Unspecified Book Proceedings Series

Semiclassical Dynamics and Exponential Asymptotics

George A. Hagedorn and Alain Joye

Abstract. We discuss various constructions of approximate solutions to the time–dependent Schrödinger equation i ℏ \frac{∂ψ}{∂t} = − \frac{ℏ^{2}}{2} Δ ψ + V ψ in IR^{d} for small values of ℏ by means of wave packets that are localized in position and momentum. After reviewing earlier constructions that yield approximations up to O(ℏ^{∞}) errors, we show that if V satisfies appropriate analyticity and growth hypotheses, similar constructions agree with exact solutions up to errors of order e^{− γ/ℏ} for some γ > 0, provided t belongs to some fixed, compact time interval. Under more restrictive hypotheses, we extend the validity of these exponential estimates up to the Ehrenfest time scale: For sufficiently small T′, |t| ≤ T′ | log(ℏ)| implies that the norm of the error is of order e^{− γ′/ℏ^{σ}} for some γ′ > 0, and σ > 0.

1. Introduction

In this paper we outline several constructions of approximate solutions to the time–dependent Schrödinger equation that are localized near a single classical path. We begin with a discussion of certain basic semiclassical wave packets from which the approximate soultions are built. We review the literature on this subject and outline the proofs of several theorems. We conclude by describing the recent exponentially accurate approximations constructed in [HJ].

While postponing precise statements and comparisons with the literature to later sections, let us describe the ideas of the paper in some more detail.

Our first goal is to construct approximate solutions ψ_{l}(x, t, ℏ) to the d–dimensional time–dependent Schrödinger equation

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

that agree with exact normalized solutions Ψ(x, t, ℏ) up to O(ℏ^{l/2}) errors. Here, l is any positive integer.

Our construction of ψ_{l}(x, t, ℏ) is technically complicated, but quite explicit. It makes fundamental use of a collection { φ_{j}(A, B, ℏ, a, η, ·) }_{j∈IN^{d}} of semiclassical wave packets that were introduced in [H2]. We describe these wave packets

\begin{matrix}c \\ ◯ \end{matrix}0000 (copyright holder)

1991 Mathematics Subject Classification. 81Q20 .

Partially Supported by National Science Foundation Grant DMS–9703751.

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

in section 2. By following the recent simplified approach of [H3], we obtain very precise information of their properties. Here A and B are d × d complex matrices that satisfy certain conditions. The quantities a and η are elements of IR^{d}. Intuitively, the function φ_{j}(A, B, ℏ, 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{ℏ}, but depend on j. The position uncertainty is determined by the matrix |A| = \sqrt{A A^{∗}}, and the momentum uncertainty is determined by |B| = \sqrt{B B^{∗}}. For fixed A, B, ℏ, a, and η, \{ φ_{j}(A, B, ℏ, a, η, ·) \} is an orthonormal basis of L^{2}(IR^{d}) as j ranges over all d-dimensional multi-indices.

In section 3 we follow [H1] and [H2] to construct the approximate solution

ψ_{l}(x, t, ℏ) = e^{iS(t)/ℏ} ∑_{|j|≤J̃(l)} c_{j}(l, t, ℏ) φ_{j}(A(t), B(t), ℏ, a(t), η(t), x), (1.2)

from these basic wave packets. This approximate solution arises from initial conditions of the coherent state form

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

with ∑_{|j|≤J} |c_{j}|^{2} = 1. It satisfies the error estimate

sup_{t∈[−T, T]} ∥ψ_{l}(x, t, ℏ) − Ψ(x, t, ℏ)∥_{L^{2}(IR^{d})} ≤ C(l) h^{l/2} (1.4)

for some constant C(l), where Ψ(x, t, ℏ) is the exact solution. Here J̃(l) = J + 3l − 3; a(t) and η(t) are the position and momentum generated by the classical flow associated with the potential V; A(t) and B(t) are determined by the classical flow, linearized around the classical trajectory; and S(t) is the classical action integral. The initial conditions a(0), η(0), and S(0) are arbitrary, whereas A(0) and B(0) must satisfy certain normalization conditions. The complex valued coefficients c_{j}(l, t, ℏ) in the expansion (1.2) satisfy a linear system of ordinary differential equations that we describe in section 3. They depend on classical quantities only and can be approximated to arbitrary order in ℏ^{1/2} by the corresponding truncated Dyson expansion.

