AN EXISTENCE FAMILY FOR HUSIMI OPERATOR

H. EMAMIRAD and Ph. ROGEON

Abstract. We prove the well-posedness of the quantum Liouville equation in L^{1}(R^{2n}), provided that the potential of the Schrödinger equation lies in some H^{s}(R^{n}). By regularizing the Wigner function we obtain the Husimi equation which is ill-posed in any L^{p}(R^{2n}) spaces. We show that for the Husimi operator the maximal and minimal extensions coincide and we construct a C-existence family in the sense of R. deLaubenfels which is a new tool for studing this operator.

1. Introduction

Suppose that in the theory of quantum mechanics the potential V is chosen in such a way that the hamiltonian

H = \frac{−ℏ^{2}}{2} Δ + V  (1.1)

(ℏ being the Planck’s constant which is equal 1 in the sequel) is self-adjoint in the Hilbert space H = L^{2}(ℝ^{n}). Let φ be the solution of the corresponding Schrödinger equation, i.e.

i \frac{∂φ}{∂t} = Hφ,  (1.2)

and φ^{∗} its complex conjugate. The correspondent density matrix ρ(x, y) = φ(x) φ^{∗}(y) defines a positive self-adjoint trace class operator, which we designate it again by ρ:

ρf = ∫_{ℝ^{n}} ρ(x, y)f(y)dy,  f ∈ H.  (1.3)

1991 Mathematics Subject Classification. Primary 82A70,81C10; Secondary 47D05. Key words and phrases. Classical Liouville equation, Wigner transform and equation, Husimi

function and equation, C-existence family.

Typeset by AMS-TEX

Furthermore , if T (H) is the Banach space of the trace class operators, endowed with the norm

∥T ∥_{tr} = ∑_{i∈I} | < ψ_{i}, T ψ_{i} > |, T ∈ T (H),

where {ψ_{i}; i ∈ I} is an ONB of H, then

∥ρ∥_{tr} = ∫_{ℝ^{n}} |φ(x)|^{2} dx.  (1.4)

It is well-known that ρ governs the classical Liouville equation

i ρ̇ = [H, ρ],  ρ(0) = ρ_{0}  (1.5)

on which the quantum theory of open system is based (see [A-L] and [Da] ). The solution of (1.5) can be expressed by so called quantum dynamical semigroup S(t)

ρ(t) = S(t)ρ_{0} = e^{−itH}ρ_{0} e^{itH}  (1.6)

which is a unitary group on T (H).

There is also another link between quantum theory of the open systems and the theory of statistical mechanics. Let us consider the Wigner equation (also called quantum Liouville equation)

\frac{∂w}{∂t} + ξ · ∇_{x}w − P (x, ∇_{ξ})w = 0,  (1.7)

where P ≡ P (x, ∇_{ξ}) is a pseudo-differential operator, which is symbolically

i[V (x + \frac{i}{2}∇_{ξ}) − V (x − \frac{i}{2}∇_{ξ})],

or equivalently

P w = (2π)^{−n} ∫_{ℝ^{2n}} e^{−i(ξ−η)y} \frac{1}{i} [V (x + \frac{y}{2} ) − V (x − \frac{y}{2} )] w(x, η)dydη
= k ∗_{ξ} w,  (1.8)

where k(x, ξ) = (2π)^{−n/2}F_{y}[\frac{1}{i} (V (x + \frac{y}{2} ) − V (x − \frac{y}{2} )) ](ξ). This equation follows from (1.2) or (1.5) via the Wigner transform [Wi]. In fact, a solution of (1.7) can be expressed by

W (x, ξ, t) = (2π)^{−n} ∫_{ℝ^{n}} e^{−iξy}ρ(x + \frac{y}{2} , x − \frac{y}{2} , t) dy.  (1.9)

If ρ is the density matrix of φ, then we call W the Wigner transform of φ and we write: W = W (φ). This function provides a phase-space ’view’ of the quantum state. e.g. if the potential V ≡ 0, then W satisfies the advection equation

\frac{∂w}{∂t} + ξ · ∇_{x}w = 0  (1.10)

( is called also collisionless transport equation in the statistical mechanics). For any g ∈ L^{2}(ℝ^{n} × ℝ^{n}), let us define g̃(x, y) = g(r, s), where

r = x + \frac{y}{2} , s = x − \frac{y}{2} .

