Supersymmetric Resolvent-Based Fourier Transform

We calculate in a numerically friendly way the Fourier transform  of a nonintegrable function, such as ( ) 1 x φ = , by replacing  with 1 −   , where  represents the resolvent for harmonic oscillator Hamiltonian. As contrasted with the non-analyticity of ( ) ( ) 1 2 2 x a x φ − + at i x a = ± in the case of a simple replacement of  by ( ) ( ) 1 2 2 2 2 ˆ ˆ p a q a − + +  , where p̂ and q̂ represent the momentum and position operators, respectively, the φ  turns out to be an entire function. In calculating the resolvent kernel, the sampling theorem is of great use. The resolvent based Fourier transform can be made supersymmetric (SUSY), which not only makes manifest the usefulness of the even-odd decomposition of φ in a more natural way, but also leads to a natural definition of SUSY Fourier transform through the commutativity with the SUSY resolvent.

  , where  represents the resolvent for harmonic oscillator Hamiltonian.As contrasted with the non-analyticity of ( ) ( ) x a x ϕ − + at i x a = ± in the case of a simple replacement of  by ( ) ( )

Introduction
Fourier transform (FT) : which is a unitary operator, is a fundamental method in function analysis and is applied to many fields in physics.The corresponding self-adjoint operator is given by the harmonic oscillator Hamiltonian ( ) ( ) where ( )( ) ( ) ) is a simultaneous eigenfunction of  and  , with their eigenvalues given by i n and n , respectively.
If a function : ϕ →   is integrable, its FT is well defined.However, if the function ϕ is not integrable, for example ( ) = , its FT should be regarded as a generalized function.To calculate the FT of ( ) x ϕ = in a numerically friendly way, one of the methods is to replace  by , and to choose  as the resolvent for  , that is [1] ( ) ( ) Considering that  includes the term proportional to 2 q , we find that ( )( )  can be Fourier transformed.
To make ( ) x ϕ = square integrable, it is sufficient to reduce the order of ( ) x ϕ (for x → ∞ ) by one, not necessarily by two.This implies that it is sufficient to choose     , as given above.However, the square root of the operator  , in general, is somewhat complicated to deal with, so we adopt an alternative approach, supersymmetrization.The supersymmetry (SUSY) can be realized by adding  in (1) to † f f [2], where † , : f f →   , representing the fermionic creation and annihilation operators, respectively, satisfying 2 0 f = , ( ) , as is analogous to (2).
The aim of this paper is replace ′  by ( ) ( ) with α ∈  chosen in an appropriate way, to finally find that the introduction of SUSY clarifies the availability of the even-odd decomposition of ϕ in a more natural way.In Section 2, we generalize the resolvent kernel for  , where  can be regarded as the specialization of the Hamiltonian  whose eigenfunction is given by the Jacobi polynomial.In calculating the resolvent kernel, the sampling theorem [3] is fully employed.In Section 3, we first reexamine the FT of ( ) x ϕ = , based on the resolvent for  .Then we compare the resolvent based method with other methods, to find that the former has some merits of being numerical calculation friendly and free of singularity for ( ) , even after analytic continuation.Analytic property is significant for calculating, for example, path integral in Minkowski space (Wick roration), and the Shannon entropy in the limit of the Rényi entropy (replica trick).We give conclusion in Section 4.

Methods
In this section, we first obtain the resolvent kernel for the Hamiltonian whose eigenfunction is given by the Jacobi polynomial.Then we calculate the resolvent kernel for  as a specialization of the former.

