Hermite Polynomials and Quasi-classical Asymptotics

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by  S. Twareque Ali

HERMITE POLYNOMIALS AND QUASI-CLASSICAL
ASYMPTOTICS
4 S.TWAREQUEALIANDMIROSLAVENGLISˇ
1
0
2 Abstract. We study an unorthodox variant of the Berezin-Toeplitz type of
quantization scheme, on a reproducing kernel Hilbertspace generated by the
n
real Hermite polynomials and work out the associated semi-classical asymp-
a
totics.
J

5
1

] 1. Introduction
h
p At the heart of most approaches to quantization lies the idea of assigning to
- functions f (the classical observables) suitable operators T (quantum observ-
h f
t ables) depending on an auxiliary parameter h (the Planck constant) in such a way
a
that as h 0, T possesses an appropriate asymptotic behaviour reflecting the
m ց f
“(semi)classical limit” of the quantum system. Typically, the functions f live on
[
a manifold equipped with symplectic structure (the phase space) and the required

1 asymptotic behaviour takes the form of the “correspondence principle”
v
ih
9 (1) T T T T T
3 f g− g f ≈ 2π {f,g}
6 where , denotes the Poissonbracket.
3 {· ·}
For complex manifolds which are not only symplectic but K¨ahler, a notable
.
1 example of such a construction is the Berezin-Toeplitz quantization, first formally
0 introducedin[9],thoughsomeideasgobacktoBerezin[6]andsimilarquantization
4
techniques had also been introduced by other authors [2, 23]. Namely, assume for
1
simplicity that the phase space Ω is simply connected, so that the K¨ahler form ω
:
v admits a global real-valued potential Ψ, i.e. ω =∂∂Ψ. Consider the L2 space
i
X
r (2) L2h ={f measurable on Ω:Z |f|2e−Ψ/hωn <∞} (h>0),
a Ω
and let L2 (the weighted Bergman space) be the subspace in L2 of functions
hol,h h
holomorphiconΩ, and P :L2 L2 the orthogonalprojection. For a bounded
h h → hol,h
measurable function f on Ω, one then defines the Toeplitz operator T on L2
f hol,h
with symbol f by
(3) T u=P (fu).
f h
This is, in fact, an integral operator: more precisely, the space L2 turns out to
hol,h
be a reproducing kernelHilbert space [3] possessinga reproducingkernel K (x,y),
h
and (3) can be rewritten as
(4) T u(x)= u(y)f(y)K (x,y)e Ψ(y)/hω(y)n.
f h −
Z
Ω
ResearchsupportedbyGACˇRgrantno.201/09/0473andRVOfundingforICˇO67985840and
NaturalSciences andEngineeringResearchCouncil(NSERC)ofCanada.
1
2 S.TWAREQUEALIANDMIROSLAVENGLISˇ
When the manifold Ω is not simply connected, one has to assume that the coho-
mology class of ω is integral, so that there exists a Hermitian line bundle with
L
the canonical connection whose curvature form coincides with ω; and the spaces
L2 (and L2) get replaced by the space of all holomorphic (or all measurable,
hol,h h
respectively) square-integrable sections of k, k = 1 = 1,2,3,.... In any case,
L⊗ h
under reasonabletechnical assumptions on Ω (e.g. for Ω compact [9], or for Ω sim-
ply connected strictly-pseudoconvex domain in Cn with smooth boundary ∂Ω and
e Ψ vanishing to exactly first order at ∂Ω [13]), the Toeplitz operators satisfy
−
(5) T T T +hT +h2T +... as h 0,
f g ≈ fg C1(f,g) C2(f,g) ց
with some bidifferential operators C such that C (f,g) C (g,f) = i f,g ,
j 1 − 1 2π{ }
implying in particular that (1) holds. The asymptotic expansion (5) even holds
in the strongest possible sense of operator norms, i.e. the difference of the left-
hand side and the sum of the first N terms on the right hand side has norm,
as an operator on L2 , bounded by a multiple of hN as h 0, for all N =
hol,h ց
1,2,3,.... Furthermore, the bidifferential operators C can be expressed in terms
j
ofcovariantderivatives,with contractionsofthe curvaturetensor andits covariant
derivatives as coefficients, thus encoding various geometric properties of (Ω,ω) in
an intriguing way.