Hence, the Wigner function can be expressed as

W (x, ξ, t) = (2π)^{− \frac{n}{2}} [F_{y}ρ̃(x, y)](ξ)  (1.11)

where

[F_{y}f](ξ) = f̂(ξ) = (2π)^{− \frac{n}{2}} ∫_{ℝ^{n}} e^{−iξ·y}f(y)dy.

Since ρ is in particular a Hilbert-Schmidt operator, its kernel ρ(·, ·) belongs to L^{2}(ℝ^{n} × ℝ^{n}) and so does W. Although the reality of W can be deduced from the selfadjointness of ρ, it is necessarily negative in some region of the phase-space (see [H-al]). These properties incited many authors ([M-R], [Ma1], [Ma2] and their references) to make their analysis in L^{2}(ℝ^{n} × ℝ^{n}) space. Nevertheless, in statistical mechanics the unknown of the problems is always the density of particles ( probability or mass ) and the physically right space invokes L_{+}^{1}(ℝ^{n} × ℝ^{n}), the positive cone of the Banach lattice L^{1}(ℝ^{n} × ℝ^{n}).

Related to this , Husimi [Hu] showed that a Gaussian smoothing of W is indeed positive and hence suitable for a probability density. In fact, let

H(x, ξ, t) ≡ [W ∗ G](x, ξ, t) = ∫_{ℝ^{2n}} W (y, η, t)G(x − y, ξ − η) dydη,  (1.12)

where G(x, ξ) = (π)^{−n} exp(−(|x|^{2} + |ξ|^{2})). For (x_{0}, ξ_{0}) ∈ ℝ^{2n}, take

ψ_{x_{0},ξ_{0}}(x) = (π)^{−n/4}e^{− \frac{|x−x_{0}|^{2}}{2}} e^{iξ_{0}x},  (1.13)

then

G(x − x_{0}, ξ − ξ_{0}) = W (ψ_{x_{0},ξ_{0}})(x, ξ) = W (ψ_{x,ξ})(x_{0}, ξ_{0}).  (1.14)

Hence

H(x, ξ) = ∫_{ℝ^{2n}} W (φ)(x_{0}, ξ_{0})W (ψ_{x,ξ,})(x_{0}, ξ_{0})dx_{0}dξ_{0}
= (\frac{1}{4π} )^{n} | ∫_{ℝ^{n}} φ(r)ψ_{x,ξ}^{∗}(r)dr|^{2} ≥ 0.  (1.15)

All these matters including many other applications of the Husimi function can be found in [L-P]. In particular, it is proved in [E-H] that the range of the Husimi transformation of L^{2}(ℝ^{2n}) is exactly the space of the coherent state vectors of a first order pseudo–differential operator in L^{2}(ℝ^{2n}) in the sense which is defined in [E-H].

Apart from the mentioned properties of the Husimi function, in the theory of wavelets the transformation

G_{x}(φ)(ξ) = ∫_{ℝ^{n}} φ(r)ψ_{x,ξ}^{∗}(r) dr

which appears in (1.15), is called Gabor transform, and its connection to uncertainty principle is reviewed in [Ch].

