Heat-kernel approach for scattering

An approach for solving scattering problems, based on two quantum field theory methods, the heat kernel method and the scattering spectral method, is constructed. This approach converts a method of calculating heat kernels into a method of solving scattering problems. This allows us to establish a method of scattering problems from a method of heat kernels. As an application, we construct an approach for solving scattering problems based on the covariant perturbation theory of heat-kernel expansions. In order to apply the heat-kernel method to scattering problems, we first calculate the off-diagonal heat-kernel expansion in the frame of the covariant perturbation theory. Moreover, as an alternative application of the relation between heat kernels and partial-wave phase shifts presented in this paper, we give an example of how to calculate a global heat kernel from a known scattering phase shift.


Introduction
In this paper, based on two quantum field theory methods, heat kernel method [1] and scattering spectral method [2], we present a new approach to solve scattering problems. This approach is not only one single approach; it is a series of approaches for scatterings. Concretely, our key result is a direct relation between partial-wave scattering phase shifts and heat kernels. By this result, each method of calculating heat kernels leads to an approach of calculating phase shifts. In a word, the approach provided in this paper converts a heat kernel method into a method of solving scattering problems. Many methods for scattering problems can be constructed, since the heat kernel theory is well-studied in both mathematics and physics and there are many mature methods for the calculation of heat kernels.
Phase shift. All information of an elastic scattering process are embedded in a scattering phase shift. This can be seen by directly observing the asymptotic solution of the radial wave equation. For spherically symmetric cases, the asymptotic solution of the free radial wave equation, − 1 This defines the partial-wave phase shift δ l (k), which is the only effect on the radial wave function at asymptotic distances [3]. Therefore, all we need to do in solving a scattering problem is to solve the phase shift δ l (k). Heat kernel. The information embedded in an operator D can be extracted from a heat kernel K (t; r, r ) which is the Green function of the initial-value problem of the heat-type equation (∂ t + D) φ = 0, determined by [1] (∂ t + D) K t; r, r = 0, with K 0; r, r = δ r − r . (1. 2) The global heat kernel K (t) is the trace of the local heat kernel K (t; r, r ): K (t) = drK (t; r, r) = n,l e −λ nl t , where λ nl is the eigenvalue of the operator D.
The main aim of the present paper is to seek a relation between the partial-wave phase shift δ l (k) and the heat kernel K (t; r, r ). By this relation, we can explicitly express a partial-wave phase shift by a given heat kernel. There are many studies on the approximate calculation of heat kernels [1,[4][5][6][7][8][9][10][11][12][13] and each approximate method of heat kernels gives us an approximate method for calculating partial-wave phase shifts.
The present work is based on our preceding work given in Ref. [14], which reveals a relation between two quantum field theory methods, heat kernel method [1] and scattering spectral method [2]. In Ref. [14], using the relation between spectral counting functions and heat kernels given by Ref. [15] and the relation between phase shifts and state densities given by Ref. [2], we provide a relation between the global heat kernel and the total scattering phase shift (the total scattering phase shift is the summation of all partial-wave phase shifts, δ (k) = l (2l + 1) δ l (k)).
Nevertheless, the result given by Ref. [14] -the relation between total scattering phase shifts and heat kernels -can hardly be applied to scattering problems, since the total phase shift has no clear physical meaning.
To apply the heat kernel method to scattering problems, we in fact need a relation between partial-wave phase shifts (rather than total phase shifts) and heat kernels. In the present paper, we find such a relation. This relation allows us to express a partial-wave phase shift by a known heat kernel. Then, all physical quantities of a scattering process, such as scattering amplitudes and cross sections, can be expressed by a heat kernel.
To find the relation between partial-wave phase shifts and heat kernels, we will first prove a relation between heat kernels and partial-wave heat kernels. The heat kernel K (t; r, r ) is the Green function of initial-value problem of the heat equation (1.2) with the operator D = −∇ 2 + V (r) and the partial-wave heat kernel K l (t; r, r ) is the Green function of initial-value problem of the heat equation (1.2) with the radial operator D l = − 1 r 2 d dr r 2 d dr + l(l+1) r 2 + V (r). By this relation, we can calculate a partial-wave heat kernel K l (t; r, r ) from a heat kernel K (t; r, r ) directly.