In section 4, we use these results to prove one of the main results of [HJ]: We construct another approximation ψ(x, t, ℏ) that is accurate up to an exponentially small error by applying the idea of optimal truncation of an asymptotic expansion. This requires a careful estimate of the l–dependence of C(l) in (1.4), which can be achieved for potentials that are analytic in a domain containing IR^{d}. This careful estimate allows us to choose l = l(ℏ) in an ℏ–dependent way, so that the error C(l(ℏ)) ℏ^{l(ℏ)/2} is minimized over all choices of l. Our analysis shows that l(ℏ) ≃ 1/ℏ. We define ψ(x, t, ℏ) = ψ_{l(ℏ)}(x, t, ℏ) and prove that for small values of ℏ, this approximation satisfies error estimates of the form

sup_{t∈[−T, T]} ∥ ψ(x, t, ℏ) − Ψ(x, t, ℏ) ∥_{L^{2}(IR^{d})} ≤ C e^{− γ/ℏ}, (1.5)

where C > 0, and γ > 0.

Finally, in section 5 we turn to the validity of these approximations for large times. Further assuming the existence of a Liapunov exponent λ > 0 for the classical dynamics, so that ∥A(t)∥ ≃ e^{λ|t|}, we can keep track of the time dependence in the estimates. This leads to an approximation ψ′(x, t, ℏ) corresponding to l(ℏ) ≃ 1/ℏ^{σ′}, such that [HJ]

(1.6) sup_{t∈[−T′| ln(ℏ)|, T′| ln(ℏ)|]} ∥ ψ(x, t, ℏ) − Ψ(x, t, ℏ) ∥_{L^{2}(IR^{d})} ≤ C′ e^{−γ′/ℏ^{σ}},

where C′, γ′, σ, and σ′ are positive, and T′ > 0 is chosen sufficiently small. This last result shows that the validity of exponential estimates can be extended up to the Ehrenfest time scale. A heuristic argument, based on the width of gaussian wave packets, shows that one cannot hope to prove such estimates beyond this time scale. We also discuss the links between the choice of T′ and the Lyapunov exponent λ.

2. Semiclassical Wave Packets

We begin this section by presenting the definition of the semiclassical wave packets φ_{j}(A, B, ℏ, a, η, x) that is given in [H3]. A more explicit, but more complicated definition is given in [H2].

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}, x^{j} = x_{1}^{j_{1}}x_{2}^{j_{2}} ⋯ x_{d}^{j_{d}}, j! = (j_{1}!)(j_{2}!) ⋯ (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 ℏ > 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.

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^{∗}.

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

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

The corresponding lowering operators A_{m}(A, B, ℏ, a, η) are their formal adjoints.

These operators satisfy commutation relations that lead to the properties of the φ_{j}(A, B, ℏ, a, η, x) that we list below. The raising operators A_{m}(A, B, ℏ, a, η)^{∗} for m = 1, 2, …, d commute with one another, and the lowering operators A_{m}(A, B, ℏ, a, η) commute with one another. However,

A_{m}(A, B, ℏ, a, η)^{∗} A_{n}(A, B, ℏ, a, η) − A_{n}(A, B, ℏ, a, η) A_{m}(A, B, ℏ, a, η)^{∗} = δ_{m, 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, ℏ, a, η, x) = π^{−d/4} ℏ^{−d/4} (det(A))^{−1/2}

× exp { − ⟨ (x − a), B A^{−1} (x − a) ⟩/(2ℏ) + i ⟨ η, (x − a) ⟩/ℏ } . (2.2)

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

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

× ( A_{d}(A, B, ℏ, a, η)^{∗})^{j_{d}} φ_{0}(A , B, ℏ, a, η, ·).

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

2. For each A, B, ℏ, a, and η, the set { φ_{j}(A, B, ℏ, 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 [H3] in the multi-dimensional case. If we set

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

then we have

( A_{m}(A, B, ℏ, a, η)^{∗} ψ) (x)

= − \sqrt{\frac{ℏ}{2}} \frac{1}{g(A, B, ℏ, a, x)} ∑_{n=1}^{d} \overline{A}_{n m} \frac{∂}{∂x_{n}} ( g(A, B, ℏ, a, x) ψ(x) ) . (2.3)

4. In [H2], the state φ_{j}(A, B, ℏ, a, η, x) is defined as a normalization factor times

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

Here H_{j}(A; y) is a recursively defined |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 scaling out the |A| and ℏ 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, ℏ, a, η, ·) are \sqrt{(j + 1/2)ℏ} |A| and \sqrt{(j + 1/2)ℏ} |B|, respectively. In higher dimensions, they are bounded by \sqrt{(|j| + d/2)ℏ} ∥A∥ and \sqrt{(|j| + d/2)ℏ} ∥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, ℏ, a, η, ·) is determined by continuity in t after an arbitrary initial choice.