In spite of the fact that the pseudo-differential operator P (x, ∇_{ξ}) commutes with the convolution, the Husimi function does not satisfy (1.7), but it does a perturbed version of it, namely

\frac{∂h}{∂t} + ξ · ∇_{x}h − P (x, ∇_{ξ})h + \frac{1}{2}∇_{x} · ∇_{ξ}h = 0.  (1.16)

In fact, if we denote by T_{0} ≡ −ξ∇_{x} and T ≡ −ξ∇_{x} − \frac{1}{2}∇_{x} · ∇_{ξ}, one can verify readly that

CT_{0}f = T_{0}Cf − \frac{1}{2}∇_{x} · ∇_{ξ}Cf = T Cf ∀f ∈ D(T_{0}),  (1.17)

where C is a bounded operator on L^{1}(ℝ^{2n}) defined by

Cf ≡ G ∗ f  ∀f ∈ L^{1}(ℝ^{2n}).  (1.18)

Consequently by acting C on (1.7), one obtains

\frac{∂h}{∂t} = C \frac{∂}{∂t}w = C(T_{0} + P)w = T Cw + P Cw
= (T + P)h

which is exactly (1.16).

The equation (1.16) is called Husimi equation , and it can be considered once more within the context of the statistical mechanics.

The fact that ∥G∥_{L^{1}(ℝ^{2n})} = 1, and

∫_{ℝ^{n}} W (x, ξ)dξ = |φ(x)|^{2}

imply that

∫_{ℝ^{2n}} H(x, ξ)dx dξ = ∫_{ℝ^{n}} |φ(x)|^{2}dx.

This together with (1.4) give us the invariants of three equations (1.2), (1.5) and (1.16), which are

∥φ(t)∥_{H} = ∥ρ(t)∥_{tr} = ∥H(t)∥_{L^{1}(ℝ^{2n})} = c_{0}.  (1.19)

for any t ∈ ℝ, where c_{0} = ∥φ_{0}∥_{H} = ∥ρ_{0}∥_{tr} = ∥H(φ_{0})(0)∥_{L^{1}(ℝ^{2n})}.

As the Stone’s theorem justifies the mathematical well-founded of the space L^{2}(ℝ^{n}) for the Schrödinger equation, the identities (1.19) ensure also the adequacy of T (H) the space of the trace class operators for the classical Liouville equation and L^{1}(ℝ_{x}^{n} × ℝ_{ξ}^{n}) for the Husimi equation. This property incited us to make an analysis of the Husimi equation in L^{1}(ℝ^{2n}).

The well-posedness of classical Liouville equation in L^{1}(ℝ^{2n}) is treated in [E-P] by requiring a minimal ammont of Sobolev regularity on the potential V. In the next section we will show the well-posedness of the quantum Liouville equation (1.7) in L^{1}(ℝ^{2n}). We prove that the Wigner operator L ≡ T_{0} + P = −ξ · ∇_{x} + P (x, ∇_{ξ}) generates a C_{0}–group e^{tL} in L^{1}(ℝ_{x}^{n} × ℝ_{ξ}^{n}) and we give the exponential bound of this group with respect of the H^{s} norm of the potential V. For studing the Husimi equation in L^{1}(ℝ_{x}^{n} ×ℝ_{ξ}^{n}) we come up against the following difficulty that the Husimi equation (1.16) is not well-posed in any L^{p}(ℝ^{2n}). In fact, by the Fourier transform we can see that there is no real number in the resolvent set of the operator B ≡ −\frac{1}{2}∇_{x} · ∇_{ξ} and even if we pertube it by a conservative operator T_{0} ≡ −ξ · ∇_{x} or by a bounded operator P (x, ∇_{ξ}) we can not remedy the ill-posdness of the Husimi operator A ≡ T_{0} + P + B.

When an abstract Cauchy problem in a Banach space X

(CP)  \{ \begin{matrix}\frac{d}{dt}u(t, φ) = Au(t, φ)  t ≥ 0 \\ u(0, φ) = φ ∈ X, \end{matrix}

is not well-posed, at least in its original formulation, one of the useful concept for dealing with it is a C-existence family which is introduced in [dL1], and, for the exponentially bounded case, in [dL2]. Roughly speaking, sometimes when A is a closed operator in X and there is no mild solution for (CP), for any φ ∈ X we find the following alternative. There exists a bounded operator C on X, which plays the role of a regularizing operator and for any φ = Cψ ∈ Im(C), Y (t)ψ ≡ u(t, Cψ) defines a mild solution for (CP). In the section 4 we define the operator C by Cf ≡ G ∗ f for all f ∈ L^{1}(ℝ^{2n}) and we show that this concept fits perfectly to our problem and one can define with these materials a mild and strong solution of the Husimi equation.

