Scattering theory without large-distance asymptotics

Abstract

In conventional scattering theory, to obtain an explicit result, one imposes a precondition that the distance between target and observer is infinite. With the help of this precondition, one can asymptotically replace the Hankel function and the Bessel function with the sine functions so that one can achieve an explicit result. Nevertheless, after such a treatment, the information of the distance between target and observer is inevitably lost. In this paper, we show that such a precondition is not necessary: without losing any information of distance, one can still obtain an explicit result of a scattering rigorously. In other words, we give an rigorous explicit scattering result which contains the information of distance between target and observer. We show that at a finite distance, a modification factor -- the Bessel polynomial -- appears in the scattering amplitude, and, consequently, the cross section depends on the distance, the outgoing wave-front surface is no longer a sphere, and, besides the phase shift, there is an additional phase (the argument of the Bessel polynomial) appears in the scattering wave function.

Published for SISSA by Springer

Received: March 25, 2014 Revised: May 28, 2014

Accepted: May 28, 2014

Published: June 16, 2014

Tong Liu,a Wen-Du Lia and Wu-Sheng Daia,b

aDepartment of Physics, Tianjin University,

Tianjin 300072, P.R. China

bLiuHui Center for Applied Mathematics, Nankai University & Tianjin University, Tianjin 300072, P.R. China

E-mail: mailto:[email protected]

Web End [email protected] , mailto:[email protected]

Web End [email protected] , mailto:[email protected]

Web End [email protected]

Abstract: In conventional scattering theory, to obtain an explicit result, one imposes a precondition that the distance between target and observer is innite. With the help of this precondition, one can asymptotically replace the Hankel function and the Bessel function with the sine functions so that one can achieve an explicit result. Nevertheless, after such a treatment, the information of the distance between target and observer is inevitably lost. In this paper, we show that such a precondition is not necessary: without losing any information of distance, one can still obtain an explicit result of a scattering rigorously. In other words, we give an rigorous explicit scattering result which contains the information of distance between target and observer. We show that at a nite distance, a modication factor the Bessel polynomial appears in the scattering amplitude, and, consequently, the cross section depends on the distance, the outgoing wave-front surface is no longer a sphere, and, besides the phase shift, there is an additional phase (the argument of the Bessel polynomial) appears in the scattering wave function.

Keywords: Integrable Equations in Physics, Scattering Amplitudes

ArXiv ePrint: 1403.5646

Open Access, c

[circlecopyrt] The Authors.

Article funded by SCOAP3. doi:http://dx.doi.org/10.1007/JHEP06(2014)087

Web End =10.1007/JHEP06(2014)087

Scattering theory without large-distance asymptotics

JHEP06(2014)087

Contents

1 Introduction 1

2 Rigorous result of scattering without large-distance asymptotics 22.1 Phase shift 22.2 Asymptotic boundary condition 42.3 Scattering wave function 52.4 Outgoing wave-front surface 52.5 Di erential scattering cross section 62.6 Total scattering cross section 62.7 Condition on potentials 7

3 Conclusions and outlook 9

1 Introduction

In conventional scattering theory, which is now a standard quantum mechanics textbook content, to seek an explicit result, one imposes a precondition that the distance between target and observer is innite. As a result, the conventional scattering theory loses all the information of the distance and the result depends only on the angle of emergence. In this paper, we will show that without such a precondition, one can still achieve a rigorous scattering theory which, of course, contains the information of distance that is lost in conventional scattering theory.

The dynamical information of a scattering problem with a spherical potential V (r) are embedded in the radial wave equation,

1 r2

r2 dRldr[parenrightbigg]+

Rl = 0. (1.1)

The scattering boundary condition in conventional scattering theory is taken to be

(r, ) = eikr cos + f () eikrr , r . (1.2)

In conventional scattering theory, in order to achieve an explicit result, two kinds of asymptotic approximations are employed [1].

1) Replace the solution of the free radial equation, i.e., eq. (1.1) with V (r) = 0, with its asymptotics:

Rl (r) = Clh(2)l (kr) + Dlh(1)l (kr) (1.3)

r

Al

1

JHEP06(2014)087

d dr

k2 l (l + 1)r2 V (r)

sin (kr l/2 + l)kr , (1.4)

[bracketrightbigg]

where h(1)l (z) and h(2)l (z) are the rst and second kind spherical Hankel functions, e2il = Dl/Cl denes the scattering phase shift l, and Al = 2ClDl.