8. The matrix elements of (x − a)^{m} satisfy

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

and

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

(2.5)

This definition is equivalent to the definition of the generalized squeezed states, obtained by the action of metaplectic operators M(F) and displacement operators D(α) on the eigenstates of the d-dimensional isotropic harmonic oscillator, as reviewed, for example, in [R, C]. The 2d×2d symplectic matrix F that parametrizes the metaplectic operators contains the same information as the matrices (A, B), whereas the vector α ∈ IR^{2d} at which the initial state is displaced is equivalent to the data (a, η). Modulo phases, the wave packet φ_{j}(A, B, ℏ, a, η, ·) corresponds to the generalized squeezed state

Φ_{j}(F, α, ·) = D(α)M(F)φ_{j}(I, I, ℏ, 0, 0, ·) (2.6)

We refer the reader to the reviews above for more details and further references.

3. Semiclassical Dynamics

We assume in this section that the potential V : IR^{d} → IR is smooth and bounded below. Without loss of generality, we may suppose that V ∈ C^{∞}(IR^{d}, IR^{+}).

Consider now the classical equations of motion

ȧ(t) = η(t),
η̇(t) = − ∇V (a(t)),
(3.1)  Ȧ(t) = i B(t),
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 (2.1)

The meaning of A(t) and B(t) in terms of linearized flow becomes clear once one realizes that the following identities hold:

A(t) = \frac{∂a(t)}{∂a(0)}A(0) + i\frac{∂a(t)}{∂η(0)}B(0)
(3.2)  B(t) = \frac{∂η(t)}{∂η(0)}B(0) − i\frac{∂η(t)}{∂a(0)}A(0)

Because V is smooth and bounded below, there exist global solutions to the first two equations of the system (3.1) for any initial condition. It then follows immediately that the remaining three equations of the system (3.1) have global solutions. Furthermore, it is not difficult ([H1], [H2]) to prove that conditions (2.1) are preserved by the flow.

We now describe the construction of approximate solutions ψ_{l}(x, t, ℏ) with errors whose norms are of order ℏ^{l/2}.

First, we decompose the potential as

V (x) = W_{a}(x) + Z_{a}(x) ≡ W_{a}(x) + (V (x) − W_{a}(x)), (3.3)

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. (3.4)

We observe ([H3]) that when A(t), B(t), a(t), η(t), and S(t) satisfy (3.1), we obtain

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

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

for all multi-indices j. In other words, our semiclassical wave packets φ_{j} exactly take into account the kinetic energy and quadratic part W_{a(t)}(x) of the potential when propagated by means of the classical flow and its linearization around the classical trajectory selected by the initial conditions.

Next, we expand the exact solution as

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

where the b_{j}(ℏ, t) satisfy an infinite linear system of ordinary differential equations

i ℏ ḃ_{k}(t, ℏ) = ∑_{j} K_{k j}^{∞}(t, ℏ) b_{j}(t, ℏ), (3.6)

whose matrix K^{∞}(t, ℏ) is obtained from the Z_{a(t)}(x) term in the hamiltonian:

K_{k j}^{∞}(t, ℏ) =

⟨ φ_{k}(A(t), B(t), ℏ, a(t), η(t), x), Z_{a(t)}(x) φ_{j}(A(t), B(t), ℏ, a(t), η(t), x) ⟩. (3.7)

In this system, we make a first approximation by replacing the function Z_{a(t)}(x) by its Taylor approximation Z_{a(t)}^{[l]}(x) of order l + 1 (reverting to multi-index notation),

Z_{a}^{[l]}(x) = ∑_{3≤|m|≤l+1} \frac{(D^{m}V)(a)}{m!} (x − a)^{m}. (3.8)

This yields an infinite linear system whose matrix is K̃(l, t, ℏ), with entries

K̃_{k j}(l, t, ℏ) =

⟨ φ_{k}(A(t), B(t), ℏ, a(t), η(t), x), Z_{a(t)}^{[l]}(x) φ_{j}(A(t), B(t), ℏ, a(t), η(t), x) ⟩, (3.9)

given by time dependent polynomials in ℏ^{1/2} of order l + 1.

