1. Generalized Functions, Gelfand Triples, Quantum Field
Theory, and the Imaginary Resolvent Theorem
Michael Maroun
March 3, 2011
1 Flavors of the Modified Feynman Integral
The modified Feynman integral (LI) is a generalization of the Feynman integral based on the
imaginary resolvent theorem (IRT). I will not give the full theorem here nor the corollary
which proves the existence of the modified Feynman integral based on the use of the imag-
inary resolvent. Instead I will give a sense for the consequences of the existence of theses
objects.
• The LI can extend and generalize the Schrodinger evolution to cases where the second
order differential equation may not be defined.
• The IRT is a generalization of the usual physically relevant unitary operator so that
given Ψt “ Uψ together with Φnt “ p1 ` i t
n
Hq´n
ϕ one has that
||Ψt ´ Φnt||L2
nÑ8
ÝÑ 0.
• The LI gives solutions to the generalized Schrodinger evolution when a Sturm-Liouville
prescription may fail.
• The LI gives the form of the Green’s function for the evolution and the Green’s function
has an expansion in the sense of generalized functions.
• The IRT together with a Gelfand triple gives a generalization of the physical Fourier
transform in the sense that just as the eigenfunctions of the one-dimensional Dirac
operator give the usual Fourier kernel, the generalized eigenfunctions associated to the
operator H have spectral measures and generalized eigenfunctions which gives a kernel
that maps the space representation to the energy representation.
1

2. It should be noted that when the spectrum of the energy operator is continuous the use
of Gelfand triples, i.e. the rigged Hilbert space is necessary to make mathematically precise
the statements of the principles of quantum mechanics. It is in this spirit that we investigate
the LI in conjunction with Gelfand triples. There is an important connection between what
has been said thus far and quantum field theory.
Conjecture 1.1 (Definition of a quantum field1
). A quantum field is a family of operator-
valued linear functionals (OVLF) such that for any specific value of the parameter, parame-
terizing the family of OVLF, the result is an operator-valued local complex measure.
(Recall that a local complex measure can be thought of as a family of linear functionals
in its own right, such that restricting the family of linear functionals results in a bona fide
complex measure on bounded subsets (discs) in C.)
Furthermore, these measures are in a sense spectral measures associated to the operator-
valued nature of the family of linear functionals. The spectral measures give generalized
eigenfunction expansions of the fields when restricted to a suitable notion of invariant sub-
domains2
of the generalized Hilbert space or Fock3
space of multi-particle states.
While the superselection principle may allow for compatibility between spontaneous sym-
metry breaking and the Wightman axioms, the importance of gauge theories would suggest
that there is more to the story. Furthermore, being forced to use the Coulomb gauge in
order to create an actual Hilbert space is unsettlingly restrictive. Thus in order to further
develop the investigation, let us consider specific examples of technical interest, which serve
to help elucidate the scope of difficulties associated with the central claims in this section.
2 Specific Relevant Examples
First we want to give examples of distribution theory for, which the distributions are thought
of as generalized functions and have singularities, which are typical of expressions for physical
systems.
1
This definition was inspired by both David Carfi’s realization that the Dirac delta functions appearing
in the Dirac calculus cannot be simply distributions, but must be a family of Dirac functionals. In addition,
the definition was further inspired by Lapidus’s notion of local complex measure from the text [La-vF].
2
Here we mean invariant in the sense of the action of operators on the domain of the operators. In
the usual sense one can accomplish this by taking the intersection of the domains of all the powers of the
operators. This is known as a maximal invariant subspace.
3
Here again there is a technicality that the Fock space may not be general enough. In this case, a natural
extension would be the tensor algebra formed from the Fock space. It is interesting to note that this space
itself is a Hilbert space, as well.
2

