Half-Space Problems for a Linearized Discrete Quantum Kinetic Equation

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Bernhoff, N. (2015)

Half-Space Problems for a Linearized Discrete Quantum Kinetic Equation.

Journal of statistical physics, 159(2): 358-379 https://doi.org/10.1007/s10955-015-1190-4

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Niclas Bernhoff

Half-Space Problems for a Linearized Discrete Quantum Kinetic Equation

Received: date / Accepted: date

Abstract We study typical half-space problems of rarefied gas dynamics, includ- ing the problems of Milne and Kramer, for a general discrete model of a quan- tum kinetic equation for excitations in a Bose gas. In the discrete case the plane stationary quantum kinetic equation reduces to a system of ordinary differential equations. These systems are studied close to equilibrium and are proved to have the same structure as corresponding systems for the discrete Boltzmann equation.

Then a classification of well-posed half-space problems for the homogeneous, as well as the inhomogeneous, linearized discrete kinetic equation can be made. The number of additional conditions that need to be imposed for well-posedness is given by some characteristic numbers. These characteristic numbers are calcu- lated for discrete models axially symmetric with respect to the x-axis. When the characteristic numbers change is found in the discrete as well as the continuous case. As an illustration explicit solutions are found for a small-sized model.

Keywords Bose-Einstein condensate · Low temperature kinetics · Discrete kinetic equation · Milne problem · Kramer problem

1 Introduction

Half-space problems have an important role in the study of the asymptotic be- havior of the solutions of boundary value problems of kinetic equations for small Knudsen numbers [4, 15, 16, 26, 27]. In this paper we study half-space problems related to a quantum kinetic equation [23, 33], for the distribution function of ex- cited atoms interacting with a Bose-Einstein condensate. Motivated by the work N. Bernhoff

Department of Mathematics and Computer Science, Karlstad University, 651 88 Karlstad, Swe- den

Tel.: +46-54-7002024 Fax: +46-54-7001851

E-mail: niclas.bernhoff@kau.se

of Arkeryd and Nouri [2] we are interested in the equation





 p¹dF

dx = C12(F) + Γ C22(F) , F(0, p) = F0(p) for p¹> 0,

(1)

where F = F (x, p) denotes the distribution function of the excitations, Γ ∈ R⁺= {x ∈ R |x ≥ 0 } is constant, x ∈ R⁺, p = p¹, p², p³
∈ R³, and F0= F₀(p) is given, with the collision integrals

C₁₂(F) = n Z

δ₀δ₃(1 + F_∗) F⁰F_∗⁰− F_∗ 1 + F⁰

1 + F_∗⁰
 dp_∗dp⁰dp⁰_∗, with

δ0= δ p_∗− p⁰− p⁰_∗
δ



p²_∗+ n − p⁰
2

− p⁰_∗
2 and δ3= δ (p∗− p) − δ p⁰− p
− δ p⁰_∗− p
,

and

C₂₂(F) = Z

δ1(1 + F) (1 + F∗) F⁰F_∗⁰− FF_∗ 1 + F⁰

1 + F_∗⁰
 dp∗dp⁰dp⁰_∗ with

δ1= δ p + p_∗− p⁰− p⁰_∗
δ



p²+ p²_∗− p⁰
2

− p⁰_∗
2 .

Here and below we use the notation F_∗⁰= F (x, p⁰_∗) etc.. The density of the conden- sate, nc, is assumed to be constant, nc= n (cf. [2]). In the Nordheim-Boltzmann [24] (or the Uehling-Uhlenbeck [28]) collision integral C22(F) binary collisions between excited atoms are considered, while in the collision integral C12(F) bi- nary collisions involving one condensate atom are considered [33].

If the distribution function F is close to an equilibrium distribution, i.e. a Planckian

P= 1

e^α(^|p|2+n)+β ·p− 1

= 1

e^α(^|p−p0|²+n₀) − 1, with α > 0, β ∈ R³, p0= −β

2, and n0= n − |p0|², cf. [2], then the non-linear equation (1) can be approximated by the linearized equation





 p¹d f

dx+ L f = 0, f = f (x, p) f(0, p) = f0(p) for p¹> 0,

(2)

where

F= P + (P(1 + P))^1/2f, F0= P + (P(1 + P))^1/2f₀, and L = L12+ Γ L22, with

L₁₂f = Z

δ₀δ₃ h

P_∗− P⁰
(P_∗⁰(1 + P_∗⁰))^1/2f_∗⁰+ P_∗− P_∗⁰
(P⁰(1 + P⁰))^1/2f⁰+ 1 + P⁰+ P_∗⁰
(P_∗(1 + P_∗))^1/2f_∗

