 1 ≤ n ≤ qp) follows from the first (with q ≤ n ≤ [q(p + 1)/2]).

To prove the first inequality in (4.7) we drop the condition m_{j} ≤ p and exactly calculate the resulting function. Dropping the upper bound on the m_{j}, we clearly have G_{p}(n, q) ≤ G(n, q), where

G(n, q) = ∑_{\begin{matrix}1 ≤ m_{1}, m_{2}, ⋯, m_{q} \\ m_{1} + m_{2} + ⋯ + m_{q} = n \end{matrix}} 1.

So, the lemma will be proved once we establish

G(n, q) = ( \begin{matrix}n − 1 \\ q − 1 \end{matrix} ) . (4.9)

To prove this, we use induction. Formula (4.9) is trivial to verify for all n ≥ 1 when q = 1 and for all n ≥ 2 when q = 2. Assume (4.9) has been verified whenever n ≥ q − 1, and suppose n ≥ q. We have

G(n, q) = ∑_{m_{1}=1}^{n−q+1} G(n − m_{1}, q − 1) = ∑_{m_{1}=1}^{n−q+1} ( \begin{matrix}n − m_{1} − 1 \\ q − 2 \end{matrix} ) ,

so the induction step is complete, since it is known (see [11], p.36) that

∑_{m_{1}=1}^{n−q+1} ( \begin{matrix}n − m_{1} − 1 \\ q − 2 \end{matrix} ) = ( \begin{matrix}n − 1 \\ q − 1 \end{matrix} ) . (4.10)

This ends the proof of the lemma.

Our next step is to obtain an estimate for G_{p}(n, q) that depends only on n.

Lemma 4.4 Let G_{p}(n, q) be defined by (4.6). There exists a constant C_{7}, such that p ≥ 1, q ≥ 1, and n ≥ 1 imply

G_{p}(n, q)) ≤ (C_{7})^{n}. (4.11)

Proof: We note that G_{p}(n, q) is zero unless q ≤ n ≤ qp. Furthermore, G_{p}(q, q) = G_{p}(qp, q) = G_{2}(2, 1) = 1 and G_{1}(n, q) = δ_{n q}. So, we need only prove existence of C_{7}, such that p ≥ 2, q ≥ 1, (p, q) ≠ (2, 1) imply

sup_{q<n<pq} (G_{p}(n, q))^{1/n} ≤ C_{7}. (4.12)

To prove this, we first study the case where q < n ≤ [q(p + 1)/2]. Since q ≤ n, we have

q^{1/n} = exp(ln(q)/n) ≤ exp(ln(n)/n) ≤ e^{1/e}.

Thus, by (4.7), the condition q < n ≤ [q(p + 1)/2], and (3.13), we have

(G_{p}(n, q))^{1/n} ≤ ( \frac{(n − 1)!}{(q − 1)! (n − q)!} )^{1/n}

≤ q^{1/n} ( \frac{(n − 1)!}{q! (n − q)!} )^{1/n}
≤ e^{1/e} \frac{n}{aq^{q/n} (n − q)^{(n−q)/n}} .  (4.13)

An explicit computation shows that

\frac{∂}{∂n} \frac{n}{q^{q/n} (n − q)^{(n−q)/n}} = \frac{n}{q^{q/n} (n − q)^{(n−q)/n}} \frac{q}{n^{2}} ( ln(q) − ln(n − q) ) .

From this, we deduce that the right hand side of (4.13) attains its maximum at n = 2q. Evaluating that maximum, we obtain

sup_{q<n≤[q(p+1)/2]} (G_{p}(n, q))^{1/n} ≤ \frac{2}{a} e^{1/e}. (4.14)

For [q(p + 1)/2] < n < pq, the number n′ = q(p + 1) − n satisifes q < n′ ≤ [q(p + 1)/2] and n′/n ≤ 1. Thus, by (4.8) and (4.14), we obtain

(G_{p}(n, q))^{1/n} = (G_{p}(n′, q))^{1/n}
≤ ( \frac{2}{a} e^{1/e} )^{n′/n}
≤ \frac{2}{a} e^{1/e}.

Thus,

sup_{[q(p+1)/2]<n<pq} (G_{p}(n, q))^{1/n} ≤ \frac{2}{a} e^{1/e}. (4.15)

Inequalities (4.14) and (4.15) imply the existence of C_{7} for which (4.12) holds. This implies (4.11),and the lemma is proved.

We now generalize Lemma 4.4 to the multi-dimensional case.

Lemma 4.5 Let F_{p}(n, q) be defined by (2.16). For all p ≥ 1, q ≥ 1, and n ≥ 1, we have

F_{p}(n, q) ≤ C_{7}^{(n+qd)}, (4.16)

where C_{7} is the constant of Lemma 4.4.

Proof: With m_{j} temporarily denoting numbers instead of multi-indices, we define

Γ_{p}(n, q) = ∑_{\begin{matrix}0 ≤ m_{1}, m_{2}, ⋯, m_{q} ≤ p \\ m_{1} + m_{2} + ⋯ + m_{q} = n \end{matrix}} 1,

which is zero unless 0 ≤ n ≤ pq.