The main aim of this paper is to explicitly express the partial-wave phase shift and scattering amplitude by a given heat kernel. As mentioned above, by our result, each method of calculating heat kernels can be converted to a method of calculating scattering problems.
In order to calculate a scattering phase shift from a heat kernel, we need off-diagonal heat kernels (i.e., heat kernels). For this purpose, in the following, we first calculate two kinds of off-diagonal heat-kernel expansions in the frames of the Seeley-DeWitt expansion and the covariant perturbation theory, respectively. It should be pointed out that many methods on the calculation of diagonal heat-kernel expansions in literature can be directly apply to the calculation of off diagonal heat kernels.
Moreover, we compare the scattering method established in the present paper with the Born approximation. By comparing the approximation results given by the three methods, the Seeley-DeWitt expansion, the covariant perturbation theory, and the Born approximation through an exactly solvable potential, we show that the method established based on the covariant perturbation theory provided in the present paper is the best approximation.
On the other hand, besides applying the heat kernel method to scattering problems, by the method suggested in the present paper, we can also apply the scattering method to the heat kernel theory. In this paper, we provide a simple example for illustrating how to calculate a heat kernel from a known scattering result; more details on this subject will be given in a subsequent work. The value of developing such a method, for example, is that though it is relatively easy to obtain a high-energy expansion of heat kernels, it is difficult to obtain a low-energy heat-kernel expansion. With the help of scattering theory, we can calculate a low-energy heat-kernel expansion from a low-energy scattering theory.
The starting point of this work, as mentioned above, is a relation between the heat kernel method and the scattering spectral method in quantum field theory. The heat kernel method is important both in physics and mathematics. In physics, the heat kernel method has important applications in, e.g., Euclidean field theory, gravitation theory, and statistical mechanics [1,13,[16][17][18]. In mathematics, the heat kernel method is an important basis of the spectral geometry [1,19]. There are many researches on the calculation of heat kernels. Besides exact solutions, there are many systematic studies on the asymptotic expansion of heat kernels, such as the Seeley-DeWitt expansion [1] and the covariant perturbation theory [10][11][12][13]. With various heat-kernel expansion techniques, one can obtain many approximate solutions of heat kernels. Scattering spectral method is an important quantum theory method which can be used to solve a variety of problems in quantum field theory, e.g., to characterize the spectrum of energy eigenstates in a potential background [2] and to solve the Casimir energy [20][21][22][23][24]. The method particular focuses on the property of the quantum vacuum.
In Sec. 2, we find a relation between partial-wave phase shifts and heat kernels. As a key step, we give a relation between partial-wave heat kernels and heat kernels. In Secs. 3 and 4, based on the relation between partial-wave phase shifts and heat kernels given in Sec. 2, we establish two approaches for the calculation of partial-wave phase shifts, based on two heat-kernel expansions, the Seeley-DeWitt expansion and the covariant perturbation theory. In Sec. 5, a comparison of the two approaches for partial-wave phase shifts established in the present paper and the Born approximation is given; especially, we compare these three methods through an exactly solvable potential. In Sec. 6, we give an example for calculating a heat kernel from a given phase shift. Conclusions and outlook are given in Sec. 7. Moreover, an integral formula and two integral representations are given in appendices A, B, and C.
2 Relation between partial-wave phase shift and heat kernel: calculating scattering phase shift from heat kernel The main result of the present paper is the following theorem which reveals a relation between partial-wave scattering phase shifts and heat kernels. This relation allows us to obtain a partial-wave phase shift from a known heat kernel directly. By this relation, what we can obtain is not only one method for scattering problems. It is in fact a series of methods for scattering problems: each heat kernel method leads to a method for solving scattering problems.