3. Example 2.1 (Gelfand-Shilov: The singular function 1
x
). Consider the pairing,
ˆ
1
x
, φpxq
˙
“ “ ”
ż
R
φpxq
x
dx,
for φpxq in the space, DpRq “ C8
0 pRq, of smooth functions with compact support. By using
analytic continuation we can define two different generalized functions, 1
x `
and 1
x ´
with ε ą 0
as follows:
1.
ˆ
1
x`
, φpxq
˙
:“
8´iż
´8´iε
φpxq
x
dx,
2.
ˆ
1
x´
, φpxq
˙
:“
8`iż
´8`iε
φpxq
x
dx,
such that, ˆ
1
x`
´
1
x´
, φpxq
˙
“
ż
|x|“1
φpxq
x
dx “ 2πiφp0q “ 2πi xδ, φpxqy .
Hence, we conclude, in the sense of generalized functions,
1
x`
´
1
x´
“ 2πi δpxq.
As another example we introduce a way of pairing even more singular generalized func-
tions.
Example 2.2. Consider the generalized function given by x´3{2
. We give a slightly different
pairing which for φ P S gives a finite result.
Proposition 2.1 (Gelfand-Shilov). The pairing,
B
1
x3{2
, φ
F
:“
8ż
0
φpxq ´ φp0q
x3{2
ă 8
proof. Indeed, as a specific case consider φpxq P S the space of Schwartz functions, S,
and in particular consider φpxq “ e´x2
. It suffices to check only one such Gaussian function
3

4. as it will generate the entire basis on S. Then,
B
1
x3{2
, e´x2
F
:“
8ż
0
e´x2
´ 1
x3{2
dx
“ lim
xÑ0
2
e´x2
´ 1
?
x
´ 4
8ż
0
?
xe´x2
dx
“ ´2
8ż
0
y´1{4
e´y
dy “ ´2 Γ
ˆ
3
4
˙
ă 8
Next we consider convolution of generalized functions, which are nice in the sense that
they are given by periodic analytic functions.
Example 2.3. Suppose u and v are generalized functions. Then their convolution is defined
by,
xu ˙ v, φy :“
Cż
R
upy ´ xqvpxq dx, φ
G
“
ż
R
ż
R
upy ´ xqvpxqφpyq dy dx
“
ż
R
ż
R
upyqvpxqφpx ` yq dy dx
and in particular let u “ sinpxq and v “ cospxq be the analytic periodic basis then,
xsinpxq ˙ sinpxq, ¨y “
Cż
R
sinpy ´ xq sinpxq dx, ¨
G
“
B
´
1
2
cospyq, ¨
F
But this is hardly the only way to define such a convolution. In fact, there are a few
technical comments in order here. The first is that the integral converges in the sense of the
Ces`aro average. Defined as,
lim
ΛÑ8
ż Λ
0
´
1 ´
x
Λ
¯α
fpxq dx,
where α ľ 0, such that the integral is finite. It is easy to check that if D α ľ 0 such that the
integral is finite then for any β ą α the integral is finite and converges to the same value.
This generalized notion of integral is necessary because although φpxq may be a test function
(e.x. an element of D), φpx ` yq need not be. Nonetheless, such a definition for the periodic
basis of functions is consistent and the definition is thus justified in the sense of averages.
4

5. Example 2.4. Let u and v be generalized functions (typically elements of the vector space
D1
pΩq “ C8
0
1
pΩq with Ω Ă R an open set) such that either,
1. One of u or v has bounded support
2. Both of u and v have support bounded from the left (or below)
3. Both of u and v have support bounded from the right (or above)
Further, let ρ P D such that, ż
supppρq
ρpxq dx “ 1
in such a way that the support of ρ coincides with the support of either one of u or v,
whichever has the (partially) bounded support. Then the convolution of the generalized func-
tions is given by:
xu ˙ v, φyρ :“ xupxq ˆ vpyq, ρpyqφpxqy “
ż
R
ż
R
upxq vpyq ρpyqφpx ` yq dx dy
It is convenient to indicate the presence of ρ passively as a subscript reminiscent of
specifying the state of an expectation value in the usual quantum mechanics.
We will sometimes refer to the fixed point of the Fourier transform map4
on the tempered
distributions S1
, as a (generalized) tempered Gaussian.
Lemma 2.1 (The tempered Gaussian). Let pT denote the Fourier transform, of T P S1
. The
tempered distribution 1 ` δ is an example of a non-trivial fixed point (up to scaling) of the
Fourier transform on the space of tempered distributions and hence a generalized Gaussian
in the following sense, informally,
z1 ` δ “ δ ` 1,
and more precisely as,
x z1 ` δ, φy :“ x1 ` δ, pφy “
ż
pφ ` pφp0q “ φp0q `
ż
φ “ xδ ` 1, φy @ φ P SpRd
q.
proof. First we show that pφp0q “
ş
φ. Recall the definition of the d-dimensional
Fourier transform on a test function,
pφpkq :“
ż
Rd
e´2πik¨x
φpxq dx.
4
in the sense of an automorphism on the space of Schwartz functions, S and their dual space, where it is
not necessarily an automorphism
5