TheBerezin-ToeplitzAnsatz abovehassubsequentlybeenextendedtoanumber
ofmoregeneralcontextsoutsidetheK¨ahlersetting,includinge.g.thatofharmonic
Bergmanspacesonsomespecialdomains[14][7][19],orwhenspacesofholomorphic
functions/sections are replaced by eigenspaces of the Spinc-Dirac operator on a
general symplectic manifold or even orbifold [11] [21] [25] [10], while numerous
other developments concerned the properties of the cochains C or miscellaneous
j
representation-theoretic aspects of the procedure [20] [24] [16] [12] [26] [27] [5].
The purpose of the present paper is to highlight an operator calculus of a com-
pletelydifferentflavour,whichnonethelessbearscertainresemblanceto(1)and(4),
and arises in a quite unexpected setting — namely, in connection with orthogonal
polynomials. Generically, the situation is the following: as explained above, the
Berezin-Toeplitz type of quantization relies on the existence of a certain L2-space
which contains a reproducing kernel Hilbert space as a subspace; the quantization
is effected by the projection operator of this subspace. Alternatively, the repro-
ducing kernel K(x,y) defines a family of vectors K(,y), y Ω in the reproducing
· ∈
kernelHilbert space,generally calledcoherent states in the literature,and then (4)
shows that the quantization may also be defined in terms of these coherent states.
However, the existence of coherent states depends only on the reproducing kernel
andnotonanyambientL2-spaceandindeed,therehavebeenproposals,somevery
recent [18, 22], to base both the theory of coherent states and geometric quantiza-
tion using a positive definite kernel alone. The present paper may be thought of
as an extension of this line of thought to Berezin-Toeplitz quantization. What is
interestinginourpresentcaseisthatitistheHermite polynomials,whichinaway
define the quantum harmonic oscillator, also define the reproducing kernel of our
problem.
To be more specific, let H (x) stand for the standard Hermite polynomials
n
(see Section 2 below for the details), and, for 0<ǫ<1, set
∞
(6) K (x,y)= ǫn H 2H (x)H (y), x,y R.
ǫ n − n n
k k ∈
nX=0
HERMITE POLYNOMIALS 3
Here H denotesthenorminL2(R,e x2dx),wherethe H formanorthogonal
n − n
k k { }
basis. Then K is a positive-definite function, and, hence, determines uniquely a
ǫ
Hilbert space of functions on R for which K is the reproducing kernel [3]; this
ǫ ǫ
H
spacefirstappearedin[1]whenstudying“squeezed”coherentstates. (Itsdefinition
mayperhapsseema bitartificialatfirstglance,butso musthaveseemed(2)when
it first came around in Berezin’s papers!) For a (reasonable) function f on R, set
(7) T u(x):= u(y)f(y)K (x,y)e y2dy.
f ǫ −
ZR
This certainly resembles the expression (4) for Toeplitz operators, however, note
that this time there is no L2 space around like (2) which would contain as a
ǫ
H
closed subspace (in fact, the set f(x)e x2/2 : f is a dense, rather than
− ǫ
proper closed, subset of L2(R)), s{o there is no pro∈jecHtio}n like P around and the
h
original definition (3) makes no sense. In particular, there is no reason a priori
evento expect (7) to be defined, not to say bounded, on some space(whereas with
(3) it immediately follows that T is not greater than the norm of the operator
f
k k
of “multiplication by f” on L2, hence T f ). It may therefore come as
f
a bit of a surprise that (7) actually yiekldsk, f≤orkf k∞L (R), a bounded operator
∞
on L2(R), and, moreover, T enjoys a nice asympto∈tic behaviour as ǫ 1, which
f
ր
we will see to correspond, in a very natural sense, to the semiclassical limit h 0
ց
in the original quantization setting.