Before introducing the notion of the C-existence family we should be certain that the operator A is closable and define an adequate close extension of A. We prove actually that the minimal and maximal extensions of A coincide. This will be the topic of the section 3.

2. Well-posedness of the Wigner equation in L^{1}(ℝ^{2n})

Now, we are going to analyse the well-posedness of the Wigner equation (1.7). With this end in view, we need to introduce the Sobolev space H^{s}(ℝ^{n}) = {f ∈ S′ | ∥f∥_{H^{s}} < ∞}, where

∥f∥_{H^{s}} = (∫_{ℝ^{n}}(1 + |ξ|^{2})^{s}|f̂(ξ)|^{2}dξ)^{\frac{1}{2}} .

We recall that for any f ∈ H^{s}(ℝ^{n}), and s > \frac{n}{2}

∥f̂∥_{1} ≤ c ∥f∥_{H^{s}},  (2.1)

where c = ∫_{ℝ^{n}}(1 + |ξ|^{2})^{−s}dξ.

Theorem 2.1. If for some s > \frac{n}{2}, the potential V lies in H^{s}(ℝ^{n}), then (a) the oprator P = P (x, ∇_{ξ}) is bounded in L^{1}(ℝ_{x}^{n} × ℝ_{ξ}^{n}) and we have

∥P (x, ∇_{ξ})w∥_{1} ≤ δ∥w∥_{1}  (2.2)

where δ = 2(2π)^{−n/2}C∥V ∥_{H^{s}};

(b) the Wigner operator L = T_{0} + P generates a C_{0}–group e^{tL} on L^{1}(ℝ_{x}^{n} × ℝ_{ξ}^{n}) which satisfies

∥e^{tL}f∥_{1} ≤ e^{δ |t|}∥f∥_{1},  ∀ t ∈ ℝ.  (2.3)

Proof. (a) The estimation (2.2) follows from

∥P (x, ∇_{ξ})w∥_{1} = ∥k ∗_{ξ} w∥_{1}

≤ sup_{x} ( ∫_{ℝ^{n}} |k(x, ξ)|dξ)∥w∥_{1}

≤ δ ∥w∥_{1}

for, k(x, ξ) = \frac{1}{i} (2π)^{−n/2} [2^{n}e^{2iξx}V̂(2ξ) − (−2)^{n}e^{−2iξx}V̂(2ξ)^{∗}] .

(b) Taking V ≡ 0 in (1.7), the advection equation (1.10) is thus governed by a unitary C_{0}–group e^{tT_{0}}f(x, ξ) = f(x − tξ, ξ). By virtue of the part (a) the operator P being bounded, (2.2) follows from perturbation theorem of the generator of a strongly continuous semigroup by a bounded operator (see [Paz]).

□

Corollary 2.2. If for some s > \frac{n}{2} the initial datum φ_{0} of the Schrödinger equation (1.2) is such that the function x → ρ_{0}(x, y) = φ_{0}(x+ \frac{y}{2} )φ_{0}^{∗}(x− \frac{y}{2} ) belongs to L^{1}(ℝ^{n}) and the functions y → ρ̃_{0}(x, y) and V both lie in H^{s}(ℝ^{n}), then the Wigner function W ∈ L^{1}(ℝ^{2n}).