6. But it then follows that
pφp0q “
ż
Rd
φpxq dx.
hence pφp0q “
ş
φ. Next we would like the reverse to be true so that,
ş
pφpkq “ φp0q. Easily
this is seen by recalling that the Fourier transform is an automorphism on SpRd
q, and so the
integral
ş
pφpkq also necessarily converges. Finally, the Dirac delta tempered distribution is
the generalized Fourier transform of the unit constant function, thus,
ż
Rd
pφpkq dk “
ż
Rd
ż
Rd
e´2πik¨x
φpxq dx dk “
ż
Rd
φpxq δpdxq “ φp0q.
hence, 1 ` δ, is a distributional fixed point5
and thus the tempered Gaussian.
As motivated above, we now make note of the following two identities,
´i {p1 ` iδq “ 1 ´ iδ
i {p1 ´ iδq “ 1 ` iδ
These are the simplest examples of a scattering matrix. The Fourier transform, transforming
the forward scattering matrix into the backward and vice versa. They are formally unitary
in the following sense,
´i ¨ i
{{p1 ´ iδq “
{{p1 ´ iδq “ 1 ´ iδ “ ´i {p1 ` iδq “ 1 ´ iδ
Hence as is well-known, the Fourier transform is a unitary map on S1
by extension of the usual
isometric automorphism (unitary) Fourier transform from S into itself. This is trivial in that
S is always contained in its dual S1
. But as we have just seen, there are elements in S1
that
are not in S. But with the use of the 4 eigenvalues of the Fourier transform 1, ´1, i, ´i, we
were able to construct a unitary-like behavior on certain tempered distributions. Obviously,
every distribution of the form T ` pT is also a fixed point of the Fourier transform, as well as
the infinite number of fixed points in S itself given by
φ0pxq “ h4np
?
πxq e´πx2
,
where hnpxq are Hermite polynomials with n P N0.
5
i.e. ˆT “ T and the equality is read in the sense of distributions
6

7. Regularization
Next, we examine some transparent shortcomings of known regularizations. Consider the
well-known dimensional regularization of a loop integral from quantum field theory. Af-
ter a Wick rotation, the (3+1)-spacetime dimensional integral becomes a (4+0)-spacetime
dimensional integral over Euclidean R4
with p P R4
and m ľ 0.
ż
R4
d4
p
p2πq4
1
pp2 ` m2q2
The dimensional regularization procedure starts by replacing the integer dimension 4 with a
continuous single complex variable that we will call ω. Hence the above is replaced by,
ż
Rω
dω
p
p2πqω
1
pp2 ` m2q2 .
By reducing the dimension ω by 1, we can separate out the radial part from the angular part
of the integral as follows.
ż
Rω
dω
p
p2πqω
1
pp2 ` m2q2 “
2 π
ω
2
Γ
`ω
2
˘
8ż
0
dp
p2πqω
pω´1
pp2 ` m2q2 “
mω´4
Γ
`
2 ´ ω
2
˘
2ωπ
ω
2
Now replace ω with the quantity 4 ´ ε and expand the remaining expression about ε “ 0.
mω´4
Γ
`
2 ´ ω
2
˘
2ωπ
ω
2
ÝÑ
p2
?
πqε
Γ
`ε
2
˘
16π2 mε
„
1
16π2
„
2
ε
´ γ ´ logpm2
q ` logp4πq ` Opεq