Jacobi Polynomial
: where ( ) ( ) represent the Jacobi polynomial and its normalization constant as , ; ; F a b c z the Gamma function and the hypergeometric function, respectively.The corresponding eigenvalue is given by The resolvent kernel for ) can be expanded using the eigenfunctions ( ) ( ) , where in the second and third equalities, use has been made of the completeness for and (4), respectively.
There seems to be no such formula as the series sum of (5) for general parameters α and β .However, it will be found that the sum can be represented as the product of two hypergeometric functions as follows.The starting point would be the following formula, which corresponds to the particular case of ( ) ( ) is defined by replacing n in (3) with ν .Before proceeding further, we discuss the validity of (6).By applying to (6) from the left, it is found that both sides of (6) satisfy the same second order differential equation for 0 x y + ≠ , due to the completeness relation of ( ) ( ) ( ) ( ) ( ) . The reason of restricting ( ) , x y to 0 x y + > is as follows.To avoid the singularity of ( ) x y δ + at 0 x y + =, ( ) , x y should be restricted to either 0 x y + > or 0 x y + < .
Moreover, to avoid the singularity of ( ) ( ) Furthermore, it should be noted that the left-hand side of ( 6) turns out to be Thus the relation of ( 6) can be rewritten as where ( ) , so that the sampling theorem [3] can be applied to ( ) ( ) . The sampling theorem states that for : supp ⋅ represents the support.Hence the validity of ( 7) is guaranteed by To show it, it is convenient to use the integral representation for ( ) cos P ν θ as [6]   ( ) ( ) ( ) Here, we have used the integral representation for the Dirac delta as ( ) we can eventually prove the relation of ( 7) by employing the sampling theorem.
Before proceeding further, we try to rewrite the summation relation in the righthand side of (8) in terms of the Dirac notation as where ( ) where : W →   represents the window function as The relation of ( 10) should be compared with [In the usual Dirac notation, k is reserved for a Fourier transformed variable, so that k  may be simply written as k .Actually, if we formally write f k  as ( ) ( ) where use has been made of the unitarity of  as † 1 − =   .In this sense, x k  can be simply written as x k .]Notice that (10) cannot be derived from ( 11) by formally setting k ∈  to n ∈  .This is because n in ( 10) can be applied only to Notice further that the following relation can be derived from ( 10): where we have used . The relation of (12) indicates that the completeness relation , so that ( ) ( ) ( ) . These completeness relations, along with the orthogonal relations, are recapitulated in Table 1, while some examples of ( ) f ν satisfying (9) are listed in Table 2. Now we go back to generalize the relation of (6).Using the integral representation for given by ( ) , , j N =  ); and so on.It should be remarked that ( ) f ν can be chosen as a more generalized function where ( ) . For the case where ( ) f ν is given by ( ) ( )

Parameters References
( ) ( ) ( ) ( ) where ( ) ( ) , we find that , and more generally to As a special case of ( ) where ν is given by a polynomial with respect to ν ).
For ( ) ( ) , representing the Gegenbauer function, we have the following relations: for , , Then it is found that the sum over n ∈  in the right-hand side of ( 13) can be replaced by the sum over n ∈  as S. Kuwata where use has been made of ( ) ( ) for all n ∈  .Once we have replaced the right-hand side of ( 13) by that of (14), it is not necessary to restrict the parameter λ to either and the righthand side of ( 14) satisfy the same second order differential equation for 0 x y + ≠ , de- spite the value of λ .By re-parameterizing λ in the right-hand side of ( 14) as 1 2 α + , the relation of ( 6) is generalized to where use has been made of ( ) ( ) ( ) for all n ∈  .The relation of (15) can be further generalized.Recall that , which is proportional to the Jacobi function.Following an analogous procedure for manipulating the Gegenbauer function above, we finally obtain [1] where use has been made of the relation

Hermite Polynomial
In this subsection, we obtain the resolvent kernel for  , whose eigenfunction is given by the Hermite polynomial ( ) then we obtain from (15), together with the asymptotic expansion as (for λ → ∞ ), the following formula: = 2 e 2 , ; ; ; a c z F a c z Φ = , the confluent hypergeometric function.Considering that , we find that the sum over n in the left-hand side of ( 17) can be formally extended to all n ∈  .Thus, ( ) ( ) ( ) satisfies the relation of ( 9) for 0 x y + > (listed in the fourth row in Table 2).
For later convenience, we divide the left-hand side of (17) into even and odd parts as for all n ∈  , we obtain from (17) where use has been made of the following formulae: The condition of y x > comes from the intersection of 0 x y + > and ( ) Substituting ( 19) into (18), and using ( ) ( ) ( ) again, we obtain the relation that is valid not only for y x > but also for ( ) for , which was derived from a somewhat more straightforward approach [1].