We make a second approximation by truncating the infinite system to a system of finitely many differential equations. We define J̃(l) = J + 3l − 3 and construct the generator K(l, t, ℏ) for this system from K̃(l, t, ℏ) by restricting the indices to |j| ≤ J̃(l) and |k| ≤ J̃(l). Our approximation is then

ψ_{l}(x, t, ℏ) = e^{iS(t)/ℏ} ∑_{|j|≤J̃(l)} c_{j}(l, t, ℏ) φ_{j}(A(t), B(t), ℏ, a(t), η(t), x), (3.10)

where A(t), B(t), a(t), η(t), and S(t) satisfy (3.1), the coefficients c_{j}(l, t, ℏ) satisfy the self-adjoint linear system

i ℏ ċ_{k}(l, t, ℏ) = ∑_{|j|≤J̃(l)} K_{k j}(l, t, ℏ) c_{j}(l, t, ℏ), |k| = 0, 1, …, J̃(l), (3.11)

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

Theorem 3.1. [H1, H2, H3] Let V ∈ C^{∞}(IR^{d}, IR^{+}) and let Ψ(x, t, ℏ) satisfy (1.1) and (1.3). Then ψ_{l}(x, t, ℏ) defined in (3.10) satisfies

sup_{t∈[−T, T]} ∥ψ_{l}(x, t, ℏ) − Ψ(x, t, ℏ)∥_{L^{2}(IR^{d})} ≤ C(l) ℏ^{l/2} (3.12)

for some constant C(l).

The proof of the statement follows from an application of the next abstract lemma, whose proof is an easy application of Duhamel’s formula (see e.g. [H3]), and by keeping track of the powers in ℏ^{1/2}.

Lemma 3.2. Suppose H(ℏ) is a family of self-adjoint operators for ℏ > 0. Suppose ψ(t, ℏ) belongs to the domain of H(ℏ), is continuously differentiable in t, and approximately solves the Schrödinger equation in the sense that

i ℏ \frac{∂ψ}{∂t} (t, ℏ) = H(ℏ) ψ(t, ℏ) + ξ(t, ℏ), (3.13)

where ξ(t, ℏ) satisfies

∥ ξ(t, ℏ) ∥ ≤ μ(t, ℏ). (3.14)

Then, for t > 0,

∥ e^{−itH(ℏ)/ℏ} ψ(0, ℏ) − ψ(t, ℏ) ∥ ≤ ℏ^{−1} ∫_{0}^{t} μ(s, ℏ) ds. (3.15)

The analogous statement holds for t < 0.

The semiclassical propagation of gaussian wave packets has been considered in many papers in both the physics and mathematics literature, with various degrees of generality and rigor; see the recent reviews [R, C]. A result generalizing theorem 3.1 was proven by Combescure and Robert in [CR]. They proved estimates on the propagation of coherent states in the semiclassical regime to any order in ℏ^{1/2} for a wider class of hamiltonians together with an efficient control of the remainder terms for large times. More precisely, they considered a hamiltonian Ĥ(t, ℏ) given by the Weyl quantization of a sufficiently regular classical time dependent hamiltonian h(x, p, t), using the squeezed states mentioned earlier as a basis of reference. They used the time dependent states Φ_{j}(F (t), α(t), x), where α(t) gave the classical trajectory, and F (t) described the linearized flow around α(t), to solve the time–dependent Schrödinger equation with the hamiltonian replaced by its quadratic approximation around α(t). Denoting the corresponding evolution operator by U(t, s) and considering a normalized initial condition of the form Ψ(x, 0, ℏ) = ∑_{|j|≤J} d_{j}Φ_{j}(F (0), α(0), x), they proved [CR] that for any l ∈ IN, there exist scalars d_{j}(l, t, ℏ), j ∈ IN^{d}, depending only on the classical dynamics, and a constant D_{l}(t) such that

∥U(t, 0)Ψ(x, 0, ℏ) − ∑_{|j|≤J̃(l)} d_{j}(l, t, ℏ)Φ_{j}(F (t), α(t), x)∥_{L^{2}(IR^{d})} ≤ D_{l}(t) ℏ^{l/2}. (3.16)

Moreover, they obtained explicit control of the time-dependence of D_{l}(t) in terms of properties of the classical flow, to which we return in section 5.

4. Exponential Asymptotics

As described in the introduction, we can construct a more accurate approximate solution ψ(x, t, ℏ) = ψ_{l(ℏ)}(x, t, ℏ) by keeping track of the l–dependence of C(l) in (3.12) and then choosing l(ℏ) in order to minimize the error. This requires careful estimation of C(l). Because of (3.8), this depends on the growth of the mth derivative of the potential as a function of m. Our main result is that the required estimate on C(l) can be obtained for a large class of analytic potentials. For t in a fixed compact interval, the precise statement is as follows:

Theorem 4.1. 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 }.