This expression is clearly divergent as ε tends to 0. The standard lore is that the 1
ε
pole
cancels with other relevant diagrammatic calculations and the physical contribution is finite.
This is all well and good however as a form of regularization it is a total failure.
Instead, consider the same logarithmically divergent loop integral and calculate the finite
angular term leaving only the divergent radial term as such,
ż
R4
d4
p
p2πq4
1
pp2 ` m2q2 “ 2π2
8ż
0
p3
pp2 ` m2q2
dp
Now in place of calculating this last divergent integral, we first change to dimensionless
variables and integrate to a cut-off Λ. This is quite standard and leads to the familiar
expression,
8ż
0
p3
pp2 ` m2q2
dp
N
mż
0
x3
px2 ` 1q2 dx,
7

8. where m is the mass of the particle in the field theory, is an intrinsic spacing and N is the
diverging Minkowski content.
As an aside, it is worth noting that the quantity N is reminiscent of the Bohr quanti-
zation ansatz that the angular momentum is quantized as n . As is well-known, the correct
quantization turned out to be npn ` 1q 2
for the square of the angular momentum operator.
For large n the two expressions are very close. In particular, for our case we separate out
the divergence due to “the size” of the set and use as a scaling that crudely represents
the completion of the rational numbers. However, a theorem by Ostrowski asserts that any
sensible notion of valuation on the set of rational numbers is either the usual absolute value
function of the real numbers or the p-adic valuation, where the rationals have been com-
pleted in a fashion that results in the p-adic numbers. The point being, we leave within the
framework of this theory a notion of measure, which is deliberately ambiguous to indicate
that the spectrum of the momenta need not be assumed to have the form of an element of
the set Rd
in d-dimensions.
Now, we consider a discrete sum of the same functional form as the radial part, the
partial sum,
N
mÿ
n“0
n3
pn2 ` 1q2
,
and form the quotient limit,
lim
NÑ8
N
mż
0
x3
px2 ` 1q2 dx
N
mÿ
n“0
n3
pn2 ` 1q2
“1
There are several reasons for the choices thus far. First, it is necessary that one create an
unitless expression for comparison with the corresponding discrete sum. Second, because
of the freedom in choice for a change of variables, essentially any finite value for the above
difference can be obtained. This is highly arbitrary and to avoid this, we make use of the
physical situation. The physical circumstances show that the divergences are due to the
radial part of the momentum integral. We make the ansatz here that the divergence is due
to our ‘measure’ of Minkowski content and not as a result of any intrinsically infinite physical
quantity. The mass of the particle m and the intrinsic physical spacing both carry physical
units and are of the same unit-dimension.
We shall call this type of regularization, continuum regularization. It is simple to
show that this difference limit will always exist because the rate of divergence of the Riemann
integral will always be the same as the rate of divergence of the associated partial sum by
8