Theorem 1
The relation between the partial-wave scattering phase shift, δ l (k), and the heat kernel, K (t; r, r ) = K (t; r, θ, ϕ, r , θ , ϕ ), is 1) where K s (t; r, r ) is the scattering part of a heat kernel, P l (cos γ) is the Legendre polynomial, and γ is the angle between r and r with cos γ = cos θ cos θ + sin θ sin θ cos (ϕ − ϕ).
Notice that only the radial diagonal heat kernel, K s (t; r, θ, ϕ, r, θ , ϕ ), appears in Eq. (2.1). Here the heat kernel K (t; r, r ) is split into three parts: K (t; r, r ) = K s (t; r, r ) + K b (t; r, r )+K f (t; r, r ), where K s (t; r, r ) is the scattering part of the heat kernel, K b (t; r, r ) the bound part, and K f (t; r, r ) the free part [14]. Note that δ l (0) = π/2 if there is a halfbound state and δ l (0) = 0 if there is no half-bound state [14].
The remaining task of this section is to prove this theorem. In order to prove the theorem, we need to first find a relation between partial-wave heat kernels and heat kernels.

Relation between partial-wave heat kernel and heat kernel
As mentioned above, the heat kernel K (t; r, r ) of an operator D is determined by the heat equation (1.2) [1]. For a spherically symmetric operator D, the heat kernel K (t; r, r ) = K (t; r, θ, ϕ, r , θ , ϕ ) can be expressed as K t; r, r = n,l,m e −λ nl t ψ nlm (r) ψ * nlm r , where λ nl and ψ nlm (r) = R nl (r) Y lm (θ, ϕ) are the eigenvalue and eigenfunction of D, determined by the eigenequation Dψ nlm = λ nl ψ nlm , where R nl (r) is the radial wave function and Y lm (θ, ϕ) is the spherical harmonics. The global heat kernel is the trace of the local heat kernel K (t; r, r ): 3) The local partial-wave heat kernel of the operator D is the heat kernel of the l-th partial-wave radial operator [14] which determines the radial equation D l R nl = λ nl R nl . The global partial-wave heat kernel is the trace of the local partial-wave heat kernel K l (t; r, r ), Now we prove that the relation between K l (t; r, r ) and K (t; r, r ) can be expressed as follows.

Proof of Theorem 1
Now, with Lemma 2, we can prove Theorem 1.
Proof. In Ref. [14], we prove a relation between total phase shifts and global heat kernels, and a relation between partial-wave phase shifts and partial-wave global heat kernels, Here the global heat kernel and the global partial-wave heat kernel are split into the scattering part, the bound part, and the free part: [14]. Starting from the global partial-wave heat kernel given by Eq. (2.6) and using the relation between partial-wave heat kernels and heat kernels given by Lemma 2, Eq. (2.7), we have  It should be noted here that the relation given by Ref. [14], Eqs. (2.15) and (2.16), only allows one to calculate the total phase shift δ (k) from a heat kernel K (t) or to calculate the partial-wave phase shift δ l (k) from a partial-wave heat kernel K l (t). Such results, however, are not useful in scattering problems, because the total phase shift δ (k) is not physical meaningful and the partial-wave heat kernel K l (t) is often difficult to obtain.
Nevertheless, the result given by Theorem 1, Eq. (2.1), allows one to calculate the partial-wave phase shift δ l (k) form a heat kernel K (t) rather than a partial-wave heat kernel K l (t). The heat kernel has been fully studied and there are many known results [1].

Heat-kernel approach for phase shift: Seeley-DeWitt expansion
In this section, we present an asymptotic expansion method for phase shifts based on the Seeley-DeWitt expansion of heat kernels.
The Seeley-DeWitt asymptotic expansion of heat kernels is an important method in the heat kernel theory. In the following, by Theorem 1, we convert the Seeley-DeWitt heat-kernel expansion into an expansion for partial-wave phase shifts.
The Seeley-DeWitt type expansion for a partial-wave scattering phase shift is δ l (k) = δ where J v (z) is the Bessel function of the first kind [25]. A detailed calculation is as follows.

