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eneralized Ornstein-Uhlenbeck processesV. Bezuglyy and B. MehligDepartment of Physics, Göteborg University, 41296 Gothenburg, Sweden

M. WilkinsonFaculty of Mathematics and Computing, The Open University, Walton Hall,Milton Keynes, MK7 6AA, United Kingdom

K. NakamuraDepartment of Applied Physics, Osaka City University, Osaka 558-8585, Japan

E. ArvedsonDepartment of Physics, Göteborg University, 41296 Gothenburg, Sweden

�Received 23 February 2006; accepted 26 April 2006; published online 17 July 2006�

We solve a physically significant extension of a classic problem in the theory ofdiffusion, namely the Ornstein-Uhlenbeck process �Ornstein and Uhlenbeck, Phys.Rev. 36, 823 �1930��. Our generalized Ornstein-Uhlenbeck systems include a forcewhich depends upon the position of the particle, as well as upon time. They exhibitanomalous diffusion at short times, and non-Maxwellian velocity distributions inequilibrium. Two approaches are used. Some statistics are obtained from a closed-form expression for the propagator of the Fokker-Planck equation for the casewhere the particle is initially at rest. In the general case we use spectral decompo-sition of a Fokker-Planck equation, employing nonlinear creation and annihilationoperators to generate the spectrum which consists of two staggered ladders. © 2006American Institute of Physics. �DOI: 10.1063/1.2206878�

. INTRODUCTION

This paper introduces a physically important extension of a classic problem in the theory ofiffusion, namely the Ornstein-Uhlenbeck process.1 Our results are obtained by spectral decom-osition of a linear operator. The spectrum of this operator consists of two ladders of eigenvaluesith, respectively, odd and even parity. The ladders of eigenvalues are staggered, that is thedd-even step is different from the even-odd step �see Fig. 1�. The corresponding eigenfunctionsre generated by a raising operator. A concise account of our work on these staggered ladderpectra appeared earlier.2 In the following we show how the results summarized in Ref. 2 werebtained. We also derive new results, not included in our earlier report: a closed-form solution forxample, and the generalization of our previous results to a continuous family of diffusion pro-esses.

. The Ornstein-Uhlenbeck process

Before we discuss our extension of the Ornstein-Uhlenbeck process, we describe its usualorm.1 This considers a particle of momentum p subjected to a rapidly fluctuating random force

f�t� and subject to a drag force −�p, so that the equation of motion is

p = − �p + f�t� . �1�

he random force has statistics �f�t��=0, �f�t�f�t��=C�t− t�� angular brackets denote ensembleverages throughout�. If the correlation time � of f�t� is sufficiently short ��1�, the equation of
otion may be approximated by a Langevin equation:
47, 073301-1022-2488/2006/47�7�/073301/21/$23.00 © 2006 American Institute of Physics

Jul 2006 to 137.108.145.11. Redistribution subject to AIP license or copyright, see http://jmp.aip.org/jmp/copyright.jsp

http://dx.doi.org/10.1063/1.2206878

http://dx.doi.org/10.1063/1.2206878

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073301-2 Bezuglyy et al. J. Math. Phys. 47, 073301 �2006�

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dp = − �pdt + dw , �2�

here the Brownian increment dw has statistics �dw�=0 and �dw2�=2D0dt. The diffusion constants

D0 =1

2�

−�

�

dt �f�t�f�0�� . �3�

his problem is discussed in many textbooks �for example, Ref. 3�.

. Generalized Ornstein-Uhlenbeck processes

Our extension arises when the force depends upon position as well as time. We consider thease where the fluctuations of the force on the particle are mainly a consequence of the spatial,ather the temporal, fluctuations of the force f�x , t�. A consequence of this difference is that thempulse �w supplied to the particle in a short time �t depends upon the momentum of the particle.f the particle is at position x0 at time t0, this impulse is

�w = �t0

t0+�t

dt f�x0 + p�t − t0�/m,t� + O��t2� . �4�

In particular, the impulse approaches zero as the speed �p� /m of the particle increases, becausehe motion of the particle effects an average over the spatial fluctuations of the force. This can beeen clearly by considering the second moment of �w. We assume that the force f�x , t� has theollowing statistics:

�f�x,t�� = 0, �f�x,t�f�x�,t�� = C�x − x�,t − t�� . �5�

The spatial and temporal correlation scales of the random force f�x , t� are � and �, respec-ively. We consider the case where �for most of the time� the momentum of the particle is largeompared to p0=m� /�, then the force experienced by the particle decorrelates much more rapidlyhan the force experienced by a stationary particle. If �t is large compared to � but small comparedo 1/�, we can estimate the variance of the impulse ��w2�=2D�p��t as follows �due to transla-ional invariance, we consider without loss of generality a particle which starts from position x

IG. 1. The spectrum of H consists of two equally spaced �ladder� spectra �n− and �n

+ which are “staggered” �that is, they

re interleaved with uneven spacings�. A and A+ do not change the parity of the eigenfunctions.

0 at time t=0�:


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��w2� = �0

�t

dt1�0

�t

dt2 �f�pt1/m,t1�f�pt2/m,t2�� = �t�−�

�

dt C�pt/m,t� + O��2� . �6�

e define the momentum diffusion constant by writing ��w2�=2D�p��t+O��t2�, and find

D�p� =1

2�

−�

�

dt C�pt/m,t� . �7�

hen p� p0 we recover D�p�=D0. When p� p0, we can approximate �7� to obtain

D�p� =D1p0

�p�+ O�p−2�, D1 =

m

2p0�

−�

�

dX C�X,0� . �8�

hen the force is the gradient of a potential V�x , t� with a correlation function having continuouserivatives, we find that D1 is zero. This case is discussed in Sec. VIII, where it is shown that�p��p�−3 provided the correlation function of V�x , t� is sufficiently differentiable. Anotherariation, also discussed in Sec. VIII, arises when the correlation function of the force exhibits aiscontinuity at t=0 �as when the potential V�x , t� is itself generated by an Ornstein-Uhlenbeckrocess�. In this case D�p��p�−2, and other exponents are also possible. We therefore consider aeneral situation where D�p��p�− and give exact results for the case

D�p� = D�p0/�p�� 9�

ith 0. We analyze the dynamics by solving a Fokker-Planck equation which determines therobability density for the Langevin process in which the momentum has diffusion constant giveny �9�. We discuss the form of this Fokker-Planck equation in Sec. II; the remainder of thisntroduction will set our work in context with earlier research on related topics.