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

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

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 (2.1) and ∑_{|j|≤J} |c_{j}|^{2} = 1. Then for any T > 0, there exist C, g and γ > 0, such that the difference between the semiclassical approximation ψ(x, t, ℏ) = ψ_{l(ℏ)}(x, t, ℏ) with l(ℏ) = [[g/ℏ]] and the exact solution Ψ(x, t, ℏ) to the Schrödinger equation (1.1) with initial condition (1.3) satisfies

sup_{|t|≤T} ∥ ψ(x, t, ℏ) − Ψ(x, t, ℏ) ∥_{L^{2}(IR^{d})} ≤ C exp { − γ/ℏ } .

(Here, and henceforth, [[α]] denotes the greatest integer less than or equal to α.)

Remarks: 1. Theorem 4.1 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 |D^{m}V (x, t))| ≤ M exp(τ|x|^{2}) m!/δ^{|m|}, for |t| ≤ T for all multiindices 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 4.1 holds. Note that the above condition essentially means that estimates provided by the use Cauchy formula hold uniformly in time. The result can probably be extended in the spirit of [CR] to accomodate time dependent hamiltonians obtained by Weyl quantization of a suitable real valued classical hamiltonian possessing an analytic extension in a neighbourhood of the real phase space.

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

3. Theorem 4.1 is optimal in the sense that the conclusion fails if the hypotheses are relaxed slightly. Indeed, consider the one-dimensional potential V (x) = exp(−1/x^{u}), for x > 0, and V (x) = 0 for x ≤ 0, where u > 0. We know from [J] 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, ℏ, 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 if t = ℏ^{δ},

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

The lower bound comes from considering small times O(ℏ^{δ}), so that the asymptotics of the evolution operator e^{−itH(ℏ)/ℏ} can be obtained easily. Note also that if we choose a(0) = −a < 0 and η(0) = η > 0, then it is easy to check that the error term is O(exp(−γ/ℏ)), for each t < a/η, where a/η is the classical time needed to reach the origin where the potential is non-trivial.

4. For all practical purposes, we can replace the c_{j}’s by the Dyson expansion up to order l(ℏ) corresponding to equation (3.11) 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. Indeed, such wave packets cannot describe tunneling, which is an exponentially small effect in ℏ. This further yields limitations on the best exponential decay rate we can obtain in case of trapped trajectories.

6. The approximation ψ(x, t, ℏ) is sharply peaked on the classical trajectory as the next estimate confirms. For any ϵ > 0, there exist sufficiently small values of the parameter g > 0, such that the wave packet ψ(x, t, ℏ) is localized near a(t) within a distance ϵ up to an error exp { −Γ_{g}/ℏ }, with Γ_{g} > 0, in the sense that

( ∫_{|x−a(t)|>ϵ} |ψ(x, t, ℏ)|^{2} dx )^{1/2} ≤ exp { −Γ_{g}/ℏ } .

Results on exponential accuracy of semiclassical constructions in a dynamical setting can be found in the work of Yajima [Y]. These results concern the propagation of wave packets of the form φ(x) = e^{iS(x)/ℏ}f(x), where S is analytic and f belongs to the set of compactly supported Gevrey functions of order s > 1. Yajima assumes the potential V is analytic and 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/ℏ^{1/(2s−1)}}, with γ > 0, for |t| < T, where T is small enough. His result also holds in the presence of an analytic vector potential. The main difference between Yajima’s approach and ours lies in the initial conditions considered. In [Y], all points on the Lagrangian manifold Λ(S, supp f) = {(x, ∇S(x)), x ∈ supp f} have a weight of order 1 in the limit ℏ → 0, whereas our coherent state (1.3) selects one point of phase space only, (a(0), η(0)). Yajima also imposes the condition that |t| be small. His results are obtained by means of the pseudo-differential techniques developed in the analytic context by Sjöstrand in [S]. It should be possible to use this theory to recover our results.

Before presenting an outline of the proof of theorem 4.1, we turn to the long time version of our result.

5. The Ehrenfest Time Scale

Theorem 4.1 can also be generalized to allow time intervals that grow like | log(ℏ)| as ℏ tends to zero, at the cost of a somewhat weaker conclusion. Our precise results are summarized by the following theorem.

Theorem 5.1. Suppose V is bounded below and analytic in

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

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

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

Suppose initial conditions A(0), B(0), a(0), η(0), S(0) = 0, and c_{j} for |j| ≤ J are specified that satisfy (2.1) and ∑_{|j|≤J} |c_{j}|^{2} = 1, and further assume there exist N > 0 and λ > 0, such that

∥A(t)∥ ≤ N exp(λ|t|). (5.1)

Then for sufficiently small T′ > 0, there exist C′, g′, γ′, σ′, and σ > 0, such that the difference between the semiclassical approximation ψ(x, t, ℏ) = ψ_{l(ℏ)}(x, t, ℏ) with l(ℏ) = [[g′/ℏ^{σ′}]] and the exact solution Ψ(x, t, ℏ) to the Schrödinger equation (1.1) with initial condition (1.3) satisfies

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

whenever |t| ≤ T′ | log(ℏ)|.