Itshouldbestressedthattheresultingasymptoticsarenotquite ofthe form(5)
and, in particular, (1) does not hold, so that our results claim no direct physical
relevance; on the other hand, the same is true as well for some of the general-
izations of the classical Toeplitz calculus mentioned two paragraphs above, while
not depriving the latter of their mathematical beauty and relevance. We hope the
sametobeatleastpartlytruealsoforourdevelopmentshereandthusjustify their
disclosure to a wider audience.
WereviewthenecessarystandardmaterialonHermitepolynomialsinSection2.
Thespaces arediscussedinSection3,andthebasicfactsabouttheoperatorsT
ǫ f
H
from(7)inSection4. TheasymptoticbehaviourisstudiedinSection5. InSection6
we observe how to recover the standard Berezin-Toeplitz quantization on C using
the Hermite Ansatz and an appropriate analogue of the Bargmann transform.
A large portion of this work was done while the second author was visiting the
first in September 2012; the hospitality of the Department of Mathematics and
Statistics of Concordia University on that occasion is gratefully acknowledged.
2. Hermite polynomials
The Hermite polynomials H (x), n=0,1,2,..., are defined by the formula
n
dn
(8) Hn(x):=(−1)nex2dxne−x2.
They can also be obtained from the generating function
(9) e2xz−z2 = ∞ znHn(x)
n!
nX=0
and satisfy the orthogonality relations
(10) Hn(x)Hm(x)e−x2dx=n!2n√πδmn.
ZR
4 S.TWAREQUEALIANDMIROSLAVENGLISˇ
It follows that the Hermite functions
(11) h (x):=(n!2n√π) 1/2H (x)e x2/2,
n − n −
n = 0,1,2,..., form an orthonormal basis in the Hilbert space L2(R) on the real
line.
The representation (9) also leads to the explicit formula
[n/2]( 1)m(2z)n 2m
−
(12) H (x)=n! − ,
n
m!(n 2m)!
mX=0 −
[x] denoting the integer part of x. From this follows the estimate
(13) H (z) √n!2ne√2nz
n | |
| |≤
valid for all complex z.
All this, of course, is quite standard and well-known (see e.g. [4], Chapter 6.1),
perhaps with the exception of the last estimate for non-real z; for completeness,
we therefore attach a proof. Observe first of all that
n! 2[n/2][n]!
(14) 2 .
rnn ≤ n[n/2]
Indeed,bothforn=2kevenandforn=2k+1odd,thecorrespondinginequalities
(2k)! 2kk! (2k+1)! 2kk!
,
p(2k)k ≤ (2k)k (p2k+1)k+21 ≤ (2k+1)k
reduce to the elementary estimate
(2k)! (2 4 6 (2k))2 =4kk!2.
≤ · · ····
Since
2mm! m j
=
nm n/2
jY=1
is a decreasing function of m for 0 m n, it follows from (14) that even
≤ ≤ 2
n! 2mm! n
, m=0,1,..., .
rnn ≤ nm h2i
Consequently,
√n!
2n2−2m nn2−m2n2−m =(2n)n−22m
m! ≤
and
[n/2]√n!2n2−2m|z|n−2m [n/2](√2n|z|)n−2m e√2n|z|,
m!(n 2m)! ≤ (n 2m)! ≤
mX=0 − mX=0 −
proving (13).
As a corollary,we also get the estimate
(15) h (z) π 1/4e√2nz Rez2/2, z C,
n − | |−
| |≤ ∈
for the corresponding Hermite functions.