. Earlier work

The motion of a damped particle subjected to a force fluctuating in both space and time wasrst studied by Deutsch,4 who addressed an entirely different aspect of the problem. Deutschonsidered the case where the momentum of the particle remains small compared to p0, and posedhe question of whether particles aggregate. He discovered that there is a phase transition betweenoalescing and noncoalescing trajectories. �Two of the authors of the present paper subsequentlyolved Deutsch’s one-dimensional model exactly,5 and results for two and three spatial dimensionsre discussed in Refs. 6 and 7�. All of these papers only considered cases where p� p0.

Sturrock8 analyzed the motion of a particle subjected to a spatially varying force field withoutamping. He introduced the concept of a momentum diffusion constant which varies as a functionf the momentum: that is, he considered the same problem as is addressed in the present paper, butn the limit of damping constant �=0. Subsequently Golubovic, Feng, and Zeng9 identified themportance of the relation D�p��p�−3 �in the case of a potential force�, and discussed the naturef the Fokker-Planck equation and its solution in the case where �=0. It was argued that thearticle exhibits anomalous diffusion and solution for the propagator of the Fokker-Planck equa-ion with initial value p=0 was proposed. Later Rosenbluth10 pointed to an error in the evaluationf this propagator. The results of Refs. 8–10 were applied to the stochastic acceleration11 ofarticles in plasmas, and subsequent contributions have concentrated on refining models for thealculation of D�p� �see, for example, Refs. 12 and 13�.

In the following we analyze the problem with the damping term, proportional to �, included.urprisingly, we find that this more general problem is more tractable: we are able, for example,

o obtain precise results concerning the problems considered in Refs. 9 and 10 by taking, in ourolutions, the limit �→0.

There is a large literature devoted to the motion of particles advected in random velocity fieldscorresponding to the large-� limit of the model we study�. In the case where the velocity field is

14,15
ndependent of time, subdiffusive motion is typically found. The advection of tracers in a

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urbulent fluid is described by models with rapidly fluctuating velocity fields.16 In our problem thenertia of the particles plays an important role. Particles suspended in a turbulent fluid can showurprising clustering properties when inertia effects are significant. These were first proposed byaxey;17 the current state of knowledge is summarized in Ref. 7. In cases where the random force

esults from motion of the surrounding fluid, it is not possible for the condition p� p0 to beealized.7

Although our generalized Ornstein-Uhlenbeck process exhibits anomalous diffusion �at shortimes�, this is not a result of power-law distributions that are built into the model. This distin-uishes it from the anomalous diffusion of Levy flights or walks, reviewed in Ref. 18, which areconsequence of power-law distributions of the step lengths or waiting times for random jumps.ur model is thus distinct from the ‘fractional Ornstein-Uhlenbeck process’ described in Ref. 18.

Finally, we remark that a brief summary of many of the results of this paper has already beenublished.2 The closed-form solution of Sec. III, the Wentzel-Kramers-Brillouin �WKB� analysis,nd most of the results for general values of were not discussed in Ref 2.

. Description of our results and outline of this paper

In order to simplify the presentation, we describe in detail only our results for the case =1,orresponding to a generic random force. Corresponding expressions for general values of arebtained using the same method, and we quote the most important results for general values of n Sec. VIII at the end of the paper.

In Sec. II, the Fokker-Planck equation for the generalized Ornstein-Uhlenbeck processes isescribed. In Sec. III we briefly discuss a particular closed-form solution, which enables us toetermine the steady state momentum distribution �which is non-Maxwellian� and some statistics,uch as the time evolution of the variance of the momentum. The results of Sec. III are notufficient to enable all statistics to be calculated, and in the general case we obtain statistics via apectral decomposition of the Fokker-Planck equation. Section IV discusses this spectral decom-osition. We transform the Fokker-Planck operator into a Hermitean operator and determine theigenvalues and eigenvectors of this “Hamiltonian” operator by generating them using a new typef raising and lowering operators, which are nonlinear second-order differential operators. Wehow that the resulting spectrum is a ladder spectrum, consisting of separate ladders for the oddnd even parity states. These are staggered: the odd-even separation differs from even-odd. Sec-ion V contains calculations of the matrix elements needed for computing correlation functionsnd expectation values. In Sec. VI we summarize our results on diffusion and anomalous diffusionor generic random forcing.

Section VII discusses a technical issue concerning our evaluation of the spectrum. When thendex of the eigenvalue is large, it is possible to apply standard WKB approximation methodsverywhere except in the vicinity of a singularity of the Hamiltonian. We show that the singularityntroduces phase shifts which explain the staggered-ladder structure of the spectrum.

Finally, in Sec. VIII we explain in more detail how other values of can arise and summarizeur results for general .

I. FOKKER-PLANCK EQUATIONS

We consider a particle with equations of motion

x = p/m, p = − �p + f�x,t� �10�

here the force f�x , t� is random, with statistics given by Eq. �5�. In the limit as the correlationime � of the force approaches zero, the equation of motion of the momentum may be approxi-ated by a Langevin equation, �2�, where the random increment dw has second moment �dw2�2D�p�dt with D�p� given by �7�. This Langevin equation for the stochastic evolution of p�t�orresponds to a Fokker-Planck equation �generalized diffusion equation� for the probability den-

3
ity of the momentum, P�p , t�. Using standard results, the Fokker-Planck equation is

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�P

�t= −

�

�p�v�p�P� +

�2

�2p�D�p�P� , �11�

here

v�p� =�dp�dt

, D�p� =�dp2�2dt

. �12�

ote that we can replace dp by dw in the expression for D�p�, because the neglected terms are ofigher order in dt, and that D�p� has already been obtained in Eq. �7�. In order to determine theorrect form of the Fokker-Planck equation it remains to determine �dp�=−�p dt+ �dw�.