Seeley-DeWitt expansion of heat kernel
The Seeley-DeWitt expansion is an asymptotic expansion of heat kernels in powers of the proper time t [13]. For a Laplace-type operator D = −∇ 2 +V , the Seeley-DeWitt expansion of heat kernels reads [26] K t; r, r = 1 where a j (r, r ) is the heat-kernel coefficient and σ (r, r ) = d 2 (r, r ) /2 with d (r, r ) the geodesic distance between r and r . Note that ∇ 2 here is the Laplace-Beltrami operator which reduces to the Laplace operator in flat space. For the Seeley-DeWitt expansion, the heat-kernel coefficient a j (r, r ) satisfies the recurrence identity [26] a j+1 (r, r ) where r (λ) describes a geodesic segment with r (0) = r and r (1) = r and V V M (r, r ) is the van Vleck-Morette determinant [26], with g (r) = det g µν (r) and d the dimension of space. For flat space, the Seeley-DeWitt expansion, Eq. (3.3), reduces to In flat space, V V M (r, r ) = 1, r (λ) = λr+ (1 − λ) r , and the recurrence identity, Eq.

Expressing phase shift by heat-kernel coefficient
In this section, in terms of the Seeley-DeWitt heat-kernel coefficients, we present an asymptotic expansion for partial-wave scattering phase shifts. It should be noted that only the radial diagonal heat kernel, K (t; r, θ, ϕ, r, θ , ϕ ), appears in Eq. (2.1) rather than the heat kernel, K (t; r, θ, ϕ, r , θ , ϕ ), so only the radial diagonal heat-kernel coefficient is needed in the calculation of phase shifts. For spherical potentials, by taking r = r in Eq. (3.6), we achieve a radial diagonal Seeley-DeWitt expansion for heat kernels: where a j (r, γ) = a j (r, r )| r=r is the radial diagonal heat kernel coefficient. The scattering phase shift can be obtained by substituting Eq. (3.9) into the relation between partial-wave phase shifts and heat kernels, Eq. (2.1),  11) and the scattering phase shift then reads (3.12) The first two contributions are (3.14) 3.3 First-order phase shift δ (1) l (k) In this section, we calculate the first-order phase shift in the frame of the Seeley-DeWitt expansion.
First-order heat-kernel coefficient a 1 (r, r ): In order to calculate the phase shift δ l (k), we need to first calculate the heat-kernel coefficient a j (r, γ).
The first-order Seeley-DeWitt heat-kernel coefficient can be obtained by the recurrence identity (3.7): where R = r + λ (r − r ). As mentioned above, in our case, only the radial diagonal heatkernel coefficient, a 1 (r, r )| r =r , is needed in the calculation of phase shifts. Therefore, for spherically symmetric cases, the first-order radial diagonal heat-kernel coefficient, from Eq. (3.15), reads Fourier transforming V (R) and working out the integral of the angle give where γ pR is the angle between p and R. Substituting Eq. (3.17) into Eq. (3.16), we have (3.20) By constructing a derivative representation we rewritten a 1 (r, γ) as By constructing a derivative representation we rewritten Eq. (3.23) as (3.24) By using the integral formula (B.1) given in Appendix B, we have Finally, we arrive at Eq. (3.1).

Second-order phase shift δ
(2) l (k) In this section, we calculate the second-order phase shift in the frame of the Seeley-DeWitt expansion.
Second-order heat-kernel coefficient a 2 (r, r ): The second-order Seeley-DeWitt heatkernel coefficient can be obtained by the recurrence identity (3.7): where a 1 (r (λ) , r ) is the first-order heat-kernel coefficient. For spherically symmetric cases, substituting Eq. (3.15) into Eq. (3.26) gives we have (3.31) The angle integrals in Eq. (3.31) can be worked out directly, Notice that only the radial diagonal heat kernel is needed in our case. For the radial diagonal case, we have and Then, the second-order radial diagonal heat-kernel coefficient becomes where x = cos γ.