Remarks: 1. Condition (5.1) is satisfied if the norm of the Hessian V^{(2)}(a(t)) is uniformly bounded, as is ensured by standard existence and uniqueness theorems for systems of ODE’s. That is the case if V is the sum of a quadratic polynomial plus an analytic function that is 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.

2. We can also deduce 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 allows us to express the maximal time validity of the Theorem in terms of the Lyapunov exponent of the classical trajectory under consideration.

Let us compare this to the long time estimates of Combescure and Robert in [CR]. A corollary of their main result is that the semiclassical approximation is valid for times as large as \frac{1}{6λ}| ln(ℏ)| for polynomially growing potential in the sense that the error term goes to zero. The above remark shows that with our construction, we get an exponential control of the approximation over the same time intervals as in [CR].

However, as the following heuristic argument shows, one expects that the actual critical time should rather be \frac{1}{2λ}. The width of our gaussian states is of order \sqrt{ℏ}∥A(t)∥, and they remain concentrated near the classical trajectory only if \sqrt{ℏ}∥A(t)∥ ≃ \sqrt{ℏ} e^{λt} remains small, i.e., up to times t ≃ | ln(ℏ)|/2λ, for small ℏ. For later times, one expects interferences between several pieces of the wave function to play a important role, especially in cases where the underlying classical dynamics displays strong chaotic properties. To our knowledge, the only rigorous results showing that this picture is correct, at least in some cases, can be found in the recent work of De Bièvre and Bonechi [DBB] where they consider the long time evolution of wave packets on a torus driven by quantized versions of the Baker and Cat maps.

Similar issues have been dealt with by Bambusi, Graffi and Paul in [BGP] who consider the approximation of the Heisenberg evolution of observables in the semiclassical limit for times up to the Ehrenfest time scale. They consider the Weyl quantization Ĥ and B̂ of a classical hamiltonian h and of a bounded observable b, assumed to possess analytic extensions in a neighbourhood of the real phase space. Under the hypotheseses that h grows at most quadratically and b decays exponentially rapidly at infinity in phase space, the authors construct an approximation B^{N}(t) to the observable at time t, B̂(t) = exp(itĤ/ℏ)B̂ exp(−itĤ/ℏ), that is determined by the classical flow and has the following properties. First, for any N, there is a T_{N}(ℏ) = O(ln(ℏ)) such that ∥B^{N}(t) − B̂(t)∥ = O(ℏ^{N−4/9} ln(ℏ)) for all times 0 ≤ t ≤ T_{N}(ℏ). This estimate for observables corresponds essentially to the result (3.16) of Combescure and Robert for coherent states. Second, if N = N(ℏ) is chosen properly, ∥B^{N(ℏ)}(t)−B̂(t)∥ = O(ℏ^{∞}) for all times 0 ≤ t ≤ T (ℏ), where T (ℏ) is any increasing function, such that T (ℏ)/| ln(ℏ)| → 0 as ℏ → 0. This estimate for observables is very close to Theorem 5.1, which holds for coherent states only. Note, however, that the hamiltonians they can accomodate consist essentially of analytic perturbations of the harmonic oscillator that decay as x and p tend to infinity.

6. Sketch of the Proof

In this section, we present the main steps of the proof of Theorem 4.1.

We define ψ_{l}(x, t, ℏ) by (1.2), where the c_{j}(l, t, ℏ) are determined by the system (3.11) and initial conditions described above. To apply Lemma 3.2 we define

ξ_{l}(x, t, ℏ) = i ℏ \frac{∂}{∂t}ψ_{l}(x, t, ℏ) − ( − \frac{ℏ^{2}}{2} Δ + V (x) ) ψ_{l}(x, t, ℏ). (6.1)

By using (3.5), we see that this can be decomposed as a sum of two terms,

ξ_{l}^{(1)}(x, t, ℏ) = ( Z_{a(t)}^{[l]}(x) − Z_{a(t)}(x) ) ψ_{l}(x, t, ℏ) (6.2)

and

ξ_{l}^{(2)}(x, t, ℏ) = − P_{{|j|>J̃(l)}} Z_{a(t)}^{[l]}(x) ψ_{l}(x, t, ℏ), (6.3)

where P_{{|j|>J̃(l)}} is the orthogonal projection onto the span of the set

{ φ_{j}(A(t), B(t), ℏ, a(t), η(t), ·) : |j| > J̃(l) }.

