Spectral decomposition and resolvent kernel
for a magnetic Laplacian in
C
n
N. Askour
a)
and Z. Mouayn
b)
Department of Mathematics, Faculte
´
des Sciences et Techniques (M’Ghila),
BP.523, Be
´
ni Mellal, Morocco
͑Received 13 March 2000; accepted for publication 19 May 2000͒
Spectral decomposition and resolvent kernel for a magnetic Laplacian in C
n
are
given. As an application we obtain the corresponding Schro
¨
dinger propagator and
wave kernel. © 2000 American Institute of Physics. ͓S0022-2488͑00͒02809-7͔
I. INTRODUCTION
In this paper we will be concerned with the magnetic Laplacian in C
n
:
⌬
˜
ϭϪ
͚
j ϭ1
n
ץ
2
ץ
z
j
ץ
z
¯
j
ϩ
͚
j ϭ1
n
z
¯
j
ץ
ץ
z
¯
j
,
with D(⌬
˜
)ϭC
0
ϱ
(C
n
,C), the space of C-valued C
ϱ
-functions with a compact support in C
n
as its
natural regular domain in the weighted Hilbert space HϭL
2
(C
n
,e
Ϫ
͉
z
͉
2
d
) endowed with the
Hermitian scalar product
͗
f,g
͘
H
ϭ
͐
C
n
f(z)g(z)e
Ϫ
͉
z
͉
2
d
(z). Here, d
(z) denotes the Lebesgue
measure in C
n
ϭR
2n
and
͉
z
͉
2
ϭ
͗
z,z
͘
the usual Euclidean norm square. The operator ⌬
˜
is obtained
from the Schro
¨
dinger operator with a uniform magnetic field ͑of length one͒ given by
H
˜
ϭ
Ϫ1
4
͚
j ϭ1
n
ͩͩ
ץ
ץ
x
j
ϩ
ͱ
Ϫ1y
j
ͪ
2
ϩ
ͩ
ץ
ץ
y
j
Ϫ
ͱ
Ϫ1x
j
ͪ
2
ͪ
,
defined on the Hilbert space L
2
(R
2n
,d
). Precisely, we have Qؠ(H
˜
Ϫn/2)ؠQ
Ϫ1
ϭ⌬
˜
, where Q is
the unitary map from L
2
(R
2n
,d
) into H defined by Qfϭe
(1/2)
͉
z
͉
2
f. In Ref. 1, the authors have
obtained explicit formulas for reproducing kernel of eigen-spaces of ⌬
˜
in the Hilbert space H.
Actually, some general spectral properties of ⌬
˜
in H are well known. Namely, ⌬
˜
is an essentially
self-adjoint operator in H. The spectrum of ⌬
˜
is the set
͕
m,mZ
ϩ
͖
of eigenvalues with infinite
multiplicities.
The purpose of the present paper is to endow the L
2
spectral theory of the operator ⌬
˜
with
fundamental tools as its spectral decomposition and resolvent kernel. As an application, we obtain
the corresponding Schro
¨
dinger propagator and wave kernel.
The paper is organized as follows. In Sec. II, we give the spectral decomposition of ⌬
˜
. In Sec.
III, we deal with the resolvent operator. In Sec. IV, we give some applications of the obtained
explicit formula for the resolvent kernel of ⌬
˜
.
II. SPECTRAL DECOMPOSITION
Let us fix some notations. For p,qZ
ϩ
, let H(p,q) denote the space of restrictions to the
sphere S
2nϪ1
ϭ
͕
C
n
,
͉
͉
ϭ1
͖
of Euclidean harmonic polynomials h(z)onC
n
which are ho-
mogenous of degree p in z and the degree q in z
¯
. The dimension d(n,p,q)ofH(p,q)isas
follows. For nϭ2,3,..., we have
a͒
Electronic mail: naskour@caramail.com
b͒
Electroinc mail: mouayn@caramail.com
JOURNAL OF MATHEMATICAL PHYSICS VOLUME 41, NUMBER 10 OCTOBER 2000
69370022-2488/2000/41(10)/6937/7/$17.00 © 2000 American Institute of Physics