The last fact we need to recall is the differential equation
H (x) 2xH (x)+2nH (x)=0
n′′ − n′ n
for H (x), which translates into another differential equation
n
h (x)+(2n+1 x2)h (x)=0
′n′ − n
HERMITE POLYNOMIALS 5
for the Hermite functions h . In other words, the (Schr¨odinger) operator
n
x2 1 1 d2
(16) A:= − I
2 − 2dx2
on L2(R) satisfies
(17) Ah =nh , n=0,1,2,...
n n
(that is, A= n ,h h ).
n h· ni n
P
3. Reproducing kernel spaces
For 0<ǫ<1, the reproducing kernels
(18) Kǫ(x,y)= ∞ ǫnHnn(!x2)nH√nπ(y) = (1 1 ǫ2)πe−1−ǫ2ǫ2(x2+y2−2ǫxy),
nX=0 −
p
wereintroducedin[1];thesecondequalityisknownasMehler’sformula. Wedenote
by the correspondingreproducingkernelspace[3]; thatis, is the completion
ǫ ǫ
of lHinear combinations of the functions K (,y), y R, with rHespect to the scalar
ǫ
· ∈
product
a K (,y ), b K (,x ) = a b K (x ,y ).
j ǫ j k ǫ k j k ǫ k j
DXj · Xk · E Xj,k
We will also use the Hilbert spaces
=e x2/2 = e x2/2f(x):f
ǫ − ǫ − ǫ
H H { ∈H }
corresponding to the reeproducing kernel
(19) Kǫ(x,y)=e−(x2+y2)/2Kǫ(x,y)= ∞ ǫnhn(x)hn(y).
nX=0
e
Since the transition from to involves only the multiplication by e x2/2,
ǫ ǫ −
H H
we will state the various facts below usually only for one of these spaces.
e
The following assertion, though not explicitly stated in [1], is fairly straightfor-
ward.
Proposition 1. One has
(20) ǫ = f(x)= fnhn(x): ǫ−n fn 2 <
H { | | ∞}
Xn Xn
e
with the norm in being given by
ǫ
H
(21) e kfk2ǫ = ǫ−n|fn|2.
Xn
Proof. Letustemporarilydenotethespaceontheright-handsideof(20)(withthe
norm given by (21)) by . From the equality
ǫ
M
ǫnhn(y)2ǫ−n = ǫn hn(y)2 =Kǫ(y,y)= 1 eǫǫ−+11x2 <
| | | | (1 ǫ2)π ∞
Xn Xn −
e p
(cf. (18)), it follows that the function
K := ǫnh (y)h =K (,y)
ǫ,y n n ǫ
·
Xn
e e
6 S.TWAREQUEALIANDMIROSLAVENGLISˇ
belongs to , for any y R. Furthermore, for any f = f h ,
Mǫ ∈ n n n ∈Hǫ
P
f,K = ǫ nf ǫnh (y)= f h (y)=f(y)
ǫ,y ǫ − n n n n
h i
Xn Xn
e
(herewehaveusedthefactthath isreal-valuedonR). ThusK isthereproducing
n ǫ
kernelfor . Since a reproducing kernelHilbert space is uniquely determined by
Mǫ e
its reproducing kernel, = , with equality of norms. (cid:3)
ǫ ǫ
M H
e
The last proposition allows for the following interpretation of the spaces
ǫ
and . Recall that the Sobolev space of order s on R can be defined as tHhe
(comHplǫetion of the) space of all f ( (R)) for which e
∈D
kfk2s :=h(I −∆)sf,fiL2(R) <∞.
By analogy, one could define “Hermite-Sobolev” spaces s(R) on R by
W
kfk2s :=h(I +A)sf,fiL2(R) <∞.
In view of (17), this is equivalent to
s(R)= f = f h : f 2 = (n+1)s f 2 < .