2) Replace the plane wave expansion in the boundary condition with its asymptotics:

Xl=0(2l + 1) iljl (kr) Pl (cos ) (1.5)

eikr cos =

Xl=0(2l + 1) il sin (kr l/2)kr Pl (cos ) , (1.6)

where jl (z) is the spherical Bessel function.

Technologically speaking, the above two treatments in conventional theory are to replace the spherical Hankel function, h(1)l (kr) and h(2)l (kr), and the spherical Bessel function, jl (kr), with their asymptotics, and, thus, inevitably lead to the loss of information of the distance r.

In this paper, we will show that the above two replacements is not necessary; without these two replacements, we can still obtain a rigorous scattering theory which contains the information of the distance between target and observer.

A systematic rigorous result of a scattering with the distance between target and observer is given in section 2. The conclusion and outlook are given in section 3.

2 Rigorous result of scattering without large-distance asymptotics

In this section, a rigorous treatment without large-distance asymptotics for short-range potentials is established. The scattering wave function, scattering amplitude, phase shift, cross section, and a description of the outgoing wave are rigorously obtained.

2.1 Phase shift

In conventional scattering theory, as mentioned above, one replaces the solution of the free radial equation, Rl (r), given by eq. (1.3) with its asymptotics, eq. (1.4), using the asymptotics of the spherical Hankel functions h(1)l (kr)

1ikr ei(krl/2) and h(2)l (kr)

1ikrei(krl/2). Obviously, such a replacement will lose information.

In the following, with Rl (r) given by eq. (1.3), rather than its asymptotics, eq. (1.4), we solve the scattering rigorously.

The rst step is to rewrite Rl (r) given by eq. (1.3) as

Rl (r) = Clh(2)l (kr) + Dlh(1)l (kr)

= Ml

1 ikr

[parenrightbigg]

Alkr sin

r

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kr l2 + l + l

1 ikr

[parenrightbigg][bracketrightbigg]

, (2.1)

where e2il = Dl/Cl and Ml (x) = |yl (x)| and l (x) = arg yl (x) are the modulus and

argument of the Bessel polynomial yl (x), respectively.

2

In order to achieve eq. (2.1), we prove the relation

Clh(2)l (x) + Dlh(1)l (x) = Ml 1 ix

[parenrightbigg]

Alx sin

x l2 + l + l

1 ix

[parenrightbigg][bracketrightbigg]

. (2.2)

Proof. The rst and second kind spherical Hankel functions, h(1)l (x) and h(2)l (x), can be expanded as [2]

h(1)l (x) = eix

l

Xk=0ikl1 (l + k)!2kk! (l k)!xk+1, (2.3)

h(2)l (x) = eix

l

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Xk=0(i)kl1 (l + k)!2kk! (l k)!xk+1. (2.4)

By the Bessel polynomial [2],

yl (x) =

l

Xk=0(l + k)! k! (l k)!

x 2

k, (2.5)

we can rewrite h(1)l (x) and h(2)l (x) as

h(1)l (x) = ei(xl/2)

1 ixyl

1 ix

,

h(2)l (x) = ei(xl/2)

1 ixyl

[parenleftbigg]

1 ix

. (2.6)

Using eq. (2.6), we have

Clh(2)l (x) + Dlh(1)l (x) = Cl

"ei(xl/2) ix yl

1 ix

[parenrightbigg]

+ e2il ei(xl/2) ix yl

1 ix

[parenrightbigg][bracketrightBigg]

. (2.7)

Writing the Bessel polynomial as yl = Mlei l, we prove the relation (2.2).

The wave function, then, by (r, ) =

Pl=0 Rl (r) Pl (cos ), can be obtained immedi-

ately from eq. (2.1),

(r, ) =

Xl=0Ml

1 ikr

[parenrightbigg]

Alkr sin

kr l2 + l + l

1 ikr

[parenrightbigg][bracketrightbigg]

Pl (cos ) . (2.8)

When the distance r is nite, the coe cient becomes MlAl and the phase becomes l + l, where Ml and l both depend on r. While, in conventional scattering theory, r , the coe cient is Al and the phase is l, and they are both independent of r.