Let us briefly comment on the first error term. 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,

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

At this point we write \frac{1}{m!} D^{m}V (ζ(x, a)) as a d–dimensional Cauchy integral and use the bound on the growth of V at infinity to obtain a bound of the type

\frac{1}{m!} |D^{m}V (ζ(x, a))| ≤ \frac{M_{0}}{δ^{|m|}} exp(2τ(x − a)^{2}). (6.4)

Then, using (1.2) with |c_{j}(l, t, ℏ)| ≤ 1, we need to estimate integrals of the form

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

with |m| = l +2, for all |j| ≤ J̃(l) ≃ 1/ℏ. We can obtain the necessary estimates for sufficiently small ℏ since we have detailed information about the basic wave packets φ_{j}. The ultimate result is that there exist constants C_{4} and C_{5}, such that |t| ≤ T implies

∥ ξ_{l}^{(1)}(·, t, ℏ) ∥ ≤ C_{4} ( C_{5} \sqrt{ℏ(l + 2)} )^{l+2} . (6.5)

The quantity ξ_{l}^{(2)}(·, t, ℏ) in (6.3) satisfies

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

and

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

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

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

where we denote the c_{j}(l, t, ℏ) collectively by the vector c(l, t, ℏ). We easily verify these facts by using (3.11), (3.9), (3.5) and (2.5). To estimate the norm of ξ_{l}^{(2)}(·, t, ℏ), we use the Dyson expansion with remainder to decompose

c(l, t, ℏ) = ∑_{q=0}^{l} c^{q}(l, t, ℏ) + r(l, t, ℏ), (6.6)

where (dropping some arguments)

c^{q}(t) = ∫_{0}^{t} ds_{1} ∫_{0}^{s_{1}} ds_{2} ⋯ ∫_{0}^{s_{q−1}} ds_{q}(iℏ)^{−q}K(s_{1})K(s_{2}) ⋯ K(s_{q}) c(0), (6.7)

and

r(t) = ∫_{0}^{t} ds_{1} ∫_{0}^{s_{1}} ds_{2} ⋯ ∫_{0}^{s_{l}} ds_{l+1}(iℏ)^{−(l+1)}K(s_{1})K(s_{2}) ⋯ K(s_{l+1}) c(s_{l+1}). (6.8)

To estimate the norm of ξ_{l}^{(2)}(·, t, ℏ) we study the j^{th} component of K̃(t)c(l, t, ℏ), with |j| > J̃(l). Because of (6.6), this coefficient is a sum of two types of terms: those that arise from c^{q}(l, t, ℏ) and those that arise from r(l, t, ℏ). We consider only the former contributions, the latter ones are dealt with in a similar fashion.

We define the infinite matrix

X̃_{j k}^{m}(t, ℏ) =

⟨ φ_{j}(A(t), B(t), ℏ, a(t), η(t), ·), (x − a(t))^{m} φ_{k}(A(t), B(t), ℏ, a(t), η(t), ·) ⟩, (6.9)

and then construct the finite matrix X^{m}(l, t, ℏ) from X̃^{m}(t, ℏ) by restricting its indices to |j| ≤ J̃(l) and |k| ≤ J̃(l).

We then have X_{j k}^{m}(l, s, ℏ) = 0 and X̃_{j k}^{m}(s, ℏ) = 0 if ||j| − |k|| > |m|. Further-

more, it follows from property 8 that there exists D < ∞, such that

sup_{|k| ≤J̃(l), J̃(l)+1≤ |j| ≤J̃(l)+l+1} | (X̃^{m_{0}}(t_{0}, ℏ)X^{m_{1}}(l, t_{1}, ℏ)

× X^{m_{2}}(l, t_{2}, ℏ) ⋯ X^{m_{q}}(l, t_{q}, ℏ))_{j k} | (6.10)

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

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

Using (3.9), (3.8), and (6.9), we note that each c^{q}(l, t, ℏ) is of order ℏ^{q/2} and that r(l, t, ℏ) is of order ℏ^{(l+1)/2}.

We now expand K(l, t, ℏ) in (6.7) and (6.8) and K̃(l, t, ℏ) in terms of X^{m}(l, t, ℏ) and X̃^{m}(t, ℏ), to obtain

K̃(t)c^{q}(t) = (iℏ)^{−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), (6.11)

and a similar term that arises from K̃(t) r(t).

Note that the values of m_{j} that occur, satisfy |m_{j}|/J̃(l) ≤ (l + 1)/(3l − 3) ≤ 1, as long as l ≥ 3.