W { n n k ks | n| ∞}
Xn Xn
Our spaces are thus obtained upon replacing (n + 1)s by ǫ n. Back in the
ǫ −
H
context of the ordinary Laplacian, they are thus analogues of the spaces
e
eǫ∆/2L2(R)= f : e−ǫ∆f,f L2(R) <
{ h i ∞}
of solutions at time t = ǫ of the heat equation ∂u = ∆u, u(x,0) = f(x) (“caloric
2 ∂t
functions”). More precisely, =e Alog√ǫL2(R) is the space of solutions at time
ǫ −
H
t= 1logǫ of the modified heat equation
−2 e
∂u
=Au, u=u(x,t), t>0,
∂t
with initial condition u(,0) L2(R).
· ∈
We conclude this section by showing that is actually a space of holomor-
ǫ
H
phic functions, like the weighted Bergman spaces mentioned in the Introduction.
(The same is true also for the ordinary spaces of caloric functions.)
Theorem 2. Each f extends to an entire function on C, and is the space
ǫ ǫ
of (the restrictions to∈RHof) holomorphic functions on C with reprodHucing kernel
ǫ2 (x2+y2 2xy)
(22) K (x,y)= ∞ ǫnHn(x)Hn(y) = e−1−ǫ2 −ǫ .
ǫ n!2n√π (1 ǫ2)π
nX=0 −
p
Proof. By the preceding proposition, we have
(23) f = f (n!2n√π) 1/2H ,
n − n
Xn
with
ǫ n f 2 = f 2 < .
− n
Xn | | k kHǫ ∞
HERMITE POLYNOMIALS 7
Consequently, for any z C, we get using the estimate (13)
∈
fnHn(z)(n!2n√π)−1/2 fn π−1/4e√2n|z|
| |≤ | |
Xn Xn
1/2
π 1/4 f ǫn e√2nz 2 .
≤ − k kHǫ(cid:16)Xn | | || (cid:17)
Since for any fixedz C, the radius ofconvergenceof ǫne2√2nz , with ǫ as the
∈ n | |
variable, is 1, the expression in the last parentheses isPfinite for 0 < ǫ < 1. Thus
the series (23) converges for any z C (and uniformly on compact subsets). This
∈
proves the first part of the theorem, and also shows that
f(z)= f,K , z C,
h ǫ,ziHǫ ∈
with
K := ǫnH (z)(n!2n√π) 1/2(n!2n√π) 1/2H ,
ǫ,z n − − n
Xn
that is,
H (w)H (z)
K (w)= ǫn n n ,
ǫ,z n!2n√π
Xn
showing that (22) is indeed the reproducing kernel for on all of C. (cid:3)
ǫ
H
4. Toeplitz-type operators
Drawing inspiration from (4), we define, for a function (“symbol”) f on R, the
“Toeplitz operator” T(ǫ), 0<ǫ<1, on L2(R) by
f
(24) T(ǫ)u(x):= u(y)f(y)K (x,y)e y2dy.
f ZR ǫ −
We will also use the analogous operators
T(ǫ)u(x):= u(y)f(y)K (x,y)dy
f ZR ǫ
(25)
e e x2+y2
= u(y)f(y)Kǫ(x,y)e− 2 dy
ZR
on L2(R) defined using the kernel K instead of K . Clearly,
ǫ ǫ
(26) T(ǫ)ue=e 1/2T(ǫ)e1/2
f − f
where we introduced the notateion
e(x):=ex2.
ItturnsoutthattheoperatorsT(ǫ)haveabitnicerexpressionintermsoftheFourier
transform, while T(ǫ) are a bit nicer from the point of view of the “semiclassical”
asymptotics as ǫ 1. In view of (26), it is always a simple matter to pass from
րe
T(ǫ) to T(ǫ) or vice versa.
In the formula (4), the reproducing kernel K (x,y) is the integral kernel of the
e h
orthogonalprojectionP ontoL2 ,i.e.ofaboundedoperatorinthecorresponding
h hol,h
space L2. On the other hand, for K we have no such interpretation, in fact the
h ǫ
space ,ofwhichK isthereproducingkernel,isdenseinL2(R)(thisisimmediate
from PHrǫoposition 1 aǫnd the fact thaet h is an orthonormal basis of L2(R)).
e e { n}∞n=0
The next two results may therefore seem somewhat surprising.