It should be emphasized that l here is the same as that in conventional scattering theory. This is because l is determined only by the coe cient Cl and Dl and yl 1ikr

[parenrightbig]

r =

1. Thus when r , Cl, Dl, and, accordingly, l remains unchanged.

The modication factors, l and Ml, are independent of potentials. When r ,

Ml (r ) = 1 and l (r ) = 0.

3

2.2 Asymptotic boundary condition

The outgoing wave is no longer a spherical wave when the observer stands at a nite distance from the target, other than that in large-distance asymptotics. The outgoing wave now becomes a surface of revolution around the incident direction, determined by the potential and the observation distance. Because the outgoing waves are di erent at di erent distances, there is no uniform expression of the asymptotic boundary condition like eq. (1.2). Here, we express the boundary condition as

(r, ) = eikr cos + f (r, ) eikrr , (2.9)

where f (r, ) depends not only on but also on r.

When the distance r is nite, however, the di erential scattering cross section is no longer the square modulus of f (r, ). Only when r , f (, ) = f () and the di er

ential cross section reduces to |f ()|2.

To calculate f (r, ), as that in conventional scattering theory, we expand the incoming plane wave eikr cos by the eigenfunction of the angular momentum. Now, we prove that the expansion of eikr cos, eq. (1.5), can be exactly rewritten as

eikr cos =

Xl=0(2l + 1) iljl (kr) Pl (cos )

=

Xl=0(2l + 1) ilMl

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1 ikr

[parenrightbigg]

1kr sin

kr l2 + l

1 ikr

[parenrightbigg][bracketrightbigg]

Pl (cos ) . (2.10)

Proof. A plane wave can be expanded as [3]

eikr cos =

Xl=0(2l + 1) iljl (kr) Pl (cos ) . (2.11)

By the relations h(1)l (x) = jl (x) + inl (x) and h(2)l (x) = jl (x) inl (x), the spherical Bessel function jl (x) can be rewritten as jl (x) = 12

hh(1)l (x) + h(2)l (x)

i, where nl (x) is the

spherical Neumann function [2]. By eq. (2.6), we have

jl (kr) = Ml

1 ikr

[parenrightbigg]

1kr sin

kr l2 + l

1 ikr

[parenrightbigg][bracketrightbigg]

. (2.12)

Substituting this result into eq. (2.11) proves eq. (2.10).

The plane wave expansion (2.10) is exact, rather than the asymptotic one, eq. (1.6), used in conventional scattering theory. In conventional scattering theory, the spherical Bessel function jl (kr) given by eq. (2.12) is replaced by its asymptotics: jl (kr)

1kr sin (kr l/2), i.e., Ml and l are asymptotically taken to be Ml 1ikr

[parenrightbig]

1 and

0; as a result, the information embedded in Ml and l is lost.

The boundary condition, eq. (2.9), then, by eq. (2.10), can be expressed as

(r, ) =

Xl=0(2l + 1) ilMl

l 1ikr

[parenrightbig]

1 ikr

[parenrightbigg]

1kr sin

kr l2 + l

1 ikr

[parenrightbigg][bracketrightbigg]

Pl (cos )

+f (r, ) eikrr . (2.13)

4

2.3 Scattering wave function

The scattering wave function can be calculated by imposing the boundary condition (2.13) on the asymptotic wave function (2.8).

Observing the outgoing part of the wave function (2.13), f (r, ) eikr/r, we can see that

the leading contribution of f (r, ) must only be a zero power of r, or else the outgoing wave is not a spherical wave when r . Thus, we can expand f (r, ) by the Bessel

polynomial, which is complete and orthogonal [4], as

f (r, ) =

Xl=0gl () yl

1 ikr

. (2.14)

JHEP06(2014)087

The reason why only yl 1ikr

[parenrightbig]

appears in the expansion (2.14) is that only the ux corre-

sponding to yl 1ikr

eikr/r is an outgoing spherical wave; or, in other words, the requirement that the scattering wave must be an outgoing wave rules out the terms including yl 1ikr

= yl 1

ikr

[parenrightbig]

.