Expanding the impulse �4� about a reference trajectory x�t�= pt /m, and using the fact thatf�x , t��=0, we obtain

��w� =1

m�

0

�t

dt1�0

t1

dt2�0

t2

dt3 exp�− ��t2 − t3�� f

�x�pt1/m,t1�f�pt3/m,t3�� . �13�

ote that throughout the three-dimensional region of integration, we have 0� t3� t2� t1��t, andhe short correlation time implies that the integrand is negligible unless �t1− t3��. The integrands therefore significant along a line rather than a surface, because t2 must lie between t1 and t3. Thentegral is therefore O��t�, rather than O��t2� which would obtain if the integrand were significantn a surface. We replace the factor exp�−��t2− t3�� by unity because ��1, and the other factor isegligible when �t2− t3� �. The integral over t2 then gives simply t1− t3. Writing t= t1− t3, in theimit ��1, the result is therefore

��w� =�t

2m�

−�

�

dt t �f

�x�0,0�f�pt/m,t�� = �t

d

dpD�p� . �14�

his implies

v�p� = − �p +d

dpD�p� . �15�

osenbluth10 has pointed out that this relation can also be obtained as a consequence of applyinghe principle of detailed balance.

With �7� and �15�, the following Fokker-Planck equation obtains:

�P

�t=

�

�p��p + D�p�

�

�p P . �16�

turrock8 introduced a related Fokker-Planck equation �without the damping term� and also gaven expression for D�p� analogous to Eq. �7�.

In the following we discuss our solution of �16� with D�p� given by Eq. �9�, for the particularase of generic random forcing �corresponding to =1�. Results for other values of are obtainedn an analogous fashion. The general case is briefly described in Sec. VIII.

II. A PARTICULAR CLOSED-FORM SOLUTION

In this section we introduce a particular solution of the Fokker-Planck equation �16� with D�p�iven by �9�. We restrict ourselves to the case of generic random forcing �corresponding to =1�:

�P=

� ��p + D1p0 � P . �17�

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Consider the distribution P�p , t� of momentum p for particles initially at rest. It satisfies thenitial condition P�p ,0�=��p� where ��p� is the Dirac �-function. For this particular initial con-ition, we have found the following closed-form solution of �17�:

P�p,t� =1

2��4/3��1/3

�3p0D1�1 − e−3�t��1/3 exp�−��p�3

3p0D1�1 − e−3�t�� . �18�

quation �18� determines how the moments of momentum grow for a particle initially at rest:

�p2l�t�� = �3D1p0

� 2/3��2l + 1�/3�

��1/3��1 − e−3�t�2l/3 �19�

or positive integers l. This result is consistent with the result obtained in Ref. 2 �Eq. �8� in thataper�. In the limit of small times �19� gives rise to anomalous diffusion

�p2l�t�� t2l/3. �20�

t large times ��t�1�, by contrast, we obtain a stationary non-Maxwellian momentum distribu-ion

P0�p� =1

2��4/3��1/3

�3p0D1�1/3 exp�− ��p�3/�3p0D1�� . �21�

he particular solution �18� generalizes in a natural way to other values of .However, in order to determine the momentum correlation function and the spatial diffusion

roperties, the particular solution �18� is not sufficient, the general solution for arbitrary initialondition is required. We have not been able to obtain the general solution to �16� in closed form.herefore, we determine it using spectral decomposition: we construct the eigenvalues �n and

igenfunctions �n of a Hermitian operator H corresponding to the Fokker-Planck equation �17�.e identify raising and lowering operators A+ and A which map one eigenfunction to anotherith, respectively, two more or two fewer nodes. We use these to obtain the spectrum and eigen-

unctions of H which in turn allow us to construct the propagator, expectation values, and corre-ation functions. This approach is described in Secs. IV and V.

V. SPECTRAL DECOMPOSITION

Introducing dimensionless variables �t�=�t and p=zp0�D1 / ��p02��1/3� we write �17� as

�P

�t�=

�

�z�z +

1

�z��

�z P � FP . �22�

t is convenient to transform the Fokker-Planck operator F to a Hermitian form which we shallefer to as the Hamiltonian operator:

H = P0−1/2FP0

1/2 =1

2−

�z�3

4+

�

�z

1

�z��

�z. �23�

ere P0�z��exp�−�z�3 /3� is the stationary solution �21� satisfying FP0=0. We solve the diffusionroblem by constructing the eigenfunctions of the Hamiltonian operator. In the following we makese of Dirac notation19 of quantum mechanics to write the equations in a compact form and tomphasize their structure.

The eigenfunctions of the Fokker-Planck equation �16� are alternately even and odd functions,

efined on the interval �−� ,��. The operator H, describing the limiting case of this Fokker-Planck

perator, is singular at z=0. We identify two eigenfunctions of H by inspection, �0+�z�

+ 3 + − − 3 −
C0 exp�−�z� /6� which has eigenvalue �0 =0 and �0�z�=C0z�z�exp�−�z� /6� with eigenvalue �0

=

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−2. These eigenfunctions are of even and odd parity, respectively �zero and one node, respec-

ively�. Our approach to determining the full spectrum is to define a raising operator A+ whichaps any eigenfunction �n

±�z� to its successor with the same parity, �n+1± �z�, having two additional

odes.

. Algebra of raising and lowering operators

We write

H = a−�z�−1a+. �24�

ere a±= ��z±z�z� /2�. We introduce the operators

A = a+�z�−1a+ and A+ = a−�z�−1a− �25�

s well as

G = a+�z�−1a−. �26�

ote that A+ is the Hermitian conjugate of A. The commutator of A and A+ is

�A,A+� = − 3�H + G� . �27�

ote also that H− G= I �where I is the identity operator�.