By defining m′_{j} = m_{j} +1, we see that every decomposition of n as n = m_{1} +m_{2} +⋯+m_{q}, with 0 ≤ m_{j} ≤ p, corresponds uniquely to a decomposition of n + q as n + q = m′_{1} + m′_{2} + ⋯ + m′_{q}, with 1 ≤ m′_{j} ≤ p + 1. Therefore, we have the identity

Γ_{p}(n, q) = G_{p+1}(n + q, q). (4.17)

We now let the m_{j} denote multi-indices and let m_{j}(k) denote the k^{th} component of m_{j}. Then, using (4.17) and Lemma 4.4, we easily obtain

F_{p}(n, q) ≤ ∑_{\begin{matrix}0 ≤ m_{j}(k) ≤ p \\ ∑_{j=1}^{q} ∑_{k=1}^{d} m_{j}(k) = n \end{matrix}} 1 = Γ_{p}(n, qd) ≤ C_{7}^{(qd+n)}.

Proof of Lemma 2.4: Suppose 2 ≤ q ≤ l + 2, and let L_{q}(l) satisfy (2.17), i.e.,

L_{q}(l) ≤ \frac{C_{1}^{q}}{h̄^{q} q!} ∑_{n=3l−2}^{q(l+1)} (C_{2} h̄ l)^{n/2} F_{l+1}(n, q).

By Lemma 4.5 and (3.13), there exist C_{8} and C_{9}, such that

L_{q}(l) ≤ \frac{(C_{1} C_{7}^{d})^{q}}{(a h̄ q)^{q}} ∑_{n=3l−2}^{q(l+1)} (C_{2} h̄ l)^{n/2} C_{7}^{n}

≤ \frac{C_{9}^{q}}{(h̄ q)^{q}} ∑_{n=3l−2}^{q(l+1)} (C_{8} h̄^{1/2} l^{1/2})^{n} .  (4.18)

Note that we can take

C_{8} = max \{ 1, \sqrt{C_{2}} C_{7} \},  (4.19)
C_{9} = max { 1, C_{1} C_{7}^{d} },  (4.20)

because we can assume C_{8} ≥ 1 and C_{9} ≥ 1 without loss of generality.

We arbitrarily choose two positive numbers α < 1 and β < 1. We then define

g = \frac{1}{C_{8}^{2}} min \{ α^{2}, \frac{β^{2}}{C_{9}^{2} C_{8}^{4} e^{2/e}} \} . (4.21)

We henceforth assume l = l(h̄) is chosen to satisfy (2.18) with this value of g. With this choice, we have

C_{8} h̄^{1/2} l^{1/2} ≤ α. (4.22) By summing a geometric series in (4.18), we see that

L_{q}(l) ≤ \frac{C_{9}^{q}}{(1 − α) (h̄ q)^{q}} ( C_{8} h̄^{1/2} l^{1/2} )^{3l−2}
= \frac{C_{9}^{q} C_{8}^{2q}}{(1 − α) (h̄ q C_{8}^{2})^{q}} ( C_{8} h̄^{1/2} l^{1/2} )^{3l−2} .  (4.23)

As a function of q,

(\frac{l}{q} )^{q} = exp ( q ln(l/q) )

is maximized at q = l/e, where it has the value e^{l/e}. Thus,

(\frac{l}{q} )^{q} ≤ e^{l/e}.

This, q ≤ l + 2, and h̄ l C_{8}^{2} ≤ α^{2} < 1 imply

\frac{1}{(h̄ q C_{8}^{2})^{q}} = \frac{1}{(h̄ l C_{8}^{2})^{q}} ( \frac{l}{q} )^{q} ≤ \frac{e^{l/e}}{(h̄ l C_{8}^{2})^{l+2}} .

From this, C_{8} ≥ 1, and C_{9} ≥ 1, we conclude

L_{q}(l) ≤ \frac{C_{9}^{l+2} C_{8}^{2(l+2)} e^{l/e}}{(1 − α)} ( C_{8} h̄^{1/2} l^{1/2} )^{l−6}
= ( e^{1/e} C_{9} C_{8}^{3} h̄^{1/2} l^{1/2} )^{l} \frac{C_{9}^{2}}{(1 − α) (h̄ l)^{3} C_{8}^{2}} .  (4.24)

By (4.21), we have

e^{1/e} C_{9} C_{8}^{3} h̄^{1/2} l^{1/2} ≤ β < 1.

Since

\frac{g}{h̄} − 1 ≤ l ≤ \frac{g}{h̄},

we see from (4.24) that if h̄ ≤ g/2, then we have

L_{q}(l) ≤ e^{−| ln(β)|l} \frac{C_{9}^{2}}{(1 − α) (g − h̄)^{3} C_{8}^{2}}
≤ \frac{2^{3} e C_{9}^{2}}{(1 − α) g^{3} C_{8}^{2}} e^{−| ln(β) |g/h̄}.  (4.25)

Note that the factor 2^{3} can be replaced by (1 + ϵ), with ϵ arbitrarily small, by taking h̄ sufficiently small.