Equating eqs. (2.8) and (2.13), using the expansion (2.14), sin = ei ei

[parenrightbig]

/ (2i),

and the orthogonality, and noting that Mlei l = yl, we arrive at

1

2ik

h(2l + 1) Alei(l/2+l)[bracketrightBig]

Pl (cos ) + gl () = 0,

(2l + 1) eil Alei(l/2+l) = 0. (2.15)

Solving these two equations gives

Al = (2l + 1) eilei(l/2+l), (2.16)

gl () =

1

2ik (2l + 1)

1 ei2l[parenrightBig]

Pl (cos ) . (2.17)

Then, we arrive at

f (r, ) = 1

2ik

Xl=0(2l + 1)

e2il 1[parenrightBig]

Pl (cos ) yl

1 ikr

. (2.18)

When taking the limit r , the modication factor the Bessel polynomial

yl 1ikr

[parenrightbig]

tends to 1, and f (r, ) recovers the scattering amplitude in conventional scattering theory: f () = f (, ) =

1

2ik

Pl=0 (2l + 1) e2il 1

[parenrightbig]

Pl (cos ).

The leading is a p-wave modication, because the s-wave modication is y0 (x) = 1.

2.4 Outgoing wave-front surface

In conventional scattering theory, the observer is at r and the outgoing wave is a

spherical wave. When the observer is at a nite distance r, the outgoing wave-front surface, however, is a surface of revolution around the incident direction, since for a spherical potential the outgoing wave must be cylindrically symmetric.

The outgoing wave-front surface is determined by the outgoing ux jsc which serves as its surface normal vector. The outgoing ux is jsc = j jin, where j = [planckover2pi1]m Im ()

5

and jin= [planckover2pi1]m Im inin

. Here we write the wave function (2.9) as = in + sc with in = eikr cos and sc = f (r, ) eikr/r.

The outgoing wave-front surface is a surface of revolution. Its generatrix, r = r (), with jsc as the normal vector, is determined by

1r ()

dr ()d =

jscjscr = tan sc, (2.19)

where sc is the intersection angle between jsc and the radial vector.

The equation of the generatrix, eq. (2.19), is a di erential equation. The integration constant can be chosen as r (0) = R, where R is the intersection between the outgoing wave-front surface on which the observer stands and the target along the z-axis. Then the solution of eq. (2.19) can be formally written as r = r (, R).

Moreover, the Gaussian curvature of the outgoing wave-front surface reads

K () = 1

r2 cos2 sc [parenleftbigg]

(1 tan sc cot ) . (2.20)

When r , sc 0, and then K = 1/r2 reduces to a curvature of a sphere.2.5 Di erential scattering cross section

The di erential scattering section is d = jsc dS/jin. The scattering ux jsc, other than

that in conventional scattering theory, is not along the radial direction. Thus,

d = jsc dS jin =

jsc

jin

JHEP06(2014)087

1 + dsc d

[parenrightbig]

jscrjin r2d , (2.21)

r2d

cos sc = 1 + tan2 sc

where jsc =

qjsc2r + jsc2 and tan sc = jsc/jscr. A straightforward calculation gives

d d =

h|f (r, )|2 + (r, )[bracketrightBig] 1+ tan2 sc

[parenrightbig]

, (2.22)

where

(r, ) = 1

k Im

f fr + eikr(1cos) [ikr(1 + cos ) 1] f + r f r

. (2.23)

2.6 Total scattering cross section

For simplicity, we only consider the leading contribution of the total scattering cross section, t (R) = 2

[integraltext]

0

|f (R, )|2 sin d, in which the outgoing wave-front surface is approximately

a sphere of radius R.

The total cross section then reads

t (R) = 4 k2

Xl=0(2l + 1) sin2 l

2 . (2.24)

In comparison with conventional scattering theory, a modication factor

[vextendsingle][vextendsingle]y

l

[vextendsingle][vextendsingle][vextendsingle][vextendsingle]yl

1 ikR

[parenrightbigg][vextendsingle][vextendsingle][vextendsingle][vextendsingle]

1 ikR

[parenrightbig][vextendsingle][vextendsingle]2

appears.