Proof. Take t = 0 in (1.11) and

w_{0}(x, ξ) = W (x, ξ, 0) = (2π)^{− \frac{n}{2}} [F_{y}ρ̃_{0}(x, y)](ξ).  (2.4)

Since

∫∫_{ℝ^{2n}} |w_{0}(x, ξ)|dxdξ ≤ (2π)^{\frac{n}{2}} ∫_{ℝ^{n}} ∥ρ̃_{0}(x, ·)∥_{H^{s}}dx < ∞,

one can choose w_{0} as an initial datum of (1.7). So that Theorem 2.1 concludes, since e^{tL} produces a unique solution of (1.7) for the initial datum w_{0}. □

3. Extensions of the Husimi operator

In this section we will prove that the Husimi operator A is closable and the minimal and maximal closed extensions of A coincide. Let us define the regularized transport operator T on S(ℝ^{2n}) by

[T f](x, ξ) ≡ [T_{0}f + Bf](x, ξ) = −ξ · ∇_{x}f(x, ξ) − \frac{1}{2}∇_{x} · ∇_{ξ}f(x, ξ).

and the Husimi operator A = T + P.

Theorem 3.1. The operator A is closable in L^{1}(ℝ^{2n}).

Proof. Since A is a bounded perturbation of T, it suffices to show that T is closable in L^{1}(ℝ^{2n}). Let {φ_{k}}_{kℕ} be a sequence of functions in S(ℝ^{2n}), converging to 0 in L^{1}(ℝ^{2n}), such that T φ_{k} converges to a function f in L^{1}(ℝ^{2n}). If we denote by T^{∗} = ξ · ∇_{x} − \frac{1}{2}∇_{x} · ∇_{ξ} the formal adjoint of T and by < ·, · > the pairing between L^{1}(ℝ^{2n}) and L^{∞}(ℝ^{2n}), then we have < T φ_{k}, ψ >=< φ_{k}, T^{∗}ψ >, for all ψ ∈ S(ℝ^{2n}) and k ∈ ℕ. Letting k → ∞ gives us < f, ψ >=< 0, T^{∗}ψ >= 0, for all ψ ∈ S(ℝ^{2n}), so by the density of S(ℝ^{2n}) in L^{1}(ℝ^{2n}) we get f = 0, as desired. □

The operator A acting on S(ℝ^{2n}) has a closure in L^{1}(ℝ^{2n}). Let us denote by A_{m} the smallest closed extension of A in L^{1}(ℝ^{2n}), which is called the minimal extension of A and can be defined also in the following manner (see [Sch]): f ∈ L^{1}(ℝ^{2n}) is in D(A_{m}) and A_{m}f = g, if there exists a sequence {φ_{k}} of functions in S(ℝ^{2n}), such that φ_{k} → f and Aφ_{k} → g in L^{1}(ℝ^{2n}).

The operator A has another closed extension A_{M} on L^{1}(ℝ^{2n}) which is called the maximal extension of A and is defined as follows (see also [Sch]): we say that f ∈ D(A_{M}) and A_{M}f = g, if f and g belong to L^{1}(ℝ^{2n}) and < f, A^{∗}φ >=< g, φ > for all φ ∈ S(ℝ^{2n}). Clearly A_{M} is an extension of A_{m} (A_{m} ⊂ A_{M}).

Theorem 3.2. The minimal and the maximal extensions of the Husimi operator A coincide.

Proof. Since A is a bounded perturbation of T, it suffices to show that D(T_{M}) ⊆ D(T_{m}) (see [Sch]).

For h ∈ D(T_{M}), there exists f ∈ L^{1}(ℝ^{2n}) such that < h, T^{∗}φ >=< f, φ >, for all φ ∈ S(ℝ^{2n}) and we have T_{M}h = f. For having h ∈ D(T_{m}) and T_{m}h = f we have juste to find a sequence {φ_{k}}_{k∈ℕ} in S(ℝ^{2n}) such that φ_{k} → h and T φ_{k} → f in L^{1}(ℝ^{2n}). For