. Eigenvalues

It can be verified that

�H,A� = 3A and �H,A+� = − 3A+. �28�

hese expressions show that the action of A and A+ on any eigenfunction is to produce anotherigenfunction with eigenvalue increased or decreased by three, or else to produce a function which

s identically zero. The operator A+ adds two nodes, and repeated action of A+ on �0+�z� and �0

−�z�herefore exhausts the set of eigenfunctions. Together with �0

+=0 and �0−=−2 this establishes that

he spectrum of H is �see Fig. 1�

�n+ = − 3n and �n

− = − 3n − 2 n = 0, . . . ,� . �29�

. Eigenfunctions

We represent the eigenfunctions by of H by kets ��n−� and ��n

+�. The actions of A and A+ are

A+��n±� = Cn+1

± ��n+1± � and A��n

±� = Cn±��n−1

± � . �30�

he normalization factor Cn+1− is determined as follows:

1 = ��n+1− ��n+1

− � = �Cn+1− �−2��n

−�AA+��n−� = �Cn+1

− �−2��n−��A,A+� + A+A��n

−� . �31�

t follows

�Cn+1− �2 = �3�− 2�n

− + 1� + �Cn−�2� . �32�

y recursion we obtain


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Cn− = �3n�3n + 2��1/2. �33�

his determines the normalization of the states �A+�n��0−�,

��n−� = Nn

−�A+�n��0−� �34�

ith

Nn− = ��

k=1

n

3k�3k + 2� −1/2

N0−. �35�

For the positive-parity states we proceed in a similar fashion and obtain

Cn+ = �3n�3n − 2��1/2. �36�

his implies

��n+� = Nn

+�A+�n��0+� �37�

ith

Nn+ = ��

k=1

n

3k�3k − 2� −1/2

N0+. �38�

he operators A+ and A differ from the usual examples of raising and lowering operators in thathey are of second order in d/dz, whereas other examples of raising and lowering operators are ofrst order in the derivative. The difference is associated with the fact that the spectrum is ataggered ladder: only states of the same parity have equal spacing, so that the raising andowering operators must preserve the odd-even parity. This suggests replacing a first-order opera-or which increases the quantum number �total number of nodes� by one with a second-orderperator which increases the quantum number by two, preserving parity.

There is an alternative approach to generating the eigenfunctions of H. This equation falls intone of the classes considered in Ref. 20, and we have written down first-order operators whichap one eigenfunction into another. However, these operators are themselves functions of the

uantum number n, making the algebra cumbersome. The approach is briefly described in the nextection.

. Schrödinger factorization

Consider the eigenvalue problem

H�� = �� 39�

ith Hamiltonian �23�. For z 0 it can be transformed by the variable change x=z3:

��3x�2 d2

dx2 + 3xd

dx− x�1

4x + � −

1

2 ��x� = 0. �40�

quation �40� is a Fuchsian linear differential equation with regular singular points of rank lesshan or equal to two. Equation �40� can therefore be factorized using a generalized Schrödingeractorization scheme �see Ref. 20 for a review of this method�.

Applying this scheme we have obtained raising and lowering operators generating the spec-±
rum �29�. The raising operator acting on �n is given by

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T±,n+1 = − 3xd

dx+

x

2− 3�n + �±� �41�

ith �+=1/3 and �−=1. The lowering operator acting on �n± is

T±,n = − 3xd

dx−

1

2x + 3�n + �±� �42�

ith �+=0 and �−=2/3. Note that the Hermitian conjugates of T±,n+1 and T±,n are

�T±,n+1�+ = − T±,n+1 + 3, �43�

�T±,n�+ = − T±,n + 3. �44�

he raising and lowering operators satisfy

T±,n+1��n±� = Cn+1

± ��n+1± � and T±,n��n

±� = Cn±��n−1

± � �45�

ith Cn± given by �33� and �36�, generating the spectrum �29�. The operators differ from A and A+

ntroduced in Sec. IV in that they are of first order in d/dx, and in that they depend on the statehey are applied to.

. CORRELATION FUNCTIONS AND MATRIX ELEMENTS

. Correlation functions

The required solutions of the Fokker-Planck equation may be expressed in terms of theropagator K�y ,z , t� which is the probability density for the scaled momentum to reach z after time

, starting from y. It satisfies the Fokker-Planck equation �t�K= FK and can be expressed in terms

f the eigenvalues �n� and eigenfunctions �n

��z�= P0−1/2�z��n

��z� of F:

K�y,z;t�� = �n�

an��y��n

��z�exp��n�t�� 46�

or t� 0. The expansion coefficients an��y� are determined by the initial condition K�y ,z ;0�

��z−y�, namely an��y�= P0

−1/2�n��y�. In terms of the eigenfunctions of H we have �for t� 0�

K�y,z;t�� = �n�

P0−1/2�y��n

��y�P01/2�z��n

��z�exp��n�t�� . �47�

quilibrium correlation functions of an observable O�z� are given by

�O�z0�O�zt��eq. = �−�

�

dz �−�

�

dy O�z�O�y�K�y,z;t��P0�y� . �48�

ince P01/2�y�=�0

+�y� this corresponds to

�O�z0�O�zt��eq. = �n�

��0+�O��n

��2 exp��n�t�� 49�

or t� 0. The momentum correlation function in equilibrium, for instance, is

�pt�p0�eq. = p02� D1

�p02 2/3

�n

��0+�z��n

−�2 exp��n−t�� 50�

+ ˆ −
or t� 0, which requires the evaluation of matrix elements ��0�z��n�.

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Consider on the other hand the time-dependence of �x2�t��, with particles initially at rest at therigin. We need to evaluate

�x2�t�� =1

m2�0

t

dt1 �0

t

dt2 �pt1pt2

� . �51�

n dimensionless variables this corresponds to

�x2�t�� =1

�2� p0D1

� 2/3 1

m2�0

t�dt1� �

0

t�dt2� �zt1�

zt2�� . �52�

he required correlation function is �assuming t2� t1� 0�

�zt2�zt1�

� = �−�

�

dz1 �−�

�

dz2 z1z2K�z1,z2;t2� − t1��K�0,z1;t1��

= �n,m

�m+ �0�

�0+�0�

��0+�z��n

−��n−�z��m

+ �exp��n−�t2� − t1�� + �m

+ t1�� . �53�

n order to evaluate �53�, the ratios of wave-function amplitudes �m+ �0� /�0

+�0� are required inddition to matrix elements of z. The matrix elements ��n

−�z��m+ � are determined in Sec. V B, while

he ratios of eigenfunctions are calculated in Sec. V C.

