Supersymmetric Resolvent-Based Fourier Transform

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1. Introduction

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

(1)

where and, through the relation

(2)

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.

2. 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.

2.1. Jacobi Polynomial

Let (where) be the Hamiltonian

The (normalized) eigenfunction for is given by

where and represent the Jacobi polynomial and its normalization constant as

(3)

with and the Gamma function and the hypergeometric function, respectively. The corresponding eigenvalue is given by

(4)

The resolvent kernel for (denoted by) can be expanded using the eigenfunctions’s (for) as

(5)

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 and. 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 as [4] [5]

(6)

where. Notice that is given by the Legendre function as

where is defined by replacing in (3) with. 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, due to the completeness relation of. The reason of restricting to is as follows. To avoid the singularity of at, should be restricted to either or. Moreover, to avoid the singularity of (for) at, the region of is not allowed.

Furthermore, it should be noted that the left-hand side of (6) turns out to be, due to the relation

Thus the relation of (6) can be rewritten as

(7)

where, so that the sampling theorem [3] can be applied to. The sampling theorem states that for

(8)

where represents the support. Hence the validity of (7) is guaranteed by showing that for (with). To show it, it is convenient to use the integral representation for as [6]

from which it is found that is vanishing for under

the conditions of and. Here, we have used the integral representation for the Dirac delta as. Noticing further that

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 right- hand side of (8) in terms of the Dirac notation as

(9)

where and

from which we obtain the orthonormality relation for all. The relation of (9) implies that the completeness relation holds, provided

it is applied to such that. Moreover, interpreting and as and, respectively, we can formally obtain from (9)

(10)

where represents the window function as

The relation of (10) should be compared with

(11)

[In the usual Dirac notation, is reserved for a Fourier transformed variable, so that may be simply written as. Actually, if we formally write as, it is found that, because

where use has been made of the unitarity of as. In this sense, can be simply written as.] Notice that (10) cannot be derived from (11) by formally setting to. This is because in (10) can be applied only to such that. Notice further that the following relation can be derived from (10):

(12)

where we have used. The relation of (12) indicates that the completeness relation holds, if it is applied to such that

, so that. These completeness relations, along with the orthogonal relations, are recapitulated in Table 1, while some examples of satisfying (9) are listed in Table 2.

Now we go back to generalize the relation of (6). Using the integral representation for (notice that) as [7]

Table 1. Orthogonal relation and completeness relation, where.

^{a} can be applied to such that. ^{b} can be applied to such that.

Table 2. Examples of satisfying (9), where and represent the Legendre and Hermite functions, respectively. Here, (for) is given by, (for); and so on. It should be remarked that can be chosen as a more generalized function where is replaced by. For the case where is given by, see Section 2.2 below.

where, we find that in Table 2 can be generalized to

, and more generally to (for, due to). As a special case of in (9), we obtain

(13)

where, with (notice that for, it turns out that is given by a polynomial with respect to). For, representing the Gegenbauer function, we have the following relations:

and

Then it is found that the sum over in the right-hand side of (13) can be replaced by the sum over as

(14)

where use has been made of for all. Once we have replaced the right-hand side of (13) by that of (14), it is not necessary to restrict the

parameter to either or. This is because and the right- hand side of (14) satisfy the same second order differential equation for, de- spite the value of. By re-parameterizing in the right-hand side of (14) as, the relation of (6) is generalized to

(15)

where use has been made of for all.

The relation of (15) can be further generalized. Recall that in Table 2 can be generalized to, which is proportional to

the Jacobi function. Following an analogous procedure for manipulating the Gegen- bauer function above, we finally obtain [1]

where use has been made of the relation

Notice the the superscripts in the left-hand and right-hand sides are ex- changed.

2.2. Hermite Polynomial

In this subsection, we obtain the resolvent kernel for, whose eigenfunction is given by the Hermite polynomial. Considering that can be given by the specialization of the Gegenbauer polynomial as [8]

(16)

then we obtain from (15), together with the asymptotic expansion as (for), the following formula:

(17)

where (amounts to the normalization constant as ). Here, , which is formally given by in (16), is related to the parabolic cylinder function as

where

with, the confluent hypergeometric function. Considering that (for) due to, and that

where, we find that the sum over in the left-hand side of (17) can be formally extended to all. Thus, satisfies the relation of (9) for (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

Recalling that for all, we obtain from (17)

(18)

where use has been made of the following formulae:

The condition of comes from the intersection of and. To obtain for (complementary to), it may be conve- nient to rewrite using another confluent hypergeometric function as

(19)

Substituting (19) into (18), and using again, we obtain the relation that is valid not only for but also for in the form

(20)

which was derived from a somewhat more straightforward approach [1] .

In a practical application, it is convenient to choose the parameter so that the -dependence of may be written as simply as possible. Considering that is given by a polynomial of of order, we can choose as 0 for. In the case of, however, cannot be chosen as 0, due to the divergence of, but can be chosen as 1. To summarize, we have

where. No such formula as (20) but 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 is to be determined in such a way that the sum over in the left-hand side can be formally extended to all integers, namely, (for). Bearing the specialization of (16) in mind, we find that the corresponding summation formula for a single Gegenbauer function is given by

(21)

Actually, the left-hand side of (21) can be rewritten as

where use has been made of, and for. Under the specialization of (16), we finally obtain from (21)

(22)

where for. The condition of in (22)

originates from the condition of in (21), which is equivalent to, with (corresponding to the case of in the first row in Table 2). The relation of (22) is listed in Ref. [10] , in which is given by using the parabolic

cylinder function. [in [10] should be read as.]

3. Results and Discussion

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

(23)

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

(24)

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

(25)

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

(26)

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

(27)

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

(28)

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

(29)

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

(30)

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

(31)

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

Table 3. Calculation of, and for, where. In the classical method 1, there is a singularity of at. As compared with other methods, it is hard enough to calculate from in the classical method 2, due to an infinite number of derivatives in.

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

(32)

which is schematically shown as

Rewriting (26) as

(33)

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

(34)

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.

4. Conclusions

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.

Acknowledgements

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

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