6

2.7 Condition on potentials

In this section, we discuss the condition on the potential V (r) that preserves the validity of the solution of the radial wave equation without the large-distance asymptotics.

The radial wave equation, eq. (1.1), can be rewritten as d2ul(r)

dr2 +

hk2 l(l+1)r2 V (r)[bracketrightBig]ul (r) = 0 by introducing Rl (r) = ul (r) /r. At the distance that the inuence of V (r) can be ignored, as pointed above, the solution of the radial wave equation with V (r) = 0 is ul (r) = yl 1ikr

[parenrightbig]

eikr. In the region that the potential

cannot be ignored, we express ul (r) as

JHEP06(2014)087

ul (r) = eh(r)yl

1 ikr

eikr (2.25)

= [1 + h (r) + ]

1 l (l + 1)2ik1r +

[bracketrightbigg]

eikr. (2.26)

r = 1 and ul (r) returns to the large-distance asymptotics: ul (r) = eh(r)eikr = [1 + h (r) + ] eikr. This requires

h (r)|r 0. Without the large-distance asymptotics, however, since ul (r) must tend

to yl 1ikr

eikr as r increases, h (r) has to decrease more rapidly than yl 1ikr

When taking the large-distance asymptotics, yl 1ikr

[parenrightbig][vextendsingle][vextendsingle]

; or,

h (r) tends to zero before the vanishing of yl 1ikr

. It can be directly seen by observing eq. (2.26) that this requirement imposes a condition on h (r): without the large-distance asymptotics, h (r) must decrease more rapidly than 1r, i.e.,

h (r) =

r1+ , (2.27)

where > 0. While, as a comparison, in the case of large-distance asymptotics, since yl 1ikr

[parenrightbig]

= 1, we only needs

hasym (r) = r , (2.28)

Next, we discuss the condition on h (r) will impose what a restriction on the potential V (r).

The equation determining h (r) can be constructed by substituting eq. (2.25) into the radial wave equation:

h (r) +

h (r)

2 2ikh (r) + 2h (r)

ddr yl 1ikr

yl 1ikr

d2dr2 yl 1ikr

+ yl 1ikr

= l (l + 1)

r2 + V (r) . (2.29)

2ik

ddr yl 1ikr

yl 1ikr

For a large r, we have

ddr yl 1ikr

yl 1ikr

=

l (l + 1) 2ik

1r2 + , (2.30)

d2dr2 yl 1ikr

yl 1ikr

=

l (l + 1) ik

1r3 + . (2.31)

7

Then by eqs. (2.29), (2.30), (2.31), and condition (2.27), we obtain a condition on the potential:

V (r)

1r2+ . (2.32)

As a comparison, in the case of large-distance asymptotics, by condition (2.28) and eq. (2.29) with yl 1, we obtain the condition for the potential in the conventional

scattering theory [5]:

V (r)

1r1+ . (2.33)

When k = 0, however, there is something di erent. The condition on the potential for k = 0 becomes stronger than that for k 6= 0.

For k = 0, the expansions (2.30) and (2.31) become

ddr yl 1ikr

yl 1ikr

JHEP06(2014)087

=

lr + , (2.34)

d2dr2 yl 1ikr

yl 1ikr

= l (l + 1)

r2 + . (2.35)

Substituting expansions (2.34) and (2.35) into (2.29) and taking k = 0 give

h (r) +

h (r) 2 + 2h (r)

l r

[parenrightbigg]

= V (r) . (2.36)

By condition (2.27) and eq. (2.36), we obtain a condition on the potential for k = 0:

V (r)

1r3+ . (2.37)

As a comparison, similarly, in the case of large-distance asymptotics with k = 0, by condition (2.28) and eq. (2.29) with yl 1 and k = 0, we have

hasym (r) +

hasym (r)

2 = l (l + 1)r2 + V (r) , (2.38)

and then we obtain the condition for the potential in the conventional scattering theory:

V (r)

1r2+ . (2.39)

It is worth to note that for l 6= 0, we need only V (r) 1/r2, but for l = 0, we need the

somewhat stronger condition (2.39).

