JOURNAL OF FUNCTIONAL ANALYSIS
14, 295-298 (1973)
Quadratic Forms and Klauder’s Phenomenon:
A Remark on Very Singular Perturbations
BARRY SIMON*?+
Dt!partment de Physique ci Luminy, Universite d’Aix-Marseille II, Marseille, France
Communicated by G. A. Hunt
Received April 2, 1973
We give a general analysis of a class of pairs of positive self-adjoint operators A
and B for which A + XB has a limit (in strong resolvent sense) as h -10 which
is an operator A, # A!
Recently, Klauder [4] has discussed the following example: Let
A be the operator -(d2/A2) +
x2 on L2(R, dx) and let B = 1 x 1-s.
The eigenvectors and eigenvalues of A are, of course, well known to
be the Hermite functions, H,(x), n = 0, l,... and E, = 2n + 1.
Klauder then considers the eigenvectors of A +
XB (A > 0)
by
manipulations with the ordinary differential equation (we consider
the domain questions, which Klauder ignores, below). He finds that
the eigenvalues E,(X)
and eigenvectors &(A) do not converge to 8,
and H, but rather
AO) + (en 4 Ho+,
J%(X) -+ gn+1
I
n = 0, 2,...,
We wish to discuss in detail the general phenomena which Klauder
has uncovered. We freely use the techniques of quadratic forms and
strong resolvent convergence; see e.g. [3], [5].
Once one decides to analyze Klauder’s phenomenon in the language
of quadratic forms, the phenomenon is quite easy to understand and
control. In fact, the theory is implicit in Kato’s book [3, VIII.31.
* Permanent Address: Departments of Mathematics and Physics, Princeton Uni-
versity, Princeton, New Jersey 08540.
t A. Sloan Foundation Fellow.
295
Copyright 0 1973 by Academic Press, Inc.
All rights of reproduction in any form reserved.
296
SIMON
It is, thus, with some hesitation that we write this note. However,
because the phenomenon is so striking, it seems worthwhile to point
out the mechanism behind it.
The first question we must face is the definition of
A + hB. We
will suppose that
A
and B are positive self-adjoint operators on a
separable Hilbert space 2 with form domains
Q(A)
and Q(B). If
h > 0 and if
Q(A) n
Q(B) = Q is dense in X, then
is easily seen to be a closed quadratic form on Q. There is, thus,
a unique self-adjoint operator so that this form is the form associated
to that self-adjoint operator [5, Theorem VIILlS]. Henceforth
(except for Example 2 below),
A
+
hB
will denote this operator. Our
main result tells us that Klauder’s phenomena occurs if and only if
Q(A) n Q(B)
is not a form core for
A.
THEOREM 1.
Let A and B be positive self-adjoint operators with
Q(A)
n Q(B) d
erase. Let A, be the (unique) positive self-adjoint operator
whose associated quadratic form is the closure of the form
on the domain Q(A) n Q(B). For h > 0, define A + AB as a sum of
quadratic forms.
Then as X 4 0, A + )tB converges to A, in strong
resolvent sens.
Proof.
Follows from Theorem VIII.3.6 or VIII.3.11 of Kato [3].
The connection between eigenvalues of
A
+
hB
and those of
A,
is
quite simple.
THEOREM
2. Let A, B, A, be as in Theorem
1.
Suppose that
Z = inf oess(A,)
and that A, has at least m eigenvahes below Z. Let E, ,..., E, be the m
lowest eigenvalues for A, .
Then, for all sq@iently small A, A -+ hB has
at least m eigenvalues E,(h),.. ., E,(h) below Z(h) and
h$ E,(X) =
Ei ,
i = J,..., m.
Proof. See
[3, Theorem VIIL3.15) or alternately [2] or [6, Appen-
dix C].
Let us close with a few examples.
A REMARK ON VERY SINGULAR PERTURBATIONS
297
EXAMPLE
1.
d2
A=--,
B = I x /+ onL2(R,
dx).
If -2 < cy < 1, then B is a small form perturbation of
A (see e.g. [4])
so it is easy to prove that
A
+
XB
converges to
A
in norm resolvent
sense as h 4 0. This last fact also holds if cy < -2 by a more com-
plicated argument (see [6]). If LY > 1, we have a different situation.
For every 9 E
Q(A)
is Holder continuous and, thus,
s
1 v(x)l2 I x /- dx < 00
implies that ~(0) = 0. Thus,
Q(A) n Q(B) C {v E D(A) / ~(0) = 0}
so that
A,
#
A
but is instead x2 plus the operator -d2/dx2 with
Dirichlet boundary conditions at x = 0. For both
A,
and
A + XB,
L2((0, co),
dx)
is an invariant subspace.
EXAMPLE
2. One can ask a slightly different question about
A+XBifD(A)nD(B)
d
is ense. Rather than ask if
A
+
AB
defined
as a sum of forms converges to
A
as h 4 0, one can ask if there is some
family of self-adjoint extensions of
A
+
XB
r
D(A)
A
D(B)
which
converges to
A
as X 4 0. In Example I, if 01 > 2 there exists no such
family since
A
+
hB
is essentially self-adjoint and its closure is the
form sum (this follows from the Weyl limit point method [l] or by
other methods [7]). But for 1 <
01 < 2, such families do exist (see [4]),
and Klauder’s phenomenon is not forced upon us if we look at
extensions of operator sums rather than the form sum. This explains
why Klauder chooses a! = 2 as the borderline while Example 1
suggests (Y = 1 as borderline.
EXAMPLE
3.
A = --d + / x j2,
B= l4-w
I
on L2(Iw”,
dx),
n>2
, .
It is known that {# 1 # E Corn, I/ s 0 near 0} is a form core for --d in
this case, so AP =
A.
Thus, Klauder’s phenomenon (in case
A
is
a harmonic oscillator) requires
B
to have a codimension 1 singularity.
As Klauder [4] remarks, this is most conviently understood in terms
of Weiner path integrals (more precisely Ornstein-Uhlenbeck path
integrals).
298
SIMON
EXAMPLE
4.
It is fitting to conclude with an example suggered by
D. W. Robinson (private communication) which shows how unsuited
the form sum can be. Let A be a smooth bounded set in IIF and let
2’ = L2(A, d%). Let
A
= --d with Neumann boundary conditions
and
B
= -A with Dirichlet boundary conditions. Then
A + hB =
(1 + h)B, so A + XB 4 B # A
as A 4 0.
ACKNOWLEDGMENTS
It is a pleasure to thank D. W. Robinson and J. Klauder for valuable discussions
and D. Kastler for the hospitality of the Department de Physique a Luminy.
REFERENCES
1.
E. A.
CODDINGTON AND N. LEVINSON, “
Theory of Ordinary Differential Equations,”
MC Graw-Hill, New York Toronto London, 1945.
2. W. M. GREFXLEE, Singular perturbations of eigenvalues Arch. Rat. Mech. Anal.
34 (1969), 143-164.
3. T. KATO, “Perturabtion Theory for Linear Operators,” Springer verlag, New York
Berlin, 1966.
4.
J.
KLAUDER, Field structure through model studies, Actu Phys. Austriaca, to
appear.
5. M. REED AND B. SIMON, “Methods of Modern Mathematical Physics,” Vol. I,
Academic Press, New York, 1972.
6. B. SIMON, Coupling constant analyticity for the anharmonic oscillator, Ann. Phys.
58 (1970), 76-136.
7. B. SIMON, Essential self-adjointness of Schrijdinger operators with singular
potentials: A generalized Kalf-Walter-Schmincke theorem,
Arch.
Rut. Mech.
Anal., to appear.