G_{k}(x, ξ) = (\frac{k^{2}}{π} )^{n} exp (−k^{2}(|x|^{2} + |ξ|^{2})) ,

let us define the operator C_{k} ∈ L(L^{1}(ℝ^{2n})) by C_{k}f ≡ G_{k} ∗ f =< f, ψ_{k} >, for all f ∈ L^{1}(ℝ^{2n}) and φ_{k} = C_{k}h. Clearly φ_{k} → h in L^{1}(ℝ^{2n}). As in (1.17) a simple calculation shows that

C_{k}T_{0}g = T_{0}C_{k}g − \frac{1}{2k^{2}} ∇_{x} · ∇_{ξ}C_{k}g ∀g ∈ D(T_{0}),

Since C_{k} commutes with ∇_{x} · ∇_{ξ}, thus

C_{k}f = T C_{k}h − \frac{1}{2k^{2}} ∇_{x} · ∇_{ξ}C_{k}h

So, for proving that T φ_{k} = T C_{k}h → f, it suffices to show that

lim_{k→∞} ∥\frac{1}{2k^{2}} ∇_{x} · ∇_{ξ}C_{k}h∥_{1} = 0  (3.1)

First we show this property for a smooth function h_{0} ∈ C_{0}^{∞}(ℝ^{2n}). This comes from

∥\frac{1}{2k^{2}} ∇_{x} · ∇_{ξ}C_{k}h_{0}∥_{1} = ∥\frac{1}{2k^{2}} [∇_{x} · ∇_{ξ}h_{0}] ∗ G_{k}∥_{1}

≤ \frac{1}{2k^{2}} ∥∇_{x} · ∇_{ξ}h_{0}∥_{1}∥G_{k}∥_{1}

= \frac{1}{2k^{2}} ∥∇_{x} · ∇_{ξ}h_{0}∥_{1} ⟶ 0 as k → ∞.

Then for an arbitary h ∈ L^{1}(ℝ^{2n}), we choose a h_{0} ∈ C_{0}^{∞}(ℝ^{2n}) such that ∥h − h_{0}∥_{1} be small enough and we write

∥\frac{1}{2k^{2}} ∇_{x} · ∇_{ξ}C_{k}h∥_{1} ≤ ∥\frac{1}{2k^{2}} [∇_{x} · ∇_{ξ}G_{k}] ∗ (h − h_{0})∥_{1} + \frac{1}{2k^{2}} ∥∇_{x} · ∇_{ξ}C_{k}h_{0}∥_{1}

≤ \frac{1}{2k^{2}} ∥∇_{x} · ∇_{ξ}G_{k}∥_{1}∥h − h_{0}∥_{1} + \frac{1}{2k^{2}} ∥∇_{x} · ∇_{ξ}C_{k}h_{0}∥_{1}.

Now since \frac{1}{2k^{2}} ∥∇_{x} · ∇_{ξ}G_{k}∥_{1} = \frac{1}{2}∥∇_{x} · ∇_{ξ}G∥_{1}, we can use the \frac{ε}{2} argument for concluding (3.1). □

4. A C-existence family for the Husimi equation

Throughout this section we will write D(A) for the domain of a linear operator A defined on a Banach space (X, ∥·∥). We will denote by [D(A)] the normed vector space with the graph norm ∥f∥_{[D(A)]} ≡ ∥Af∥ + ∥f∥. We designate L(Y) for the space of linear bounded operators from the space Y into itself and C ∈ L(X). Basic material on mild and strong C-existence families may be found in [dL2], along with more extensive references. For understanding these notions we will compare them with the notions of the mild and strong solutions for (CP).

Definition 4.1. (a) A strong solution of (CP) is u(t, φ) such that

t ↦ u(t, φ) ∈ C([0, ∞), [D(A)]) ∩ C^{1}([0, ∞), X),

satisfying (CP).

(b) A mild solution of (CP) is u(t, φ) such that t ↦ u(t, φ) ∈ C([0, ∞), X), and for all t ≥ 0, ∫_{0}^{t} u(s, φ)ds ∈ D(A), with

A (∫_{0}^{t} u(s, φ)ds) = u(t, φ) − φ (t ≥ 0), u(0, φ) = φ.