From the above discussion we learn that the condition on the potential in the scattering theory without large-distance asymptotics is stronger than that in the scattering theory with large-distance asymptotics. Without large-distance asymptotics, the inuence of the potential must be small enough at a nite distance r0; when r > r0, the inuence due to the potential can be safely neglected. In the region of r > r0, the solution is determined by the free radial wave equation.

8

Now we estimate the magnitude of r0. From eq. (2.26), we have

ul (r) = 1 + h (r) l (l + 1)2ik1r +

[bracketrightbigg]eikr. (2.40)

As analyzed above, h (r), which reects the inuence of the potential, must decrease more rapidly than

[vextendsingle][vextendsingle][vextendsingle]l(l+1)2ik 1r

. When r < r0 the inuence of the potential cannot be neglected and when r > r0 the inuence of the potential can be neglected, so r0 can be estimated by

|h (r0)| =

[vextendsingle][vextendsingle][vextendsingle][vextendsingle]

. (2.41)

Substituting the condition on h (r), eq. (2.27), and the condition on V (r), eq. (2.32), into eq. (2.29) gives / [2k (1 + )]. Then substituting eq. (2.27) and into

eq. (2.41) gives

r0

[bracketleftbigg]

Note that the case of l = 0 does not contribute to the correction from the solution without large-distance asymptotics.

In principle, di erent potentials correspond to di erent r0. The range r0 given here is indeed an upper limit of the range of the inuence of the potential without large-distance asymptotics, since the potential considered here is given by condition (2.27).

3 Conclusions and outlook

We show that one can obtain a rigorous scattering theory without the precondition r .

A rigorous scattering theory contains the information of the distance between target and ob-server is presented. The conventional scattering theory can be recovered by setting r .

In comparison with conventional scattering theory, there is an additional factor the l-th Bessel polynomial appears in the l-th partial-wave contribution. The leading modication is p-wave.

Quantum scattering theory plays an important role in many physical area and is intensively studied. Nevertheless, all studies are based on conventional scattering theory. Based on our result, we can further consider many scattering-related problems. For example, at low temperatures, the thermal wavelength has the same order of magnitude as the interparticle spacing, so the scattering in a BEC transition [6, 7] and in a transport of spin-polarized fermions [8, 9] may need to take the e ect of the distance into account. The scattering spectrum method is important in quantum eld theory [1013]; a scattering spectrum method without asymptotics can also be discussed. Moreover, the relation between scattering spectrum method and heat kernel method, which is given by ref. [14] based on refs. [15, 16], can also be improved by the exact result of the scattering theory without innite-distance asymptotics. Moreover, a related inverse scattering problems can also be systematically studied, and the result can be applied to, e.g., the interference pattern of Bose-Einstein condensates [17] and the Aharonov-Bohm e ect [18].

9

[vextendsingle][vextendsingle][vextendsingle]

l (l + 1) 2ik

1 r0

[vextendsingle][vextendsingle][vextendsingle][vextendsingle]

1/. (2.42)

JHEP06(2014)087

(1 + )

1l (l + 1)

Acknowledgments

We are very indebted to Dr. G. Zeitrauman for his encouragement. This work is supported in part by NSF of China under Grant No. 11075115.

Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

References

[1] R.G. Newton, Scattering theory of waves and particles, Dover Publications, U.S.A. (1982).

[2] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions: with formulas, graphs, and mathematical tables, Dover Publications, U.S.A. (1964).

[3] C.J. Joachain, Quantum collision theory, North-Holland Publishing Company, Amsterdam The Netherlands (1975).

[4] H.L. Krall and O. Frink, A new class of orthogonal polynomials: the Bessel polynomials, Trans. Amer. Math. Soc. 65 (1949) 100.

[5] L.E. Ballentine, Quantum mechanics: a modern development, World Scientic, Singapore (1998).

[6] P.B. Arnold and G.D. Moore, Transition temperature of a dilute homogeneous imperfect Bose gas, http://dx.doi.org/10.1103/PhysRevLett.87.120401

Web End =Phys. Rev. Lett. 87 (2001) 120401 [http://arxiv.org/abs/cond-mat/0103228

Web End =cond-mat/0103228 ] [http://inspirehep.net/search?p=find+J+Phys.Rev.Lett.,87,120401

Web End =INSPIRE ].