Heat-kernel approach for phase shift: covariant perturbation theory
In this section, based on the heat-kernel expansion given by the covariant perturbation theory [10][11][12], by the relation between partial-wave phase shifts and heat kernels given by Eq. (2.1), we establish an expansion for scattering phase shifts. The covariant perturbation theory type expansion for a partial-wave scattering phase shift is δ l (k) = δ (1)

2)
where Y ν (z) is the Bessel function of the second kind [25]. A detailed calculation is as follows.

Covariant perturbation theory for heat-kernel expansion
The heat-kernel expansion is systematically studied in the covariant perturbation theory [10][11][12]. The heat-kernel expansion given by the covariant perturbation theory reads [4,6] K t; r, r = K (0) t; r, r + K (1) t; r, r + K (2) t; r, r + · · · = r| e −H 0 t + (−t) where is the zero-order (free) heat kernel. Substituting the zero-order heat kernel (4.4) into Eq. (4.3), we obtain the first two orders of a heat kernel, For the spherical potentials V (r) = V (r), K (1) (t; r, r ) and K (2) (t; r, r ) given by Eqs. (4.5) and (4.6) become exp − 1 4τ r 2 + y 2 − 2r y cos γ r y (4πτ ) 3/2 (4.7) and where γ is the angle between r and r , γ ry is the angle between r and y, γ r y is the angle between r and y, γ yz is the angle between y and z, and γ zr is the angle between z and r .

First-order phase shift δ
(1) l (k) In this section, we calculate the first-order phase shift in the frame of the covariant perturbation theory.

Second-order phase shift δ
(2) l (k) In this section, we calculate the second-order phase shift in the frame of the covariant perturbation theory.
The second-order phase shift δ (2) l (k) can be obtained by substituting the second-order heat kernel given by the covariant perturbation theory, Eq. (4.8), into the relation between partial-wave phase shifts and heat kernels, Eq. (2.1), and taking r = r: where (4.25) Using Eq. (4.11), we rewrite the integral I 3 as d cos γP l (cos γ) dΩ y dΩ z P l 1 (cos γ ry ) P l 2 (cos γ yz ) P l 3 (cos γ zr ) . Without loss of generality, we choose r = (r , 0, 0) and then we have γ zr = θ z . The integral with respect to Ω z can be then worked out by use of the integral formula, Eq. (A.1), given in Appendix A: dΩ z P l 2 (cos γ yz ) P l 3 (cos γ zr ) = dΩ z P l 2 (cos γ yz ) P l 3 (cos θ z ) = P l 2 (cos θ y ) 4π 2l 2 + 1 δ l 2 ,l 3 , (4.27) The integral with respect to Ω y also can be integrated directly by Eq. (A.1), dΩ y P l 1 (cos γ ry ) P l 2 (cos θ y ) 4π 2l 2 + 1 δ l 2 ,l 3 = P l 1 (cos θ r ) 4π Then, performing the integral with respect to γ (γ = θ r when r = (r , 0, 0)) in Eq. (4.26), we have To perform the integral with respect to r, by using the integral representation (B.1) given in Appendix B, we rewrite  Then, the integral with respect to r can be worked out, . . (4.34) Using the expansion (4.11) and the orthogonality of the Legendre polynomials, we have . .

Comparison with Born approximation
The approach for scattering problems established in the present paper is not only one single approach. Every method of calculating heat kernels can be converted into a method for scattering problems. As applications, in Secs. 3 and 4, we suggest two methods for scattering phase shifts, based on two heat kernel methods: the Seeley-DeWitt expansion and the covariant perturbation theory. In scattering theory, there are many approximation methods, such as Born approximation, WKB method, eikonal approximation, and variational method. [27].
In this section, we compare our two methods with the Born approximation.