. Matrix elements Š�m+ �z��n

−‹

To evaluate the matrix elements Zmn= ��m+ �z��n

−� we proceed in three steps: we first evaluate

0n, in a second step the matrix elements Zmn are related to Z0,n−m for m�n. Third, Zmn isvaluated for m n.

. Matrix elements Š�0+�z��n

−‹

Consider first Z0n= ��0+�z��n

−�. These matrix elements are obtained by recursion. To evaluate

Z0,n+1 = ��0+�zA+��n

−�/Cn+1− �54�

e write zA+= zG+ z�A+− G�= z�H− I�+ z�A+− G�. It follows

��0+�zA+��n

−� = ��n− − 1�Z0n + ��0

+�z�A+ − G��n−� . �55�

sing �A+− G�=−za− and �z2 , a−�=−2z we obtain

��0+�zA+��n

−� = ��n− + 1�Z0n. �56�

his corresponds to the recursion

Z0n = �− 1�n

�k=0

n−1

�3k + 1�

��k=0

n−1

3�k + 1��3k + 5�

Z00. �57�

ith Z00=3−5/12�� /��2/3� �found by direct evaluation of an integral� we obtain

Z0n = �− 1�n3−5/12��2/3�

��n + 1/3�. �58�
�2� ��n + 1��n + 5/3�

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−‹ for mÏn

Consider now the case m�n. Let

Jmn = ��0+�Amz�A+�n��0

−� = ��0+�Am�z,A+��A+�n−1��0

−� + ��0+�AmA+z�A+�n−1��0

−� .

e use �z , A+�=−��z�−1a−+ a−�z�−1� to write

Jmn = − ��0+�Am�z�−1a−�A+�n−1��0

−� − ��0+�Ama−�z�−1�A+�n−1��0

−� + ��0+�AmA+z�A+�n−1��0

−�

= Jmn�1� + Jmn

�2� + Jmn�3� �59�

nd evaluate the three terms separately. The third one gives

Jmn�3� = ��0

+�AmA+z�A+�n−1��0−� = �Cm

+ �2��0+�Am−1z�A+�n−1��0

−� = �Cm+ �2Jm−1n−1. �60�

onsider next the first term: Am�z�−1a−�A+�n−1= Am−1a+�z�−1a+�z�−1a−�A+�n−1= Am−1a+�z�−1G�A+�n−1,nd thus

��0+�Am�z�−1a−�A+�n−1��0

−� = ��n−1− − 1��0

+�Am−1a+�z�−1�A+�n−1��0−� . �61�

sing Am−1a+�z�−1�A+�n−1= Am−1a+�z�−1a−�z�−1a−�A+�n−2= Am−1G�z�−1a−A+n−2 we obtain the recursion

Jmn�1� = ��n−1

− − 1��m−1+ − 1�Jm−1n−1

�1� = 3n�3m − 2�Jm−1n−1�1� . �62�

Now consider Jmn�2�. Using a−�z�−1�A+�n−1= A+�z�−1a−A+n−2 it follows

Jmn�2� = �Cm

+ �2Jm−1n−1�1� . �63�

his implies

Jmn�2� =

�Cm+ �2

��n−1− − 1��m−1

+ − 1�Jmn

�1� =m

nJmn

�1� . �64�

ote also that J0n�3�=0, as well as J0n

�2�=0. This gives J0n�1�=J0n �consistent with �64��. We obtain

Jmn�1� = �

k=1

m

3�n − m + k��3k − 2�J0n−m, �65�

hich results in

Jmn�1� = �− 1�m+n��2/3�3m+n+5/6

6�

��n + 1��m + 1/3��n − m + 1/3��n − m + 1�

. �66�

Equation �66� allows us to write down an inhomogeneous recursion for Jmn:

Jmn = 3m�3m − 2�Jm−1n−1 + �1 + m/n�Jmn�1� , �67�

here the inhomogeneous term is given by �66�. Iterating this recursion we obtain

Jmn = �l=0

m � �k=l+1

m

3k�3k − 2� �1 +l

n − m + l Jln−m+l

�1� , �68�

hich results in


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Jmn = �− 1�m−n3m+n+1��2/3�2

4�2 �m + n + 1��n + 1��m + 1/3��n − m + 1/3�

��n − m + 2�J00. �69�

his determines Jmn for nm.


−‹ for m>n

For m n we use instead

Jmn = ��0+�AmzA+n��0

−� = ��0+�Am−1�A, z�A+n��0

−� + ��0+�Am−1zA�A+�n��0

−� �70�

nd proceed as before. We find that Jmn=0 for m n+1. For m=n+1 we obtain

Jn+1n = ��0+�An�z�−1a+�A+�n��0

−� + ��0+�Ana+�z�−1�A+�n��0

−� + ��0+�An�z�A�A+�n��0

−�

= Jn+1n�1� + Jn+1n

�2� + Jn+1n�3� . �71�

onsider first Jn+1n�3� :

Jn+1n�3� = �Cn

−�2Jnn−1�3� = 3n�3n + 2�Jnn−1

�3� . �72�

econd, using An�z�−1a+�A+�n= An−1a+�z�−1A�A+�n we determine Jn+1n�1� :

Jn+1n�1� = �Cn

−�2Jnn−1�2� = 3n�3n + 2�Jnn−1

�2� . �73�

hird,

Jn+1n�2� = ��n

+ − 1��n−1− − 1�Jnn−1

�2� = �3n + 1�3nJnn−1�2� . �74�

e deduce that

Jn+1n�1� =

3n + 2

3n + 1Jn+1n

�2� �75�

nd obtain the recursion

Jn+1n = Jn+1n�1� + Jn+1n

�2� + Jn+1n�3� = 3n�3n + 2�Jnn−1 + �1 +

3n + 2

3n + 1 Jn+1n

�2� . �76�

Iterating �74� we obtain

Jn+1n�2� = ��

k=1

n

3k�3k + 1� J10�2� =

3�3

2�9n��2/3��n + 1��n + 4/3�J00 �77�

nd thus from �76�

Jn+1n =�3

2�9n��2/3��2 + n��n + 4/3�J00. �78�

omparing this result to �69� we find that �69� gives the correct result for m=n+1, although it waserived assuming m�n. Normalizing to obtain Zmn our final result is

Zmn = �− 1�m−n35/6

6��m + n + 1��2/3�

��n + 1��m + 1/3��m + 1��n + 5/3�

��n − m + 1/3��n − m + 2�

�79�

or nm−1 and zero otherwise.