[7] B.M. Kastening, Bose-Einstein condensation temperature of homogenous weakly interacting Bose gas in variational perturbation theory through seven loops,http://dx.doi.org/10.1103/PhysRevA.69.043613

Web End =Phys. Rev. A 69 (2004) 043613 [http://arxiv.org/abs/cond-mat/0309060

Web End =cond-mat/0309060 ] [http://inspirehep.net/search?p=find+J+Phys.Rev.,A69,043613

Web End =INSPIRE ].

[8] W. Mullin and K. Miyake, Exact transport properties of degenerate, weakly interacting, and spin-polarized fermions, J. Low Temp. Phys. 53 (1983) 313.

[9] V. Mineev, Theory of transverse spin dynamics in a polarized Fermi liquid and an itinerant ferromagnet, http://dx.doi.org/10.1103/PhysRevB.72.144418

Web End =Phys. Rev. B 72 (2005) 144418 .

[10] S.J. Rahi, T. Emig, N. Graham, R.L. Ja e and M. Kardar, Scattering theory approach to electrodynamic Casimir forces, http://dx.doi.org/10.1103/PhysRevD.80.085021

Web End =Phys. Rev. D 80 (2009) 085021 [arXiv:0908.2649] [http://inspirehep.net/search?p=find+J+Phys.Rev.,D80,085021

Web End =INSPIRE ].

[11] A. Forrow and N. Graham, Variable phase S-matrix calculations for asymmetric potentials and dielectrics, http://dx.doi.org/10.1103/PhysRevA.86.062715

Web End =Phys. Rev. A 86 (2012) 062715 [arXiv:1210.0777] [http://inspirehep.net/search?p=find+J+Phys.Rev.,A86,062715

Web End =INSPIRE ].

[12] A. Lambrecht, P.A.M. Neto and S. Reynaud, The Casimir e ect within scattering theory, http://dx.doi.org/10.1088/1367-2630/8/10/243

Web End =New J. Phys. 8 (2006) 243 [http://arxiv.org/abs/quant-ph/0611103

Web End =quant-ph/0611103 ].

[13] S.G. Rajeev, A dispersion relation for the density of states with application to the Casimir e ect, http://dx.doi.org/10.1016/j.aop.2010.12.012

Web End =Annals Phys. 326 (2011) 1536 [arXiv:1011.4659] [http://inspirehep.net/search?p=find+J+Ann.Phys.,326,1536

Web End =INSPIRE ].

[14] H. Pang, W.-S. Dai and M. Xie, Relation between heat kernel method and scattering spectral method, nttp://dx.doi.org/10.1140/epjc/s10052-012-2021-2

Web End =Eur. Phys. J. C 72 (2012) 1 .

[15] W.-S. Dai and M. Xie, The number of eigenstates: counting function and heat kernel, http://dx.doi.org/10.1088/1126-6708/2009/02/033

Web End =JHEP 02 (2009) 033 [arXiv:0902.2484] [http://inspirehep.net/search?p=find+EPRINT+arXiv:0902.2484

Web End =INSPIRE ].

10

JHEP06(2014)087

[16] W.-S. Dai and M. Xie, An approach for the calculation of one-loop e ective actions, vacuum energies, and spectral counting functions, http://dx.doi.org/10.1007/JHEP06(2010)001

Web End =JHEP 06 (2010) 001 [arXiv:1007.3214] [http://inspirehep.net/search?p=find+EPRINT+arXiv:1007.3214

Web End =INSPIRE ].

[17] W.-M. Liu, B. Wu and Q. Niu, Nonlinear e ects in interference of Bose-Einstein condensates, http://dx.doi.org/10.1103/PhysRevLett.84.2294

Web End =Phys. Rev. Lett. 84 (2000) 2294 .

[18] F. Nicoleau, An inverse scattering problem with the Aharonov-Bohm e ect,http://dx.doi.org/10.1063/1.533403

Web End =J. Math. Phys. 41 (2000) 5223 .

JHEP06(2014)087

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