Remark We can choose α and β to satisfy \frac{β}{e^{1/e}} ≤ α < \frac{1}{e^{1/e}} . Then, since \frac{β^{2}}{C_{9}^{2} C_{8}^{4} e^{2/e}} ≤

\frac{β^{2}}{e^{2/e}} ≤ α^{2}, we obtain the conclusion of Lemma 2.4 with

g = \frac{β^{2}}{C_{9}^{2} C_{8}^{6} e^{2/e}} ,  (4.26)
γ_{1} = \frac{| ln(β)| β^{2}}{C_{9}^{2} C_{8}^{6} e^{2/e}}  (4.27)
C_{3} = \frac{2^{3} e^{1+6/e} C_{9}^{8} C_{8}^{10}}{(1 − e^{−1/e}) β^{6}} ,  (4.28)

where C_{8} and C_{9} are related to C_{1}, C_{2}, and C_{7} by (4.19) and (4.20). Note that C_{7} is purely combinatorial and has no time dependence.

5 Proof of Theorem 1.2

To prove Theorem 1.2, we revisit the proof of Theorem 1.1 and make the T dependence of all constants explicit. We then allow T to grow with h̄ with the restriction that our approximation remain close to the actual solution as h̄ → 0.

Lemmas 4.3, 4.4, and 4.5 have no time dependence, and the time dependence of Lemmas 2.1, 2.2, 2.3, 3.1, 4.1, and 4.2 has been made explicit in their conclusions.

Since V is bounded below and energy is conserved, |a(t)| grows at most linearly with time. Thus, there exist v_{1} > 0 and v_{0} > 0, such that v(t) defined by (2.33) satisifies

v(t) ≤ v_{0} exp(v_{1}t). (5.1)

From assumption (1.13), it follows that

w(T) = sup_{0≤|t|≤T} ∥A(t)∥ ≤ N exp(λT). (5.2)

In the sequel, we denote all innessential constants that do not depend on T by the same symbol c.

Consider Lemma 2.2. For z ∈ C_{δ}(ζ(x, a(t))), we can prove the existence of ρ > 0 such that

∥ \frac{D^{m}V (ζ(x, a(t)))}{m!} (x − a(t))^{m} ψ_{l}(x, t, h̄) ∥

≤ c exp(ρ|t|) (c \sqrt{h̄ (l + 2)} exp(λ|t|))^{l+2} .  (5.3)

Indeed, with our bound on |V (z)| we have

\frac{D^{m}V (ζ(x, a(t)))}{m!} ≤ c exp(ρ|t|) exp(c|x − a(t)|)

instead of (3.2). With this and the estimate exp(cy) ≤ exp(cy^{2}) + exp(c) for all c > 0 and y > 0, we see that the estimates in the proof of Lemma 2.2 are still valid. This yields (5.3). Consequently, the constants C_{4} and C_{5} appearing in (2.24) satisfy

C_{4}(T) = c exp(ρT)
C_{5}(T) = c exp(λT),

where ρ > 0. By using estimates (5.1) and (5.2) in (2.34) and (2.35), we see that when we apply Lemma 2.4, the constants C_{1} and C_{2} satisfy

C_{1}(T) = c exp(ρT)
C_{2}(T) = c exp(2λT).

We still must determine the T dependence of C and γ of Theorem 1.1. We do this by determining the T dependence of g, γ_{1} and C_{3} in Lemma 2.4. Equations (4.19) and (4.20) yield C_{8} and C_{9}, in terms of which the above constants are determined. We find that

C_{8}(T) = c exp(λT)
C_{9}(T) = c exp(ρT).

Consequently, from (4.26), (4.27) ,(4.28), we have

g(T) = c exp(−νT),  ν = 2ρ + 6λ
γ_{1}(T) = c exp(−νT),
C_{3}(T) = c exp(μT),  μ = 8ρ + 10λ.

To arrive at these conclusions, we imposed various conditions. Those conditions were (3.5), that now requires h̄ ≤ c exp(−2λT), (2.36), than now requires c exp(2λT) exp(−νT) < 1, and h̄ ≤ g(T)/2 used in (4.25), that now requires h̄ ≤ c exp(−νT).

These are all satisfied, provided we take

T (h̄) ≤ T′ | ln(h̄)|,

with T′ > 0 sufficiently small. Using this in our estimate for the error term ξ(t, h̄), we see that for sufficiently small T′, |t| ≤ T′ | ln(h̄)| implies

∥ψ(x, t, h̄) − Ψ(x, t, h̄)∥_{L^{2}(IR^{d})} ≤ C′ exp(−γ′/h̄^{σ}),

for some γ′ > 0, σ > 0, and C′ > 0.

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