8 S.TWAREQUEALIANDMIROSLAVENGLISˇ
Theorem 3. The operators T(ǫ) and T(ǫ) are densely defined for any f C (R).
f f ∈ ∞
Furthermore, Tf(ǫ) is bounded for f ∈Le∞(R), with
T(ǫ) C f
k f k≤ ǫk k∞
for some constant C depending only on ǫ, 0<ǫ<1.
ǫ
Proof. By (18),
T(ǫ)u(x)= (fu)(y)e−1ǫ2ǫ2(x2−2ǫxy+y2)−y2 dy
f ZR − (1 ǫ2)π
−
ǫ2 (x y)2 dy p
= (fu)(y)e−1 ǫ2 −ǫ
(27) ZR − (1−ǫ2)π
p dt
= (fu)(ǫx 1 ǫ2t)e t2
−
ZR −p − √π
ǫx
=(δ√1−ǫ2(fu)∗e−1)(cid:16)√1 ǫ2(cid:17),
−
where we have introduced the dilation operator
δ u(x):=u(rx).
r
In other words, introducing also the operator
Gu:=u e 1
−
∗
of convolution with the Gaussian e 1, we obtain
−
(28) T(ǫ) =δ Gδ M ,
f ǫ/√1 ǫ2 √1 ǫ2 f
− −
where
M :u fu
f
7→
denotes the operator of “multiplication by f”. If f C (R) and u (R), the
∞
space of smooth functions on R with compact supp∈ort, then fu ∈ D , the
Schwartz space on R. Since dilations map into itself while ∈ D ⊂ S
S
e 1/4
(29) Gf = fˆ − ∨
2√π
(cid:16) (cid:17)
(here ˆ and denote the Fourier transform and the inverse Fourier transform,
∨
respectively) also maps into itself, we conclude that
S
T(ǫ)u for any f C (R) and u (R).
f ∈S ∈ ∞ ∈D
Since isdenseinL2 and L2,thisprovesthefirstpartofthetheoremforT(ǫ).
D S ⊂
The assertion for T(ǫ) is then immediate from (26) and the fact that e1/2
D ⊂ D
and e 1/2L2 L2.
− ⊂ e
The second part follows from (28), because M f and
f
k k≤k k∞
kδǫ/√1−ǫ2Gδ√1−ǫ2k=(4πǫ)−1/2 =:Cǫ <∞
by an elementary argument and standard properties of the Fourier transform. (cid:3)
Theorem 4. For f L the operator T(ǫ) is bounded on L2(R).
∈ ∞ f
e
HERMITE POLYNOMIALS 9
Proof. By (19)
T(ǫ)u= ǫn fu,h h .
f h ni n
Xn
e
Thus, for any 0<ǫ<1,
T(ǫ)u 2 = ǫ2n fu,h 2 fu,h 2 = fu 2 f 2 u 2,
k f k Xn |h ni| ≤Xn |h ni| k k ≤k k∞k k
e
so T(ǫ) f . (cid:3)
k f k≤k k∞
Wee remark that the same argument as in the last proof also shows that T(ǫ)
f
is bounded, for any f L , on the weighted space L2(R,e x2dx).
∞ −
∈
5. “Semiclassical” asymptotics
The Parsevalidentity
f = f,h h , f L2(R),
n n
h i ∈
Xn
shows that, at least in the weak sense (i.e. as distributions on R R),
×
(30) h (x)h (y)=δ(x y).
n n
−
Xn
Thus formally
T(ǫ)u=fu for ǫ=1,
f
that is, the operator T(ǫ) reduces just to the multiplication operator M on L2(R)
f f
(in the sense explained above) for ǫ = 1. This brings forth naturally the question
of the finer description of the behaviour of T(ǫ) as ǫ 1, in particular, whether
f ր
one has any analogue of the “semiclassical limit” formulas like (1) or (5) in the
traditional procedures.