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. Ratios of eigenfunctions

In this section we show how to evaluate �n+�0� /�0

+�0�. For z 0, the eigenfunctions �n+�z� are

f the form Nn+gn�z�exp�−z3 /6�, where gn�z� is polynomial in z3, of the form

gn�z� = gn�0� + gn

�1�z3 + ¯ . �80�

e determine how A+ and H act on these polynomials. To this end we define

A�+ = ez3/6A+e−z3/6 = ��z − z2�z−1��z − z2� , �81�

H� = ez3/6He−z3/6 = ��z − z2�z−1�z. �82�

his implies

A�+ − H� = − ��z − z2�z = z3 − z�z − 1 �83�

nd thus

�A�+ − H��gn = − gn�0� + O�z3� . �84�

sing �n+=−3n we obtain

H�gn = − 3ngn�0� + O�z3� �85�

rom the eigenvalue equation. Taking �84� and �85� together we have

A�+gn = − �3n + 1�gn�0� + O�z3� . �86�

quation �86� implies

gn+1�0� = − �3n + 1�gn

�0�. �87�

ith �30� it follows

�n+1+ �0� = Nn+1

+ gn+1�0� = − �3n + 1�Nn+1

+ /Nn+�n

+�0� = −� 3n + 1

3�n + 1��n

+�0� . �88�

ur final result for the ratio of wave-function amplitudes is therefore

�n+�0�/�0

+�0� = �− 1�n��3��2/3�2�

��n + 1/3��n + 1�

. �89�

I. EQUILIBRIUM CORRELATIONS, DIFFUSION, AND ANOMALOUS DIFFUSION

In the following the momentum correlation function in equilibrium and the time dependencef �x2�t�� are determined.

. Momentum correlation function in equilibrium

The correlation function of momentum in equilibrium is obtained from �49�. We have

�pt�p0�eq. = p02� D1

�p02 2/3

�n

Z0n2 exp��n

−t�� . �90�

sing �79� we obtain


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�ptp0�eq. =��4/3�

31/3��5/3�� p0D1

� 2/3

e−2�tF21�1

3,1

3;5

3;e−3�t �91�

or t 0. Here F21 is a hypergeometric function.21 It follows that �ptp0�eq. decays as exp�−2�t� atarge times as opposed to exp�−�t� in the Ornstein-Uhlenbeck process.

. Diffusion at long times

We now turn to �x2�t��. This expectation value is calculated using Eqs. �52�, �53�, �79�, and89�. We have

�x2�t�� = � p0D1

� 2/3 1

m2�2�k=0

�

�l=k−1

��k

+�0��0

+�0�Z0lZklTkl �92�

ith

Tkl�t�� = �0

t�dt1� �

t1�

t�dt2� e�l

−�t2�−t1��+�k+t1� + �

0

t�dt1� �

0

t1�dt2� e�l

−�t1�−t2��+�k+t2� = 2

�k+�1 − e�l

−t�� − �l−�1 − e�k

+t��l

−�k+��k

+ − �l−�

.

�93�

We define

Akl ��k

+�0��0

+�0�Z0lZkl =

32/3��2/3�2

12�2

�k + l + 1��k + 1/3��l + 1/3��l − k + 1/3��k + 1��l + 5/3��l − k + 2�

�94�

or k l−1 and zero otherwise. In order to determine �x2�t��, the sum

S�t�� = �kl

AklTkl�t�� 95�

s required. Note that Akl=0 for k� l−1. Consider the behavior of �95� at large values of t�. Werite the k=0 term separately

T0l = −2t�

�l− +

2

�l−2 �e�l

−t� − 1� . �96�

his gives

�x2�t�� = 2Dxt + � p0D1

� 2/3 1

m2�2�l=0

�

Z0l2 2

�l−2 �e�l

−t� − 1� + � p0D1

� 2/3 1

m2�2�k=1

�

�l=k−1

��k

+�0��0

+�0�Z0lZklTkl�t�� .

�97�

t large t� the secular term dominates and diffusion is thus recovered. The diffusion constant isbtained as2

Dx =�p0D1�2/3

m2�5/3

�3−5/6

2��2/3�2F32�1

3,1

3,2

3;5

3,5

3;1 . �98�


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. Anomalous diffusion at short times

Next we consider short times. In order to evaluate �95� at small values of t�, we replace theums in �95� by integrals:

S�t�� = �0

�

dl �0

l

dk T�k,l,t�A�k,l� . �99�

n order to evaluate �99� we use the asymptotic form of the coefficients Akl:

A�k,l� 32/3��2/3�2

12�2

k + l

k2/3l4/3�l − k�5/3 . �100�

he coefficients A�k , l� exhibit nonintegrable divergence k→ l. In view of this divergence we makese of a sum rule of the Akl:

�k=0

l+1

Akl = 0. �101�

t can be derived by considering

�k=0

l+1

�k+�z��k

+�z��l−� = �

k=0

�

�k+�z��k

+�z��l−� = �

k=0

�

�z��k+��k

+�z��l−� = �z�z��l

−� , �102�

hich vanishes for z=0. Replacing sums by integrals the sum rule amounts to

�0

l

dk A�k,l� = 0. �103�

Equation �103� allows us to write

S�t�� = �0

�

dl �0

l

dk �T�k,l;t�� − limk→l

T�k,l;t��A�k,l� . �104�

e find that the divergence of A�k , l� is reduced to an integrable divergence by the fact that�k , l ; t�−limk→l T�k , l ; t�=O�k− l�. Approximately, T�k , l ; t� is given by

T�k,l;t� = − 23k�1 − exp�− 3lt�� − 3l�1 − exp�− 3kt��

27lk�l − k�. �105�

hanging the integration variables in �104� to x=3lt and xy=3kt, we have k=xy / �3t� and lx / �3t�, and