Comparison of first-order contribution
For clarity, we relist some results given by the above sections in the following.
The first-order phase shift given by the Seeley-DeWitt expansion given in Sec. 3 reads (5.1) The first-order phase shift given by the covariant perturbation theory given in Sec. 4 reads As a comparison, the first-order phase shift given by the Born approximation reads [27] δ (1) Obviously, the leading contributions of these three methods are the same (in the Born approximation, the first-order contribution is in fact arctan − (π/2) ∞ 0 rdrV (r) J 2 l+1/2 (kr) , but the higher contribution can be safely ignored in the first-order contribution).

Comparison of second-order contribution
The second-order phase shift given by the Seeley-DeWitt expansion given in Sec. 3 reads (5.4) The second-order phase shift given by the covariant perturbation theory given in Sec. 4 (Eqs. (4.2) and (4.42)) reads The second-order phase shift given by the Born approximation [27] reads (5.6) It can be directly seen that the leading contribution of the second-order Born approximation and the leading contribution of the second-order covariant perturbation theory are the same. The second-order contribution given by the Seeley-DeWitt expansion is different form the other two methods. This is because the Born approximation and the covariant perturbation theory are both small V (r) expansions, but the Seeley-DeWitt expansion is a small t asymptotic expansion.

Comparison through an exactly solvable potentialčžV (r) = α/r 2
In this section, we compare the three methods, the Seeley-DeWitt expansion, the covariant perturbation theory method, and the Born approximation, through an exactly solvable potential: Using these three approximation methods to calculate an exactly solvable potential can help us to compare them intuitively. The phase shift for the potential (5.7) can be solved exactly, In order to compare the methods term by term, we expand the exact result (5.8) as δ l = δ (5.10) First order: The first-order contribution given by the Seeley-DeWitt method, the covariant perturbation theory, and the Born approximation can be directly obtained by sub-stituting the potential (5.7) into Eqs. (3.1), (4.1), and (5.3), respectively: (5.13) Comparing with the direct expansion of the exact solution, Eqs. (5.9) and (5.10), we can see that all results are good approximations, and the result given by the covariant perturbation theory is the best result. It can be checked that the second term in the result given by the Seeley-DeWitt method, Eq. (5.11), is small; the maximum value of the second term appears at l = 0, which is about 17% of the first term.
Second order: The second-order contribution given by the Seeley-DeWitt method, the covariant perturbation theory, and the Born approximation can be directly obtained by substituting the potential (5.7) into Eqs. (3.2), (4.2), and (5.6), respectively: (5.16) Comparing with the second-order contribution, Eq. (5.10), we can see that, like that in the case of first-order contributions, the result given by the covariant perturbation theory is the best result. Numerical tests show that the result given by the above three methods are very close to each other.
Based on the above comparison, we can conclude that the method based on the covariant perturbation theory is better than other two methods.

Calculating global heat kernel from phase shift
The key result of this paper is a relation between partial-wave phase shifts and heat kernels. Besides solving a scattering problem from a known heat kernel, obviously, we can also calculate a heat kernel from a known phase shift. Here, we only give a simple example with the potential α/r 2 . A systematic discussion on how to calculate heat kernels and other spectral functions, such as one-loop effective actions, vacuum energies, and spectral counting functions, from a solved scattering problem will be given elsewhere.

