Fourier transform (FT) by

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 by

where and, through the relation

The validity of (2) is verified by noticing that the Hermite polynomial (multipled by) is a simultaneous eigenfunction of and, with their eigen- values given by and, respectively.

If a function is integrable, its FT is well defined. However, if the function is not integrable, for example, its FT should be regarded as a generalized function. To calculate the FT of in a numerically friendly way, one of the methods is to replace by such that, and to choose as the resolvent for, that is [1]

Considering that includes the term proportional to, we find that behaves like for. Thus can be Fourier transformed.

To make square integrable, it is sufficient to reduce the order of (for) by one, not necessarily by two. This implies that it is sufficient to choose, not necessarily, as given above. However, the square root of the operator, in general, is somewhat complicated to deal with, so we adopt an alter- native approach, supersymmetrization. The supersymmetry (SUSY) can be realized by adding in (1) to [2] , where, representing the fermionic creation and annihilation operators, respectively, satisfying, , and, with. The modified Hamiltonian can be decomposed into, where is called a supercharge. Under the modifica- tion, it is natural to transform to, as is analogous to (2).

The aim of this paper is replace by

with chosen in an appropriate way, to finally find that the introduction of SUSY clarifies the availability of the even-odd decomposition of 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, 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 signi- ficant 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.

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.

In this section, we first deal with the FT based on the resolvent for. In a matrix representation of as

the supercharge can be written as

where and. The corresponding SUSY Hamiltonian is given by

which amounts to, where (can be simply denoted by, because commutes with all the elements generated by, and). Under the transformation, it is natural to transform FT as

In this case, turns out to be unitary due to the self-adjointness of, and is related to through

By the commutativity, so is, it follows from (23) and (25) that

where the second relation can de derived from the conjugate of the first relation (recall that is unitary, so that).

The resolvent for can be written using as

The validity of (27) is verified by. Recall that in Section 2, a convenient choice of the resolvent parameter in is given by 0 (or 1) for an odd (or even) function. This corresponds to the choice of in (27) as 1, with to which is applied being given by

where. It should be noted that the in (28) is the eigenfunction of, with its eigenvalue being unity, that is

where represents the space inversion

The relation can be formally derived from and , together with for all.

As a simple application, let us reconsider the FT of, in which.

Although the in this case does not belong to, we can formally apply to, with the result that can be Fourier transformed. A series of calculations yields

where the 's (for) are given by

For, see Table 3.

Notice that for, as is expected from the property that behaves like the multiplication by in the limit of. Bearing in mind that we have the relation

by the commutativity, so that, then we again obtain

Recalling that is an odd function of, we find that the first (second) element in (for) in (30) is given by an odd (even) function. It should be noticed that this property holds for a general in (28), not necessarily for

. The reason is as follows. From, together with (29), it is re- quired that

where, projection on the even or odd parity space. Thus, it is found that the first (second) element in is parity odd (even).

In the latter half of this section, we discuss the FT of 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

Rewriting (26) as

we find that can be chosen as such that depends on only (so that depends on only), in order to calculate in quite a simple way (we call such a case a classical method). To further simplify the calculation by, the functional form of is given by a polynomial of. Considering the condition of, we find that the simplest form of and can be written as

The calculation of, and is summarized in Table 3, together with the corresponding calculation in another classical (named classical 2, discussed in the next-next paragraph) and the resolvent methods.

Although all the methods give the same result as (31), there is an essential difference in between the classical 1 and resolvent methods from an analytical point of view. While is an entire function, has a pole at. The non-analyti- city of in the classical method is revealed when the is evaluated as in the limit of:

where is given by

In calculating from the inverse FT of, the limit operation is necessary, because (inverse) FT is given by an improper integral. After the analytic continuation of and from to, it is found that

where. Actually, for, for simplicity, we have

where, so that it is confirmed that the relation of (34) holds for. Notice that is an entire function, because has a compact

support so that its (inverse) FT turns out to be an entire function. Thus it is found that whether or not the relation of holds for all depends on the property that is an entire function (the identity theorem in complex analy- sis).

Some may further point out that in the classical method, for can be made an entire function by choosing [hence by (33)] as

in which a series of calculations is summarized in Table 3. Although the is indeed an entire function, it is hard enough to calculate from (especially in a numerical way), compared to the resolvent method, because includes an infinite number of derivatives. Even if we try to regard as an integral transform, it fails due to the divergence of the corresponding integral kernel. Actually, we obtain from

which indicates that (for) is divergent in a usual sense.

Regarding the analyticity and numerical simplicity in calculating FT of, it seems that, based on the above discussion, there is no way other than the resolvent based method.

We have obtained, using the resolvent for the harmonic oscillator Hamiltonian, the FT of a non-integrable function, such as. As compared with the classical methods in Table 3, the resolvent method has some merits of being numerical calcula- tion friendly and free of singularity for. In calculating the resolvent kernel, the sampling theorem is of great use. The introduction of SUSY to not only makes transparent the usefulness of the even-odd decomposition of the in a more natural way, but also leads to a natural definition of SUSY FT.

For future study, various extensions of the present work are possible. One extension is to deal with other unitary transforms, for example, the Hankel transform, whose eigenfunction is given by the Laguerre polynomials Using the resolvent for the corres- ponding Hamiltonian, we can obtain an analogous result. Another is to generalize to, the Clifford algebra over [in (28) cor- responds to]. 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.

The author is indebted to H. Fujisaka for useful discussions. This work was supported in part by HCU grant.

Kuwata, S. (2017) Supersymmetric Resolvent-Based Fourier Transform. Journal of Modern Physics, 8, 133-146. http://dx.doi.org/10.4236/jmp.2017.81012