i

dp_∗dp⁰dp⁰_∗

and L22f =

Z δ1

h

PP_∗− P⁰(1 + P + P_∗)
(P_∗⁰(1 + P_∗⁰))^1/2f_∗⁰+ PP_∗− P_∗⁰(1 + P + P_∗)
(P⁰(1 + P⁰))^1/2f⁰+ P(1 + P⁰+ P_∗⁰) − P⁰P_∗⁰
(P_∗(1 + P_∗))^1/2f_∗+

P_∗(1 + P⁰+ P_∗⁰) − P⁰P_∗⁰
(P(1 + P))^1/2fi

dp∗dp⁰dp⁰_∗. It can be shown (cf. [2] for L12and, for example, [16] for the linearized Boltzmann operator) that the linearized operators L12and L22, and so also L, (all acting in the velocity space) are symmetric and positive semi-definite operators on L².

In the paper [2], Arkeryd and Nouri studied the Milne problem for the lin- earized equation (2), with Γ = 0, F = P (1 + f ), and a cut-off at λ > 0 in the integrand of L, such that |p| , |p∗| , |p⁰| , |p⁰_∗| ≥ λ . The corresponding linearized half-space problems for the Boltzmann equation is well-studied [3, 17, 20], see also [4] and references therein.

In this paper we discretize the variable p and obtain a general discrete model for Eq.(1), which is similar to the discrete Boltzmann equation (a general discrete velocity model, DVM, for the Boltzmann equation) [14]. It is a well-known fact that the Boltzmann equation can be approximated up to any order of DVMs [11, 25, 18], which motivated us to introduce discrete models also for this equation. By the discretization, Eq.(1) reduces to a system of ordinary differential equations.

We find that the discrete linearized quantum kinetic equation (the discrete version of Eq.(2)) has the same structure as the linearized discrete Boltzmann equation.

This means that the linearized operator is symmetric and positive semi-definite, and that the null-space is non-trivial. One difference is that the mass flow is not constant (with respect to the variable x) as for the discrete Boltzmann equation.

However, this cause us no difficulties, in difference to in the continuous case in [2], since the structure will still be the same. A classification of well-posed half-space problems for the homogeneous, as well as the inhomogeneous, linearized discrete Boltzmann equation has been made in [5] (which is a continuation of the paper [9]), based on the dimensions of the stable, unstable and center manifolds of the singular points (Maxwellians for DVMs). We establish similar results in our case.

This means, that we, in addition to adding the Nordheim-Boltzmann (or Uehling- Uhlenbeck) collision integral C22(F), also can introduce an inhomogeneous term and more general boundary conditions. Similar results can also be established for the discrete Nordheim-Boltzmann (or Uehling-Uhlenbeck) equation and the discrete anyon Boltzmann equation (see Remark 6 and 7 in Section 4).

Furthermore, we have, for axially symmetric discrete models with respect to the x-axis, made a table of some characteristic numbers, from which we, by The- orem 1, can obtain the dimensions of the stable, unstable and center manifolds of the singular points (Planckians in our case). This includes determining when the characteristic numbers change, not only in the discrete, but also in the continuous case (cf. [7, 5] for DVMs and [17] for the Boltzmann equation).

Nonlinear half-space problems for the Boltzmann equation have also been studied for small perturbations of the singular points (Maxwellians for the Boltz- mann equation), see for example [6, 21, 22, 29] for the discrete Boltzmann equation

and [30, 19, 32] for the continuous Boltzmann equation. In the discrete case similar results to the ones in [6] can be obtained for the quantum kinetic equation (1).

We want to make clear that the aim of the paper is not the study of the gen- eral half-space problem we obtain, since it is already well studied for the discrete Boltzmann equation [9, 5]. The novelty of the paper is instead the introduction of discrete models for the equation for the distribution function of excited atoms in- teracting with a Bose-Einstein condensate and the studies of those models. These studies includes that we by the right linearization end up with a system having sim- ilar properties as the one obtained for the discrete Boltzmann equation. It makes it, as mentioned above, possible to extend the results in [2] obtained for the con- tinuous equation. The same is true also for the discrete Nordheim-Boltzmann (or Uehling-Uhlenbeck) equation and the discrete anyon Boltzmann equation (see Re- mark 6 and 7 in Section 4). However, in concrete situations, it will look different depending on which equation we study. One difference is the characteristic num- bers, studied for axially symmetric models in Section 5. Our experience from the Boltzmann equation, make us believe that these numbers (also calculated in the continuous case) are as important in the continuous case as in the studied discrete case. Another difference between our equation and the Boltzmann equation, is that in our case we will have a non-constant mass-flow. To illustrate this we created a model that we solved explicitly and was able to give an explicit expression for the non-constant mass flow for (see Section 6).