The latter asymptotics can be found by the usual Laplace (or stationary phase,
or WJKB) method, see e.g. Ho¨rmander [17, 7.7]. Namely, assume for simplicity
that f C (R) and u (R). We have seen§in (27) that
∞
∈ ∈D
T(ǫ)u(x)= (fu)(y)e−(y1−ǫǫx2)2 dy
f ZR − (1 ǫ2)π
−
p dt
= (fu)(ǫx 1 ǫ2t)e t2 .
−
ZR −p − √π
Let us temporarily write, for the sakeof brevity, fu=F. Standardestimates used
in the stationary phase method show that the integration over y outside a small
neighbourhood of x gives an exponentially small contribution as ǫ 1, while in
ր
the integral over that neighbourhood F can be replaced by its Taylor expansion.
Thus we arrive at
F(ǫx 1 ǫ2t)e−t2 dt ∞ F(k)(x) (ǫx 1 ǫ2t)ke−t2 dt
ZR −p − √π ≈Xk=0 k! ZR −p − √π
= ∞ F(j+l)(x)(ǫ 1)lxl( 1 ǫ2)j tje t2 dt
−
jX,l=0 j!l! − −p − ZR √π
∞ F(2k+l)(x) Γ(k+ 1)
= (ǫ 1)lxl(1 ǫ2)k 2
(2k)!l! − − Γ(1)
kX,l=0 2
10 S.TWAREQUEALIANDMIROSLAVENGLISˇ
as ǫ 1. Writing 1 ǫ2 =(1 ǫ)(2 (1 ǫ)) and using the binomial theorem to
ր − − − −
get powers of (1 ǫ) only, we finally get
−
(31) T(ǫ)u(x) ∞ (1 ǫ)k+l+m(fu)(2k+l)(x)xl(−1)l+m2k−m mk
f ≈ − l!k!4k (cid:0) (cid:1)
k,lX,m=0
as ǫ 1. In particular,
ր
f f
(32) T(ǫ)u=fu+(1 ǫ) ′′ xf u+(f xf)u + u +O((1 ǫ)2).
f − h(cid:16) 2 − ′(cid:17) ′− ′ 2 ′′i −
Asimilarapproachcould,ofcourse,beappliedalsotoT(ǫ);however,weproceed
f
touseadifferentargument,whichnotonlyrecoverstheformula(31)(uponpassing
e
from T(ǫ) to T(ǫ) via the relation (26)) but is also shorter and applicable in other
situations.
e
Recall the Schr¨odinger (“number”) operator
x2 1 1 d2
A= − I
2 − 2dx2
which is an (unbounded) self-adjoint operator on L2(R) satisfying Ah = nh ,
n n
n=0,1,2,....
Theorem 5. We have
T(ǫ) =ǫAM ,
f f
where ǫA =eAlogǫ is understoodinethesenseof the spectraltheorem. Consequently,
as ǫ 1,
ր
(33) T(ǫ)u ∞ (logǫ)kAk(fu).
f ≈ k!
Xk=0
e
Proof. Let us keep our shorthand F =uf, assuming for simplicity that F (R).
∈D
Then
F(y)K (x,y)dy = F(y) ǫnh (x)h (y)dy
ǫ n n
ZR ZR Xn
e
= ǫn F,h h (x)
n n
h i
Xn
= F,h ǫAh (x)
n n
h i
Xn
= ǫA F,h h (x)
n n
(cid:16) Xn h i (cid:17)
(logǫ)k
=(ǫAF)(x)= (AkF)(x).
k!
Xk
Recalling that F =fu gives the result. (cid:3)
Of course, using the familiar series
∞ (1 ǫ)j
logǫ= −
− j
Xj=1
one could easily pass in (33) from powers of logǫ to powers of (1 ǫ).
−