6 1. INTRODUCTION

F7 FRn u(ξ) =

1

(2π)

n

2

R

e

√

−1t(Ru)(ξ,

t)dt.

F8 FRn gives an automorphism of each of the following topological vector spaces:

S(Rn)

⊂

L2(Rn)

⊂ S

(Rn).

Here, F7 gives the plane wave expansion of the Fourier transform by means of the

Radon transform R defined by

Ru(ξ, t) :=

Rn

u(x)δ( x, ξ − t)dx.

In F8, we denote by

S(Rn)

the space of rapidly decreasing

C∞-functions

on

Rn

(the Schwartz space endowed with the Fr´ echet topology), and by S

(Rn)

the dual

space consisting of tempered distributions.

F4 is the Plancherel theorem, and F8 gives the Paley–Wiener theorem for the

Schwartz space

S(Rn)

(and its dual S

(Rn)).

By F1 and F2, the inversion formula

f(x) = FRn ◦ FRn

−1

f(x)

=

Rn

(FRn

−1f)(ξ)k(x,

ξ)dξ

gives an expansion of a function f into joint eigenfunctions k(x, ξ) of the commuting

self-adjoint operators pj (1 ≤ j ≤ n).

Moreover, the property F5 characterizes the operator FRn up to scalar. We

pin down this algebraic statement in two ways as follows:

Proposition 1.2.1. Let A be a continuous operator on L2(Rn) satisfying the

following identities:

(1.2.2) A ◦ xj = pj ◦ A, A ◦ pj = −xj ◦ A (1 ≤ j ≤ n).

Then, A is a scalar multiple of FRn .

Proposition 1.2.2. Let A be a continuous operator on

L2(Rn)

satisfying

(1.2.3) A ◦ xixj = pipj ◦ A, A ◦ pipj = xixj ◦ A (1 ≤ i, j ≤ n).

Then, A is of the form A = aFRn + bFRn

−1

for some a, b ∈ C.

Here, (1.2.3) is obviously a weaker condition than (1.2.2).

We did not go into details about the domain of definition for (1.2.2) and (1.2.3)

in the above propositions. The domain could be D := {f ∈

L2(Rn)

: xjf, pjf ∈

L2(Rn)

(1 ≤

∀j

≤ n)}, on which we regard the identities Axjf = pjAf those of

distributions in the case of Proposition 1.2.1. Likewise for Proposition 1.2.2.

Intertwining characterization of FC

Back to the setting in Section 1.1, we consider the differential operator Pj (of

second order!). Then, it turns out that the intertwining relation between Pj and

the multiplication by the coordinate function xj again characterizes our operator

FC up to scalar:

Theorem 1.2.3 (see Theorem 2.5.4). 1) There exists a unitary operator FC

on

L2(C)

satisfying the following relation:

(1.2.4) A ◦ Pj = 4xj ◦ A, A ◦ xj = 4Pj ◦ A (1 ≤ j ≤ n).