The remaining part of this paper is organized as follows. In Section 2 we in- troduce a general discrete model for Eq.(1) and derive some of its properties. By a transformation around a Planckian, we obtain a linearized operator and a non- linear part presented in Section 3. It is shown that the system has the same struc- ture (the linearized operator and the non-linear part have similar properties) as the corresponding system for DVMs of the Boltzmann equation. Then some results for the linearized discrete Boltzmann equation can be applied for the problem of our study. These results are presented in Section 4. In Section 5 some character- istic numbers, from which we, by Theorem 1, can obtain the dimensions of the stable, unstable and center manifolds of the singular points (Planckians), are ob- tained for axially symmetric discrete models with respect to the x-axis. When the characteristic numbers change, are determined both in the discrete as well as the continuous case. A linearized half-space problem (with Γ = 0) is explicitly solved for a small-sized discrete model in Section 6.

2 Discrete model

We introduce a general discrete model for Eq.(1) p¹_idF_i

dx = C12i(F) + Γ C22i(F) , x ∈ R+, i = 1, ..., N, (3) whereP = {p1, ..., pN} ⊂ R^d is a finite set, Fi= Fi(x) = F (x, pi), where F = F(x, p) is the distribution function of the excitations, and Γ ∈ R+ is constant.

For generality, we allow p to be of dimension d, rather than of dimension 3. We assume that

p¹_i 6= 0, for i = 1, ..., N.

The collision operators C12i(F) are given by C_12i(F) =

N

∑

j,k,l=1

(δil− δi j− δik)Γ_jk^l((1 + Fl) FjF_k− F_l(1 + Fj) (1 + Fk)) , where

δ_{i j}= 1 if i = j 0 if i 6= j , with Γ_jkⁱ = 1 if

pi= p_j+ p_kand |pi|²= pj

2+ |p_k|²+ n, (4)

and Γ_jkⁱ = 0 otherwise. Furthermore, the collision operators C_22i(F) are given by C_22i(F) =

N

∑

j,k,l=1

Γ_{i j}^kl((1 + Fi) (1 + Fj) FkF_l− F_iF_j(1 + Fk) (1 + Fl)) , with Γ_{i j}^kl= 1 if

pi+ pj= pk+ pland |pi|²+ pj

2= |pk|²+ |pl|², (5) and Γ_{i j}^kl= 0, otherwise.

Remark 1 For a function g = g(p) (possibly depending on more variables than p), we will, as we consider the discrete case, identify g with its restrictions to the points p ∈P, i.e.

g= (g1, ..., gN) , with gi= g (pi) . Then Eq.(3) can be rewritten as

BdF

dx = C12(F) + Γ C22(F) , with x ∈ R and B = diag(p¹1, ..., p¹_N). (6) The collision operator C12(F) in (6) is also given by the expression

C₁₂(F) = neLF+ n eQ(F, F), (7) where

 eLF

i =

N

∑

j,k=1

2Γ_{i j}^kF_k− Γ_jkⁱF_iand

Qe_i(F, G) =

N

∑

j,k=1

Γ_jkⁱQⁱ_jk(F, G) − 2Γ_{i j}^kQ^k_{i j}(F, G), with

Qⁱ_jk(F, G) =1

2(FjG_k+ GjF_k− F_i(Gj+ G_k) − Gi(Fj+ F_k)) , and the collision operator C22(F) in (6) is given by the expression

C22(F) = Q(F, F) + bQ(F, F, F), (8)

where

Q_i(F, G) =1 2

N

∑

j,k,l=1

Γ_{i j}^kl((G_kH_l+ H_kG_l) − (G_iH_j+ H_jG_i)) and

Qb_i(F, G, H) = 1 2

N

∑

j,k,l=1

Γ_{i j}^kl((Fi+ Fj) (G_kH_l+ H_kG_l) − (F_k+ F_l) (GiHj+ HjGi)) . A function φ = φ (p) is a collision invariant, if and only if,