Definition 4.2. (a) A mild C-existence family for A is a strongly continuous family of operators {Y (t)}_{t≥0} ⊆ L(X) such that for any φ ∈ X, t ≥ 0, ∫_{0}^{t} Y (s)φds ∈ D(A), with

A (∫_{0}^{t} Y (s)φ ds) = Y (t)φ − Cφ.

(b) The mild C-existence family {Y (t)}_{t≥0} is a strong C-existence family for A, or simply C-existence family for A if {Y (t) |_{[D(A)]}}_{t≥0} is contained in L([D(A)]) and is strongly continuous, with

(∫_{0}^{t} AY (s)φ ds) = Y (t)φ − Cφ,

for all φ ∈ D(A).

Here are some consequences of having a C-existence family.

Lemma 4.3 [dL2, Theorem 2.6 and Proposition 2.9].

(a) If there exists a mild C-existence family, {Y (t)}_{t≥0} for A, then (CP) has a mild solution for any φ = Cψ ∈ Im(C), u(t, Cψ) ≡ Y (t)ψ, and the sequence of the solutions u(t, Cψ_{n}) converges to zero, uniformly on compact subsets of [0, ∞), whenever ψ_{n} → 0.

(b) If there exists a strong C-existence family, {Y (t)}_{t≥0} for A, then (CP) has a strong solution for any φ = Cψ ∈ C(D(A)), u(t, Cψ) ≡ Y (t)ψ, and the both u(t, Cψ_{n}) and Au(t, Cψ_{n}) converge to zero, uniformly on compact subsets of [0, ∞), whenever ψ_{n} and Aψ_{n} both converge to zero.

(c) Suppose A is closed and there exists δ ∈ ℝ such that A has no eigenvalues in (δ, ∞). Then all mild solutions of (CP) are unique.

In the sequel we construct a C-existence family for the Husimi operator A. In Theorem 2.1 we have construct the C_{0}-group e^{tL} generated by the Wigner operator L on L^{1}(ℝ^{2n}). Now, let us consider the family {Y (t)}_{t∈ℝ} defined by Y (t) ≡ Ce^{tL}, where the operator C is given by (1.18). Since C is a bounded operator in L^{1}(ℝ^{2n}) and

L ∫_{0}^{t} e^{sL}ϕ ds = e^{tL}ϕ − ϕ (∀t ∈ ℝ), ϕ ∈ L^{1}(ℝ^{2n})

by acting C, we obtain

Y (t)ϕ − Cϕ = CL ∫_{0}^{t} e^{sL}ϕ ds

= AC ∫_{0}^{t} e^{sL}ϕ ds

= A ∫_{0}^{t} Y (s)ϕ ds.

This together with the facts that ψ(t) = ∫_{0}^{t} e^{sL}ϕ ds ∈ D(L) for any ϕ ∈ X and CLψ = ACψ infer that Cψ = ∫_{0}^{t} Y (s)ϕ ds = ∫_{0}^{t} Ce^{sL}ϕ ds ∈ D(A). Consequently the family {Y (t)}_{t∈ℝ} is a mild C–existence family and from the strong continuity of e^{tL} it follows that it is even a strong C–existence family.

From these results we can infer that

Theorem 4.4. Under the assumptions of Corollary 2.2, the Husimi function H(x, ξ, t) given in (1.12) is a strong solution of the equation (1.16).

Proof. The estimation (2.3) implies that ∥Y (t)∥ ≤ e^{δ|t|}, i.e. all mild solutions of (CP) are exponentially bounded. Hence there is no eigenvalues in (δ, ∞). Since A is closed the assertion (c) of Lemma 4.3 yields the uniqueness of the solution Y (t)w_{0} starting with the initial datum w_{0}(x, ξ) given by (2.4) which lies in L^{1}(ℝ^{2n}). This produces exactly the Husimi function. □

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