A�x,y,t� =310/3��2/3�2

12�2 t8/3x−8/3 1 + y

y2/3�1 − y�5/3 . �106�

he Jacobian of the transformation is J=x / �9t2�. In the new variables,

T�x,y ;t� = − 2t2

x�a�x� − a�xy�

1 − y� �107�

here

a�x� = �1 − exp�− x��/x . �108�

sing


w

T

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tc

V

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limy→1

T�x,y ;t� =� − 2t2

x� �

�ya�xy��

y=1= − 2

t2

xxa��x� �109�

e finally obtain

T�k,l;t� − limk→l

T�k,l;t� = − 2t2

x�a�x� − a�xy�

1 − y− xa��x�� . �110�

his gives

S�t� = − C t8/3�0

�

dx x−8/3�0

1

dy �a�x� − a�xy�1 − y

− xa��x�� 1 + y

y2/3�1 − y�5/3 �111�

ith C=31/3��2/3�2 / �2�2�. Equation �111� implies anomalous spatial diffusion at short times:

�x2�t�� = Cx�p0D1�2/3m−2t8/3 �112�

ith

Cx = − C�0

�

dx x−8/3�0

1

dy �a�x� − a�xy�1 − y

− xa��x�� 1 + y

y2/3�1 − y�5/3 . �113�

This anomalous diffusion is analogous to that described by Golubovic, Feng, and Zeng9 in thease where there is no damping �they considered the case where =3 in their paper�. Our resultsive the prefactor as well as the scaling behavior. It is noteworthy that we are able to solve theroblem discussed in Ref. 9 by solving a more complex set of equations exactly, and taking aimit.

In Ref. 2, Fig. 2b shows plots of Eqs. �97� and �112� in comparison with numerical simula-ions of the equations of motion �10�, exhibiting anomalous spatial diffusion at short times, and arossover to diffusion at long times.

II. WKB ANALYSIS

The staggered ladder spectrum discussed in Sec. IV is surprising, especially in view of the facthat for large quantum number n we expect that the eigenfunctions of the Hamiltonian �23� might

e obtained by WKB theory. In this section we show how to determine the spectrum �29� of Hsing asymptotic WKB analysis. We show how phase shifts associated with the singularity at z0 of the Hamiltonian �23� are the source of the staggered spectrum. It turns out that the WKBrocedure gives rise to the exact eigenvalues.

In dimensionless coordinates, the classical Hamilton function corresponding to �23� is

Hcl =1

2−

z3

4− p2/z . �114�

olving Hcl=� for p we obtain p�z ,��= ± 12��2−4��z−z4, while the velocity is

z = �Hcl/�p = − 2p/z . �115�

he classical trajectories are figure-of-eight orbits, illustrated in Fig. 2.The WKB wave function is of the form

f�z� = �z/p�z,��1/2 exp�±i�z

dzp�z,�� . �116�

he phase of the wave function can be determined as follows. We discuss separately the behaviorsf the wave function at the origin z0=0 and in the vicinity of the regular turning point zt.p.= �2

1/3
4�� . In the latter case, the wave function at z�zt.p. connected with the turning point is

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f t.p.�z� = Ct.p.�z/p�z,��1/2 sin��z

zt.p.

dz p�z,�� +�

4 . �117�

Consider now the behavior of the wave function near the origin. The Hamiltonian has aingularity at z=0. We find an exact solution of the equation when �=0. This equation has aontinuous spectrum, but we identify solutions f+�z� and f−�z� which correspond, respectively, toven and odd solutions of the full equation for � 0 �with discrete spectrum�. Close to z=0, theamping term in the Hamiltonian is negligible, and for z 0 the eigenfunctions resemble solutionsf the equation

�zz−1�z f�z� = − �f�z� �118�

here ��=−�� is a positive constant. Write f =F�, and find that

�

�z�F� + �zF

z = 0 �119�

o that F�+�zF=Cz for some constant C. Thus we find G�z�=F�z�−C /� satisfies G�+z�G=0,hich has solution G�z�=Ai�−�1/3z�, and a similar solution constructed from Bi�x� �here Ai�y�

nd Bi�y� are Airy Ai and Bi functions21�. The general solution is

f�z� = A1Ai��− �1/3z� + A2Bi��− �1/3z� . �120�

We must find solutions of this form which resemble the behavior of the eigenfunctions of thequation with � 0 which obey the boundary conditions

d2f+�0�dz2 = 0 and f−�0� = 0. �121�

he functions Ai� and Bi� have the following forms in the neighborhood of z=0

Ai��y� = c1� y2

+ O�y5� − c2�1 +y3

+ O�y6� , �122�

FIG. 2. �Color online� The trajectories of the classical Hamiltonian �114� are figure-of-eight orbits.

2 3


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Bi��y� = �3c1� y2

2+ O�y5� + �3c2�1 +

y3

3+ O�y6� �123�

ith y=−�1/3z, c1=3−2/3 /��2/3�, and c2=3−1/3 /��1/3�. So the positive-parity solution corre-ponds to the choice

A1+ = − �3 and A2

+ = 1 �124�

hile

A1− = �3 and A2

− = 1. �125�

t large values of y the corresponding wave functions are of the form

f±�y� �− y�1/4 sin�2

3�− y�3/2 +

�

4±

�

3 . �126�

oting that, near z=0, �0zdzp�z ,��= �2/3��−y�3/2 and �z / p�z ,��1/2� �−y�1/4, the solution coming

rom the origin is

fO,±�z� = CO,±�z/p�z,��1/2 sin��0

z

dz� p�z�,�� +�

4±

�

3 . �127�

he forms �117� and �127� should be smoothly connected for 0�z�zt.p.. This requires

�0

zt.p.

dz p�z,�� +�

2±

�

3= �n + 1�� , �128�

or n=0,1 , . . . together with Ct.p.= �−1�nCO,±. Writing

S�� = �0

zt.p.

dz p�z,�� =�

12�− 4� + 2� , �129�

e find the quantization condition

S��±� = �n + 12 �

13�� . �130�

sing �129� the quantization condition takes the form

� = �− 3n even parity

− 3n − 2 odd parity,� �131�

hich corresponds exactly to the spectrum �29� obtained by algebraically diagonalizing H.