9. virtue of the definition of the integral. In the limit, N Ñ 8 and Ñ m, we obtain for the
original integral,
ż
R4
d4
p
p2πq4
1
pp2 ` m2q2
reg
:“ ´
π2
2
“
2ψp1 ´ iq ` 2ψp1 ` iq ´ iψp1q
p1 ´ iq ` iψp1q
p1 ` iq
‰
« ´9.7071,
where ψpxq and ψp1q
pxq are the digamma and trigamma functions, respectively. Lastly, there
is a very useful and significant discontinuity in this quantity as a function of . If instead
we take the limit Ñ 8 the entire sign of the integral changes! This limit is equivalent to
the limit m Ñ 0. Physically, this corresponds to a theory of massless fermions and as is
well-known physically, the sign indeed does reverse.
Local Representations
Several Theorems regarding the existence of representations and the global form of these
representations have been proved by A. Jaffe et al. Here we examine the local representations
of quantum fields in the context of the neutrix calculus of van der Corput and Hadamard,
the operational calculus of J. Mikusinski and integral transforms of generalized functions.
First, consider a generalized convolution, which is not commutative, in the sense that
f ˙ g ‰ g ˙ f. In the context of quantum field theory, we have the formal expression,
“
appq, a:
pqq
‰
“ δpp ´ qq
It is common to invoke the smeared fields as employed by A. Jaffe et al, J. Klauder and so on
but this prohibits local manipulation. While the physical content of the fields do not depend
upon the specific local data, the physicist’s ability to make statements concerning the system
is very much a local phenomena or even paradigm. Aferall, all experience is first built with
the macroscopic classical world and corresponding statements about physical systems are
made using the local statements of the standard calculus since the time of Newton. Hence,
it is desirable to have an analogous local generalized calculus. From the above equation,
we re-interpret the point-wise multiplication of the appq and a:
pqq as a non-commutative
multiplication such that,
“
appq, a:
pqq
‰
:“
ż
Drzpp´q, εqs
apz ´ ωq a:
pωq dω ´
ż
D:rzpp´q, εqs
a:
pz ´ ωq apωq dω
“
1
2πi
„
1
p ´ q ´ iε
´
1
p ´ q ` iε

“ δpp ´ qq,
in the sense of generalized functions such that Drzpp ´ q, εqs and D:
rzpp ´ q, εqs are open
sets and δ is defined on Drzpp´q, εqs
Ş
D:
rzpp´q, εqs. It may happen through applications
9

10. of an operational calculus that appq and a:
pqq have no representation in terms of a locally
integrable function. Nonetheless, they are useful local representations for purposes of local
manipulation.
The Resolvents
Theorem 2.1. If the space of test functions DpRd
q “ SpRd
q, the space of Schwartz functions
and if for every choice of the sequence δn we have, δn P SpRd
q, @ n such that δn
nÑ8
ÝÑ δpxq,
the Dirac delta measure, then,
1
1 ` δp
pxq
“ 1,
where 1 is the usual Lebesgue measure on Rd
, 0 ĺ p ă 8, x P Rd
with d P N and the equality
is in the sense of distributions.
proof. Because of the Gaussian basis in SpRd
q, it suffices to check the Gaussian func-
tion e´a}x}2
together with a Gaussian delta sequence. By way of a direct calculation, we
have,
B
1
1 ` δ
, e´ax2
F
:“ lim
nÑ8
C
1
1 `
an
π
e´nx2 , e´ax2
G
“ lim
nÑ8
8ż
´8
e´ax2
1 `
an
π
e´nx2
“ lim
nÑ8
1
?
n
8ż
0
y´1
2 e´ a
n
y
1 `
an
π
e´y
“ lim
nÑ8
c
π
n
Φ
ˆ
´
c
n
π
,
1
2
,
a
n
˙
,
where Φpz, s, aq is the Lerch zeta function, defined by the following integral and sum repre-
sentations,
Φpz, s, aq “
8ÿ
k“0
zk
pk ` aqs
“
8ż
0
ts´1
e´a t
1 ´ z e´y
dy for paq ą 0, psq ą 0, z ă 1.
Note that this gives the identity,
Φpz, s, aq “ a´s
` z Φpz, s, a ` 1q
10