S. Kuwata
In a practical application, it is convenient to choose the parameter ν so that the y -dependence of ( ) ( ) , h x y ε ν may be written as simply as possible.Considering that ( ) n H y is given by a polynomial of y of order n , we can choose ν as 0 for 1 ε = .
In the case of 0 ε = , however, ν cannot be chosen as 0, due to the divergence of ( ) ( ) , but can be chosen as 1.To summarize, we have , h x y has been listed in Ref. [9].
At the end of this subsection, we deal with the sampling-theorem based summation formula for a single Hermite function of the form where the coefficient ν γ ∈  is to determined in such a way that the sum over n in the left-hand side can be formally extended to all integers, namely, Bearing the specialization of ( 16) in mind, we find that the corresponding summation formula for a single Gegenbauer function is given by ( Actually, the left-hand side of (21) can be rewritten as where use has been made of ( ) ( ) Under the specialization of (16), we finally obtain from (21) where originates from the condition of 0 1 x < < in (21), which is equivalent to cos x θ = , with π 2 θ < (corresponding to the case of 2 k = in the first row in Table 2).The relation of ( 22) is listed in Ref. [10], in which

Results and Discussion
In this section, we first deal with the FT based on the resolvent for  .In a matrix representation of † 2 2 , : where ( ) e .
In this case, ′  turns out to be unitary due to the self-adjointness of ′  , and is By the commutativity [ ] , it follows from (23) and ( 25) where the second relation can de derived from the conjugate of the first relation (recall The resolvent for  can be written using ( ) The validity of ( 27) is verified by ( )( ) ( ) ( ) Recall that in Section 2, a convenient choice of the resolvent parameter a in a  is given by 0 (or 1) for an odd (or even) function.This corresponds to the choice of α in (27) as 1, with ( ) where ( ) ( ) ( ) , together with ( ) ( ) ( ) As a simple application, let us reconsider the FT of ( ) L ×   , we can formally apply ( ) to φ , with the result that ( ) can be Fourier transformed.A series of calculations yields where the k φ 's (for 1, 2, 3 k = ) are given by For ( ) D x , see Table 3.
Notice that ( ) , as is expected from the property that  behaves like the multiplication by x in the limit of x → ±∞ .Bearing in mind that we have the relation  ± Γ = ±  , projection on the even or odd parity space.Thus, it is found that the first (second) element in k φ is parity odd (even).
In the latter half of this section, we discuss the FT of ( ) 1 x ϕ = in another method.
Some may point out that the result of (31) can be derived more efficiently from a method where  is replaced by ( ) which is schematically shown as we find that 1  can be chosen as such that depends on q only (so that 2  depends on p only), in order to calculate 1 ϕ  in quite a simple way (we call such a case a classical method).To further simplify the calculation by 2  , the functional form of 2  is given by a polynomial of p .Considering the condition of ( ) ( ) Although the Clifford FT, in itself, is defined in various ways [11] [12] [13] [14], mainly due to the non-commutativity of the algebra, the resolvent based calculation will still be of use, despite the non-commutativity. necessarily  is called a supercharge.Under the modifica- tion which we obtain the orthonormality relation mn m n δ = for all , m n ∈  .The relation of (9) implies that the completeness relation 1 it is applied to f such that [ ] supp π,π f ⊆ −  .Moreover, interpreting f  and n  as f  and n  , respectively, we can formally obtain from (9) superscripts , α β in the left-hand and right-hand sides are ex- changed.

2 ′
. It should be noted that the φ in (28) is the eigenfunction of ≅ − ⊕    , with its eigenvalue being unity, that is

.
an odd function of x , we find that the first (second) element in k φ (for 1, 2, 3 k = ) in (30) is given by an odd (even) function.It should be noticed that this property holds for a general ( ) The reason is as follows.From [ ] .As compared with other methods, it is hard enough to calculate 3ϕ from 2 ϕ in the classical method 2, due to an infinite number with other unitary transforms, for example, the Hankel transform, whose eigenfunction is given by the Laguerre polynomials Using the resolvent for the corresponding Hamiltonian, we can obtain an analogous result.Another is to generalize :

Table 1 .
Orthogonal relation and completeness relation, where x ∈  .

Table 2 .
Examples of ( ) ν represent the Legendre and Hermite functions, respectively.Here, 