III. RESULTS FOR OTHER VALUES OF �

Up to now we have only considered the case of generic random forcing �where the constant

1 in Eq. �8� is not zero�, corresponding to the case =1 in Eq. �9�. In this section we explain twoases where other values of arise and briefly describe results for arbitrary positive values of ,nalogous to the results obtained in Sec. VI.

First consider the case where the force is the gradient of a potential, f�x , t�=�V�x , t� /�x. Wessume that the potential has mean value zero and correlation function C�X ,T�= �V�x+X , t
T�V�x , t��. Assuming that C�X ,T� is sufficiently differentiable at T=0, the diffusion constant is

Iif

T=

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D�p� =1

2�

−�

�

dt �V

�x�pt/m,t�

�V

�x�0,0�� =

− m

2�p��−�

�

dX�2C�X2 �X,mX/p�

=− m

2�p��−�

�

dX � �2C�x2 �X,0� +

mX

p

�3C�2X�T

�X,0� +m2X2

2p2

�4C�2X�2T

�X,0� + O�X3�� . �132�

ntegration by parts shows that the integral over the first term of the expansion is zero, and thentegral over the second term is zero by symmetry. The leading-order contribution in �p�−1 comesrom the third term. Integrating this term by parts twice gives

D�p� − m3

2�p�3�−�

�

dX�2C�T2 �X,0� . �133�

hus in the case of a potential force with a sufficiently smooth correlation function we have 3.

An exceptional case which is worthy of comment is when the potential V�x , t� is itself gen-rated from a set of Ornstein-Uhlenbeck processes Aj�t� by writing V�x , t�=� jAj�t�� j�x�, wherehe � j�x� are elements of some suitable set of basis functions. In this case the correlation functionf V�x , t� is of the form c�x�exp�−��t�� for some function c�x��. Then the second term in thexpansion on the final line of Eq. �132� does not vanish by symmetry and we find D�p�� p�−2, thats =2.

For general positive values of the Hamiltonian �22� is replaced by

H =1

2−

1

4�z�2+ +

�

�z

1

�z��

�z. �134�

ts ground state

�0+ = 0 and �0

+�z� = C0+e−�z�+2/�4+2� �135�

nd first excited state

�0− = − 1 − and �0

−�z� = C0−z�z�e−�z�+2/�4+2� �136�

re found by inspection. Raising and lowering operators can be introduced in a manner analogouso Eqs. �24�–�26�. We write

H = a−�z�−a+ �137�

ith a±=�z±z�z� /2. The operators

A = a+�z�−a+ and A+ = a−�z�−a− �138�

atisfy

�H,A� = �2 + �A and �H,A+� = − �2 + �A+ �139�

nd act as lowering and raising operators. For the spectrum of H we obtain

�n+ = − �2 + �n and �n

− = − �2 + �n − 1 − . �140�

hese expressions replace �29�. Note also that the commutator of A and A+ is

�A,A+� = − �2 + ��H + G� �141�

ˆ ˆ+ −ˆ− ˆ ˆ ˆ
here G=a �z� a and H−G= I. The normalization of the eigenstates

i

a

Tf

T

dO

w

w

Tr


Downloaded 30

A+��n−� = Cn+1

− ��n+1− � , �142�

A��n−� = Cn

−��n−1− � �143�

s determined as in Sec. IV A 2. We obtain

�Cn+1− �2 = �2 + ��n + 1��2 + �n + 3 + 2� �144�

nd

�Cn+1+ �2 = �2 + ��n + 1��2 + �n + 1� . �145�

he results of Sec. V for the matrix elements Zmn= ��m− �z��n� and for �n

+�0� /�0+�0� generalize as

ollows:

Zmn = �− 1�n−m �2 + �−�1+�/�2+�

��

2 + �m + n + 1�

��

2 + − m + n ��n + 1�� 1

2 + + m

��2 − m + n��3 + 2

2 + + n ��m + 1�

,

�146�

�n+�0�/�0

+�0� = �− 1�n��2 + �n + 1�/�2 + ��n + 1��1/�2 + ��

. �147�

his allows us to obtain, for example, the diffusion constant

Dx =1

m2� p02D

2

�4+ 1/�2+� �2 + �−�4+3�/�2+��F32�

2 + ,

2 + ,1 +

2 + ;3 + 2

2 + ,3 + 2

2 + ;1

sin� �

2 + ��3 + 2

2 + 2

�148�

escribing the dynamics at large times. Upon substituting =0, Eq. �148� reproduces the standardrnstein-Uhlenbeck result, and it gives �98� for =1.

For the short-time anomalous diffusion we obtain

�x2�t�� = Cx��p0D�2/�2+�m−2�t�6+2�/�2+� �149�

ith

Cx = − C�0

�

dx x−�6+2�/�2+��0

1

dy �a�x� − a�xy�1 − y

− xa��x�� 1 + y

y�1+�/�2+��1 − y��4+�/�2+� ,

�150�

here a�x� is the same as in �113�, and

C =2�2 + �−2/�2+�

��

2 + 2 . �151�

his reproduces �112� for =1 and concludes our summary of results for other than generic
andom forcing.

A

Mi

1

1

1

1

1

1

1

1

1

1

2

2


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CKNOWLEDGMENTS

B.M. acknowledges financial support from Vetenskapsrådet. B.M., M.W., and K.N. thankarko Robnik for inviting them to the 2005 summer school in Maribor where this work was

nitiated.

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5 M. B. Isichenko, Rev. Mod. Phys. 64, 961 �1992�.6 B. Falkovich, K. Gawedzki, and M. Vergassola, Rev. Mod. Phys. 73, 913 �2001�.7 M. R. Maxey, J. Fluid Mech. 174, 441 �1987�.8 R. Metzler and J. Klafter, Phys. Rep. 339, 1 �2000�.9 P. A. M. Dirac, The Principles of Quantum Mechanics �Oxford University Press, Oxford, 1930�.0 L. Infeld and T. E. Hull, Rev. Mod. Phys. 23, 21 �1951�.1 M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, 9th ed. �Dover, New York, 1972�.


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