φ_i= φ_j+ φ_k, (9)

for all indices such that Γ_jkⁱ 6= 0, if Γ = 0, with the additional condition

φ_i+ φj= φk+ φl, (10)

for all indices such that Γ_{i j}^kl 6= 0, if Γ 6= 0. We have the trivial collision invariants (”the physical collision invariants”)

φ¹= p¹, ..., φ^d= p^d, φ^d+1= |p|²+ n (11) including all linear combinations of these. We want to stress that by Remark 1 and in correspondence with Eqs.(9) , (10) the collision invariants φⁱ= φⁱ(p) in Eq.(11) are vectors.

In the discrete case, in difference to the continuous case, there can be spurious (or non-physical) collision invariants. We consider below (even if this restriction is not necessary in our general context) only normal discrete models. That is, discrete models without spurious collision invariants, i.e. any collision invariant is of the form

φ = φ (p) = −α

|p|²+ n

− β · p (12)

for some constant α ∈ R and β ∈ R^d. Construction of normal discrete kinetic models and especially DVMs have been extensively studied, see for example [10, 12, 13] and references therein. A Maxwellian distribution or just Maxwellian is on the form

M= e^φ= e^−α(^|p|2+n)^{−β ·p} or

M_i= e^φi= e^−α(^|pi|²+n)−β ·p_i, i = 1, ..., N,

where φ = (φ1, ..., φN) is a collision invariant, and a Planckian distribution or just Planckian is given by

P= M

1 − M = 1

M⁻¹− 1= 1

e^α(^|p|2+n)+β ·p− 1

or

P_i= Mi

1 − Mi

= 1

e^α(^|pi|²+n)+β ·pi− 1for i = 1, ..., N.

One can easily see that hH,C₁₂(F)i =

n

N

∑

i, j,k=1

Γ_jkⁱ((1 + F_i) F_jF_k− F_i(1 + F_j) (1 + F_k)) (H_i− H_j− H_k) , (13) and so

 log F

1 + F,C12(F)

= n

N

∑

i, j,k=1

Γ_jkⁱ(1 + Fi) (1 + Fj) (1 + F_k)

 F_j 1 + Fj

F_k 1 + Fk

− F_i 1 + Fi

  log F_i

1 + Fi

− log

 F_j 1 + Fj

F_k 1 + Fk



≤ 0, with equality if and only if

F_i 1 + Fi

= F_j 1 + Fj

F_k 1 + Fk

, (14)

for all indices such that Γ_jkⁱ 6= 0. Here and below, we denote by h·, ·i the Euclidean scalar product in Rⁿ. Hence, there is equality in Eq.(14), if and only if, F

1 + F is a Maxwellian, or equivalently, if and only if, F is a Planckian.

Moreover, one can easily obtain that

hH,C₂₂(F)i =1 4

N

∑

i, j,k,l=1

Γ_{i j}^kl(Hi+ Hj− H_k− H_l)

((1 + F_i) (1 + F_j) F_kF_l− F_iF_j(1 + F_k) (1 + F_l)) , (15) and so

 log F

1 + F,C₂₂(F)

=1 4

N

∑

i, j,k=1

Γ_{i j}^kl(1 + F_i) (1 + F_j) (1 + F_k) (1 + F_l)

 log

 F_i 1 + Fi

F_j 1 + Fj



− log

 F_k 1 + Fk

F_l 1 + Fl



 F_k 1 + Fk

F_l 1 + Fl

− F_i 1 + Fi

F_j 1 + Fj



≤ 0, (16) with equality if and only if

F_i 1 + Fi

F_j 1 + Fj

= F_k 1 + F_k

F_l

1 + F_l, (17)

for all indices such that Γ_{i j}^kl6= 0. There is equality in Eq.(16), if and only if, F 1 + F is a Maxwellian, or equivalently, if and only if, F is a Planckian.

By the relations (13) and (15)

hφ ,C12(F) + Γ C22(F)i , (18)

is zero, independently of our choice of non-negative function F, if and only if, φ is a collision invariant, and so (for normal models) the equation

hφ ,C12(F) + Γ C22(F)i = 0, has the general solution (12).