11. But then the limit,
lim
nÑ8
c
π
n
Φ
ˆ
´
c
n
π
,
1
2
,
a
n
˙
“ lim
nÑ8
„c
π
a
´ Φ
ˆ
´
c
n
π
,
1
2
,
a
n
` 1
˙
.
It is now convenient to write,
Φ
ˆ
´
c
n
π
,
1
2
,
a
n
` 1
˙
“
8ÿ
k“0
`
´
an
π
˘k
pk ` 1 ` a
n
q
1
2
.
By the squeeze theorem we have,
lim
nÑ8
ˇ
ˇ
ˇ
ˇ
ˇ
8ÿ
k“0
`
´
an
π
˘k
pk ` 1 ` a
n
q
ˇ
ˇ
ˇ
ˇ
ˇ
ĺ lim
nÑ8
ˇ
ˇ
ˇ
ˇ
ˇ
8ÿ
k“0
`
´
an
π
˘k
pk ` 1 ` a
n
q
1
2
ˇ
ˇ
ˇ
ˇ
ˇ
ĺ lim
nÑ8
ˇ
ˇ
ˇ
ˇ
ˇ
8ÿ
k“0
ˆ
´
c
n
π
˙k
ˇ
ˇ
ˇ
ˇ
ˇ
and the asymptotic properties of the summation on the lower bound together with the sum
identity on the upper bound give,
lim
nÑ8
„c
π
n
log
ˆ
1 `
c
n
π
˙
ĺ lim
nÑ8
ˇ
ˇ
ˇ
ˇ
ˇ
8ÿ
k“0
`
´
an
π
˘k
pk ` 1 ` a
n
q
1
2
ˇ
ˇ
ˇ
ˇ
ˇ
ĺ lim
nÑ8
«
1
1 `
an
π
ff
,
hence
0 ĺ lim
nÑ8
ˇ
ˇ
ˇ
ˇ
ˇ
8ÿ
k“0
`
´
an
π
˘k
pk ` 1 ` a
n
q
1
2
ˇ
ˇ
ˇ
ˇ
ˇ
ĺ 0.
Thus finally we conclude,
B
1
1 ` δ
, e´ax2
F
“
c
π
a
“ x1, e´ax2
y.
It would appear at first that this result is highly restrictive as it requires both the test
functions and the delta sequence to be of Schwartz type. However, in light of the Gelfand
triple, S Ď L2
Ď S1
, where S as a topological vector space is ‘rigged’ with a continuous
imbedding (the inclusion map or canonical injection) into L2
together with S being dense in
L2
gives a natural more robust meaning to the above results.
In particular, we follow Maurin, Gelfand and Vilenkin et. al in reconciling how operators
on S1
act. The resolution is quite natural in the theory of distributions in the sense that
whenever sense can be made of a self-adjoint operator A : S Ñ S : φ ÞÑ ψ for the pairing:
xT, ψy “ xT, Aφy “ xAˆ
T, φy “ xS, φy
where S, T P S1
, φ, ψ P S and Aˆ
: S1
Ñ S1
a suitable extension of the self-adjoint operator.
In particular the extension is such that the pairing x¨, ¨y is compatible with the inner product
on L2
, keeping in mind that L2
can be replaced with any Hilbert space and in particular as
pointed out by Maurin, a direct integral of Hilbert spaces.
11

12. A Purely Speculative Note
The notion of the spectral measures mentioned in the conjecture is not the usual notion of
spectral measure since it is most likely the case that for each distinctly different physical
system the measures will need to be defined in such different ways that the ad hoc procedures
may not be related in any obvious way.
Nonetheless, it is my belief that in order that a physical theory be mathematically consis-
tent and physically robust, it must be able to over come significant changes in the dynamics
due to changes in the energy that the physical phenomena is observed at. Currently, the
method for doing this is Wilson’s renormalization group procedure. But renormalization,
while it gives a reasonable interpretation to phase transitions and spontaneous symmetry
breaking, is not a construct that is independent of energy and interaction type. In particular,
it is well-known that non-renormalizable theories like gravity cannot be incorporated into
this framework6
.
Therefore, one may wonder as to whether the relationship between the operator-valued
spectral measures of different theories may fall into an ultimate flow of measures as described
briefly in [Lap3]. While it is entirely speculative, it would be quite beautiful and spectacular
if the dynamics of different energy regimes and the differences between various interactions of
the known forces of nature fell into a universal dynamics connected to the very fabric of the
geometry of the universe itself, and embodied in the very mathematics of the mathematical
notions of volume, distance, dimension, content, and capacity. That is, the interplay between
the different mathematical notions of ‘set size’, and the geometry physical reality could finally
be made precise.
A Appendix
Definition A.1 (The Rigged Hilbert Space). A rigged Hilbert space is a pair pH, Φq with H
a Hilbert space, Φ a dense subspace, such that Φ is given a topological vector space structure
for which the inclusion map i is continuous. Using the Riesz representation theorem, one
identifies H with its dual space Hˆ
, the adjoint to i is the map iˆ
: H “ Hˆ
Ñ Φˆ
. The
duality pairing between Φ and Φˆ
has to be compatible with the inner product on H, i.e.
xu, vyΦˆΦˆ “ pu, vqH whenever u P Φ Ă H and v P H “ Hˆ
Ă Φˆ
.
Note that in the special case where the topological vector space Φ is itself a Hilbert
space (and hence isomorphic to its own dual), the isomorphism to Φˆ
is not the same as
the composition of the inclusion i with its adjoint iˆ
, i.e. iˆ
i : Φ Ă H Ñ H “ Hˆ
Ñ Φˆ
is
different than the identity mapping, id : Φ » Φˆ
Ñ Φˆ
» Φ.
6
Loop quantum gravity and other novel space-time lattice regularizations not withstanding
12

