Positivity 5: 359–382, 2001.
© 2001 Kluwer Academic Publishers. Printed in the Netherlands.
Positivity and Negativity of Solutions to a
Schrödinger Equation in R
, JACQUELINE FLECKINGER-PELLÉ
CEREMATH and UMR MIP, Université des Sciences Sociales, 21 Allee de Brienne, F–31042
Toulouse Cedex, France,
E-mail: email@example.com and
Fachbereich Mathematik, Universität Rostock, Universitätsplatz 1, D–18055 Rostock, Germany.
(Received 13 April 1999; accepted 6 March 2000)
Abstract. Weak L
)-solutions u of the Schrödinger equation, −u + q(x)u − λu = f(x)
), are represented by a Fourier series using spherical harmonics in order to prove the
following strong maximum and anti-maximum principles in R
(N 2): Let ϕ
denote the positive
eigenfunction associated with the principal eigenvalue λ
of the Schrödinger operator A =− +
q(x)• in L
). Assume that the potential q(x) is radially symmetric and grows fast enough near
inﬁnity, and f is a ‘sufﬁciently smooth’ perturbation of a radially symmetric function, f ≡ 0and
0 f/ϕ C ≡ const a.e. in R
.Thenu is ϕ
-positive for −∞ <λ<λ
(i.e., u cϕ
c ≡ const > 0) and ϕ
-negative for λ
+ δ (i.e., u −cϕ
with c ≡ const > 0), where
δ>0 is a number depending on f . The constant c>0 depends on both λ and f .
Mathematics Subject Classiﬁcations (2000): Primary 35B50, 35J10; Secondary 35P30, 81Q10
Key words: positive or negative solutions, pointwise bounds, principal eigenvalue, positive eigen-
function, strong maximum and anti-maximum principles, spherical harmonic functions
Positivity or negativity of weak L
-solutions to a linear partial differential equation
with the Schrödinger operator,
−u + q(x)u− λu = f(x) in R
has been a subject of a number of research articles and monographs, see e.g. Alzi-
ary, Fleckinger and Taká
c , Alziary and Taká
c , Davies , Reed and Simon
[17, Sect. X.4], [18, Sect. XIII.12], and many others. Here, f is a given function
satisfying 0 f ≡ 0inR
(N 1), and λ stands for the spectral parameter.
This author expresses her gratitude to the University of Rostock for the support during her visit.
Research supported in part by Deutsche Forschungsgemeinschaft (Germany). A part of this
research was performed when this author was a visiting professor at CEREMATH/UMR MIP,
e des Sciences Sociales, Toulouse, France.