Conclusions and outlook
In this paper, based on two quantum field theory methods, the heat kernel method [1] and the scattering spectral method [2], we suggest a method for calculating the scattering phase shift. The method suggested in the present paper is indeed a series of different methods of calculating scattering phase shifts constructed from various heat kernel methods.
The key step is to find a relation between partial-wave phase shifts and heat kernels. This relation allows us to express a partial-wave phase shift by a heat kernel. Then, each method of the calculation of heat kernels can be converted to a method of the calculation of phase shifts.
As applications, we provide two methods for the calculation of phase shifts, corresponding to two heat kernel methods, the Seeley-DeWitt expansion and the covariant perturbation theory.
Furthermore, as emphasized above, by this approach, we can construct various methods for scattering problems with the help of various heat kernel methods. In subsequent works, we shall construct various scattering methods by using various heat-kernel expansions.
In this paper, as a coproduct, we also provide two kinds of off-diagonal heat-kernel expansions, based on the technique developed in the Seeley-DeWitt expansion and the covariant perturbation theory for diagonal heat kernels, since the heat kernel method for scatterings established in the present paper is based on the off-diagonal heat kernel rather than the diagonal heat kernel. It should be emphasized that many methods for calculating diagonal heat kernels can be directly applied to the calculation of off-diagonal heat kernels. That is to say, the method for calculating the diagonal heat kernel often can also be converted to a method for calculating off-diagonal heat kernels and scattering phase shifts, as we have done in the present paper. Therefore, we can construct scattering methods from many methods of diagonal heat kernels, e.g., [1,9,28].
The heat kernel theory is well-studied in both mathematics and physics. Here, as examples, we list some methods on the calculation of heat kernel. In Refs. [29][30][31], the author calculates the heat-kernel coefficient with different boundary conditions. In Ref. [32], using the background field method, the author calculates the fourth and fifth heat-kernel coefficients. In Refs. [33][34][35], the author calculates the third coefficient by the covariant technique. In Refs. [9,36], by the string-inspired worldline path-integral method, the authors calculate the first seven heat-kernel coefficients. In Ref. [28], a direct, nonrecursive method for the calculation of heat kernels is presented. In Ref. [37], the first five heat-kernel coefficients for a general Laplace-type operator on a compact Riemannian space without boundary by the index-free notation are given. In Refs. [10][11][12][13]38], a non-Seeley-DeWitt expansion of a heat kernel is established. In Refs. [39][40][41], a covariant pseudo-differentialoperator method for calculating heat-kernel expansions in an arbitrary space dimension is given. iochum2012spectral An important application of the method given by this paper is to solve various spectral functions by a scattering method. The problem of spectral functions is an important issue in quantum field theory [15,42,43]. A subsequent work on this subject is a systematic discussion on calculating heat kernels, effective actions, vacuum energies, etc., from a known phase shift. We will show that, based on scattering methods, we can obtain some new heat-kernel expansions. It is known that though there are many discussions on the high energy heat-kernel expansion, the low-energy expansion of heat kernels is relatively difficult to obtain. While, there are some successful low-energy scattering theories; by using the relation given in this paper, we can directly obtain some low-energy results for heat kernels.
Starting from the result given by the present paper, we can study many problems. The method presented in this paper can be applied to low-dimensional scatterings. One-and two-dimensional scatterings and their applications have been deeply studied, such as the transport property of low-dimensional materials [44][45][46]. We will also consider a systematic application of our method on relativistic scatterings. The relativistic scattering is an important problem, e.g., the collision of solitons in relativistic scalar field theories [47] and the Dirac scattering in the problem of the electron properties of graphene [48,49]. We can also apply the method to low-temperature physics. There are many scattering problems in lowtemperature physics, such as the scattering in the problem of the transition temperature of BEC [50,51] and the transport property of spin-polarized fermions in low temperatures [52,53].
The application of the method to inverse scattering problems is an important subject of our subsequent work. The inverse scattering problem has extremely important significance in physics [54,55]. In practice, for example, the inverse scattering method can be applied to the problem of BEC [56] and the Aharonov-Bohm effect [57].
In Ref. [42], we provide a method for solving the spectral function, such as one-loop effective actions, vacuum energies, and spectral counting functions in quantum field theory.
The key idea is to construct the equations obeyed by these quantities. We show that, for example, the equation of the one-loop effective action is a partial integro-differential equation. By the relation between partial-wave phase shifts and heat kernel, we can also construct an equation obeyed by phase shifts.
Moreover, in conventional scattering theory, an approximately large-distance asymptotics is used to seek an explicit result. In Ref. [58], we show that such an approximate treatment is not necessary: without the large-distance asymptotics, one can still rigorously obtain an explicit result. The result presented in this paper can be directly applied to the scattering theory without large-distance asymptotics.