Moreover, as c_{k}(0) is non-zero only for |k| ≤ J, we need only consider the j^{th} coefficient of K̃(t)c(t) for |j| > J̃(l), due to the properties of X^{m} and X̃^{m}. So, the only relevant values of n in (6.11) must satisfy n ≥ 3l − 2. This condition is also satisfied for all values of n in the term that arises from K̃(t) r(t), since the corresponding sum begins with n = 3l + 6.

The analyticity assumption implies the existence of a v(t) > 0 such that

|D^{m}V (a(t))| ≤ v(t)m! / δ^{|m|}, (6.12)

where δ is the width of the analyticity strip. Furthermore, since ∥A(t)∥ depends continuously on t, there exists w(T), such that sup_{t∈[−T, T]} ∥A(t)∥ ≤ w(T). Moreover, the norm of the vector c(t) is 1 since K(l, t, ℏ) 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, for the term stemming from K̃(t) r, c(t) has at most (J̃(l) + 1)^{d} non-zero entries, each of which is bounded by 1.

Finally, we introduce

F_{p}(n, q) = ∑_{\begin{matrix}1 ≤ |m_{1}|, |m_{2}|, ⋯, |m_{q}| ≤ p \\ |m_{1}| + |m_{2}| + ⋯ + |m_{q}| = n \end{matrix}} 1, (6.13)

which is 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.

For l ≥ 3, (3.9), (3.8), (6.9), (6.11), and (6.10) imply the following estimate when j satisfies J̃(l) + 1 ≤ |j| ≤ J̃(l) + l + 1:

| (K̃(t) c^{q}(t))_{j} |

≤ \frac{v(t) ℏ(q + 1) (J + 1)^{d}}{∫_{0}^{T} v(s) ds} \frac{(∫_{0}^{T} v(s) ds)^{q+1}}{ℏ^{q+1} (q + 1)!}

× ∑_{n=3l−2}^{(l+1)(q+1)} ( Dw(T)\sqrt{ℏ J̃(l)}/δ )^{n} F_{l+1}(n, q + 1), (6.14)

and a similar estimate for the term arising from (K̃(t) r(t))_{j}, with q replaced by l + 1 and J by J̃(l).

The theorem is essentially proved if we can estimate quantities of the type

L_{q}(l) = \frac{C_{1}^{q}}{ℏ^{q} q!} ∑_{n=3l−2}^{q(l+1)} (C_{2}ℏl)^{n/2} F_{l+1}(n, q), (6.15)

for some constants C_{1} and C_{2}, uniformly for 2 ≤ q ≤ l + 2, and optimize the result over choices of l. The following lemma tells us that this is possible.

Lemma 6.1. 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 ℏ^{∗}, that depend only on g, C_{1}, and C_{2}, such that

0 < ℏ < ℏ^{∗}, l(ℏ) = [[ g/ℏ]], and 2 ≤ q ≤ l(ℏ) + 2 (6.16)

imply

L_{q}(l(ℏ)) ≤ C_{3} exp { − γ_{1}/ℏ } . (6.17)

Remark: The key step in the proof of this Lemma is the estimate F_{p}(n, q) ≤ C_{7}^{qd+n} which holds for all p ≥ 1, q ≥ 1 and n ≥ 1 with some constant C_{7}, uniform in the dimension d.

By Lemma 6.1, both | (K̃(t) c^{q}(t))_{j} | and | (K̃(t)r(t))_{j} | are bounded by

C_{3} exp { −γ_{1}/ℏ } for an appropriate choice of l(ℏ) = [[ g/ℏ]] and sufficiently small ℏ. Each of the l + 2 terms in (6.6) contributes a term of this type to ξ_{l}^{(2)}(x, t, ℏ), so

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

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

We shrink g if necessary to make C_{5}^{2} g < 1 and set l = l(ℏ) in (6.5). This yields a similar estimate

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

for some γ_{3} > 0.

We combine these two estimates and apply Lemma 3.2 to obtain (1.5) with γ = min {γ_{2}, γ_{3}}. This proves the theorem.

To prove Theorem 5.1, we simply reexamine the proof of Theorem 4.1 and make the T dependence of all of its constants explicit. We then allow T to grow with ℏ with the restriction that our approximation must remain close to the actual solution as ℏ → 0.

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[Y] Yajima, K., Gevrey Frequency Set and Semi-classical Behavior of Wave Packets, in Schrödinger Operators, The Quantum Mechanical Many Body Problem, Lecture Notes in Physics, 403, ed. by E. Balslev. Berlin, Heidelberg, New York: Springer-Verlag, 1992, pp 248–264.

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

 E-mail address: hagedorn@math.vt.edu

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

 E-mail address: alain.joye@ujf-grenoble.fr