13. Definition A.2 (Positive Part of a Map). We call the positive part of a map V : Rd
Ñ R,
V` :“ maxpV, 0q.
Definition A.3 (Negative Part of a Map). We call the negative part of a map V : Rd
Ñ R,
V´ :“ maxp´V, 0q.
Definition A.4 (Form Boundedness). Given a semi-bounded self-adjoint operator A and a
symmetric densely defined operator B, we say that the operator B is A-form bounded with
bound less than 1 iff QpAq Ď QpBq and D 0 ă γ ă 1 and δ ą 0 such that,
|qBpφq| ĺ γqApφq ` δ||φ||2
@ φ P QpAq,
where qA is the quadratic form associated to the operator A and QpAq is the domain of the
quadratic form. The infimum of all such positive numbers γ is called the A-form bound of B
and is strictly less than one.
The Imaginary Resolvent Theorem (IRT) is a generalization of the unitary operator and
Feynman integral.
Theorem A.1 (Lapidus (Imaginary Resolvent) [JoLa]). Let the potential V : Rd
Ñ R be
Lebesgue measurable and such that the positive part of the function V` is in L1
locpRd
q and the
negative part V´ is H0 “ ´1
2
2
-form bounded with bound less than 1, futher let ϕ P L2
pRd
q
then @ t P R,
Φnpt, xq :“
“`
r1 ` i t
n
H0s´1
r1 ` i t
n
V s´1
˘n
ϕ
‰
pxq
defines a function which is in L2
pRd
q and is given for almost every x P Rd
by the modified
Feynman integral (LI).
Furthermore, there is a function ψ : R ˆ Rd
Ñ C such that ψpt, ¨q P L2
pRd
q @ t P R and
||ψpt, ¨q ´ Φnpt, ¨q||L2 Ñ 0, as n Ñ 8,
where the limit is uniform in t on all bounded subsets of R. In fact, the function ψpt, ¨q is
given by the action of the unitary group e´itH
on ϕ, that is,
ψpt, xq “
`
e´itH
ϕ
˘
pxq, for L.a.e. x,
where the operator H :“ H0 ` V is the form sum of the self-adjoint operators H0 and V .
Finally, for ϕ P DpHq, ψpt, xq is the unique solution in the sense of semi-groups of the
Schrodinger equation
Bψ
Bt
“ ´iHψ, ψp0, ¨q “ ϕ
with initial state ϕ.
13

14. Theorem A.2 (Reed-Simon [ReSi1]). The operator, H :“ ´1
2
2
´ αδpxq with α ą 0, is
essentially self-adjoint in L2
pRn
q, n ľ 4.
Corollary A.1 (Albeverio-Hœgh-Krohn [AGHH]). There are no point interactions (bound
states) in Rn
, for n ą 3. That is, the discrete spectrum of the operator H :“ ´1
2
2
´ αδpxq
denoted, σdispHq is given by σdispHq “ H for α ą 0 and n ą 3. The remainder of the
spectrum is continuous, i.e. σpHq “ σcontpHq “ r0, 8q, on account of the operator, in
dimensions greater than 3, being bounded from below and essentially self-adjoint.
References
[AG] Al-Gwaiz, M. A., Theory of Distributions. Chapman & Hall/CRC Press/Marcel
Dekker, New York, 1992.
[AGHH] Albeverio, S., Gesztesy, F., Hoegh-Krohn, R. and Holden, H., Solvable Models in
Quantum Mechanics, 2nd
Ed. AMS Chelsea Publishing, Providence, 2005 (reprint).
[AnTr] Antoine, J-P. and Trapani, C., Partial Inner Product Spaces: Theory and Applica-
tions. Springer, New York, 2009.
[BiLa] Bivar-Weinholtz, A. and Lapidus, M. L., Product formula for resolvents of normal
operators and the modified Feynman integal. Proc. Amer. Math. Soc. 110 2 (1990),
449-460.
[Bo] Bohm, A., Quantum Mechanics: Foundations and Applications. Springer, New York,
1993.
[Do] Donoghue, W. F., Distributions and Fourier Transforms. Academic Press, New York,
1969.
[FrJo] Friedlander, F. and Joshi, M., Introduction to the Theory of Distributions. Cam-
bridge U. Press, Cambridge, 1999.
[GeSh1] Gel’fand, I. and Shilov, G., Generalized Functions: Properties and Operations. I,
Academic Press, New York, 1964.
[GeSh2] Gel’fand, I. and Shilov, G., Generalized Functions: Properties and Operations. II,
Academic Press, New York, 1964.
[GeSh3] Gel’fand, I. and Shilov, G., Generalized Functions: Properties and Operations. III,
Academic Press, New York, 1964.
14

15. [GeVi] Gel’fand, I. and Vilenkin, N., Generalized Functions: Applications of Harmonic
Analysis. IV, Academic Press, New York, 1964.
[Ha] Haag, R., On quantum field theories. Dan. Mat. Fys. Medd. 29 12 (1955), 1-39.
[Hu] Hughes, R., A version of the Trotter product formula for quadratic-form perturba-
tions. J. London Math. Soc. 2 30 (1984), 322-334.
[JeJoNi] Jefferies, B., Johnson, G. W., and Nielsen, L., Feynman’s operational calculi: spec-
tral theory for noncommuting self-adjoint operators. Math. Phys. Anal. Geom. 10
(2007), 65-80.
[JoLa] Johnson, G. and Lapidus, M., The Feynman Integral and Feynman’s Operational
Calculus. Oxford U. Press, New York, 2000.
[La1] Lapidus, M., Generalization of the Trotter-Lie formula. Integral Equations Operator
Theory 4 (1981), 366-415.
[La2] Lapidus, M., Product formula for imaginary resolvents with application to a modified
Feynman integral. J. Funct. Anal. 63 3 (1985), 261-275.
[Lap3] Lapidus, M., In Search of the Riemann Zeros. American Mathematical Society, Prov-
idence, 2008.
[La-vF] Lapidus, M. and van Frankenhuijsen, M., Fractal Geometry, Complex Dimensions
and Zeta Functions. Springer, New York, 2013.
[Ma] Maurin, K., General Eigenfunction Expansions and Unitary Representations of
Topological Groups. Polish Scientific Publishers, Warsaw, 1968.
[ReSi1] Reed, M. and Simon, B., Methods of Modern Mathematical Physics. v1, Functional
Analysis. Academic Press, New York, 1980.
[ReSi2] Reed, M. and Simon, B., Methods of Modern Mathematical Physics. v2, Fourier
Analysis, Self-Adjointness. Academic Press, New York, 1980.
[ReSi4] Reed, M. and Simon, B., Methods of Modern Mathematical Physics. v4, Analysis of
Operators. Academic Press, New York, 1978.
[Ru] Rudin, W., Functional Analysis. McGraw-Hill, New York, 1991.
[Tr] Treves, F., Topological Vector Spaces, Distributions and Kernels. Academic Press,
New York, 1967.
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