3 Linearized operator Given a Planckian

P= 1

e^α(^|p|2+n)+β ·p− 1

= 1

e^α(^|p−p0|²+n₀) − 1, (19) with α > 0, β ∈ R^d, p0=β

2 and n0= n − |p0|², we denote

F= P + R^1/2f, with R = P(1 + P), (20) in Eq.(6), and obtain

Bd f

dx+ L f = S ( f ) ,

where L = L12+ Γ L₂₂is the linearized collision operator (N × N matrix) given by L₁₂f = −2nR^−1/2Q(P, Re ^1/2f) − neLR^1/2f (21) and

L₂₂f = −R^−1/2

2Q(P, R^1/2f) + bQ(R^1/2f, P, P) + 2 bQ(P, R^1/2f, P)

. (22) The nonlinear part S ( f ) = S12( f , f ) + Γ S₂₂( f , f , f ) is given by

S₁₂( f , g) = nR^−1/2Q(Re ^1/2f, R^1/2g) (23) and

S₂₂( f , g, h) = R^−1/2

Q(R^1/2f, R^1/2g) + bQ(P + R^1/2f, R^1/2g, R^1/2h)+

Q(Rb ^1/2f, P, R^1/2h) + bQ(R^1/2f, R^1/2g, P) . (24) In more explicit forms, the operators (21) and (23) read

(L12f)_i= n

N j,k=1∑

Γ_jkⁱLⁱ_jkf− 2Γ_{i j}^kL_{i j}^k f R^1/2_i

, i = 1, ..., N, (25)

where

Lⁱ_jkf = (1 + Pj+ P_k) R^1/2_i f_i− (P_k− P_i) R^1/2_j f_j− (P_j− P_i) R^1/2_k f_k, and

S_12i( f , g) = n

N

∑

j,k=1

Γ_jkⁱSⁱ_jk( f , g) − 2Γ_{i j}^kS^k_{i j}( f , g) R^1/2_i

, i = 1, ..., N, with

Sⁱ_jk( f , g) =1 2



R^1/2_j R^1/2_k ( fjg_k+ gjf_k) − R^1/2_i R^1/2_j ( fig_j+ gif_j) −

R^1/2_i R^1/2_k ( fig_k+ gif_k) . Moreover, the operators (22) and (24) read, in more explicit forms,

(L22f)_i=

N

∑

j,k,l=1

Γ_{i j}^kl R^1/2_i

(P_{i j}^klf_i+ P^kl_jif_j− P_kl^{i j}f_k− P_lk^{i j}f_l), i = 1, ..., N (26)

where

P_{i j}^kl= (P_j(1 + P_k+ P_l) − P_kP_l) R^1/2_i , and

S22i( f , f , f ) =

N

∑

j,k,l=1

Γ_{i j}^kl R^1/2_i



S^kl_{i j}( f , f , f ) − S^{i j}_kl( f , f , f )

, i = 1, ..., N,

with

S^kl_{i j}( f , f , f ) = (1 + P_i+ P_j) R^1/2_k R^1/2_l f_kf_l+

R^1/2_i f_i+ R^1/2_j f_j



P_kR^1/2_l f_l+ PlR^1/2_k f_k+ R^1/2_k R^1/2_l f_kf_l . By Eqs.(4),(25), and the relations

P_j(1 + Pj)(P_k− P_i) = P_k(1 + P_k)(Pj− P_i) = Pi(1 + Pj)(1 + P_k), P_i(1 + P_j+ P_k) = P₂P₃= P_i(1 + P_j)(1 + P_k)

for Γ_jkⁱ 6= 0, we obtain the equality

hg, L₁₂f i= n

N

∑

i, j,k=1

Γ_jkⁱP_i(1 + Pj) (1 + Pk)



 f_i R^1/2_i

− fj

R^1/2_j

− f_k R^1/2_k







 g_i R^1/2_i

− g_j R^1/2_j

− g_k R^1/2_k



.

Similarly, by Eqs.(5),(26), and the relations

P_iP_j(1 + Pk)(1 + Pl) = PkP_l(1 + Pi)(1 + Pj), P_{i j}^kl = PkP_l(1 + Pj)

√1 + P_i

√P_i

for Γ_{i j}^kl6= 0, we obtain the equality

hg, L₂₂f i= 1 4

N

∑

i, j,k,l=1

Γ_{i j}^klP_iP_j(1 + Pk)(1 + Pl)



 f_i R^1/2_i

+ f_j R^1/2_j

− f_k R^1/2_k

− f_l R^1/2_l







 g_i R^1/2_i

+ g_j R^1/2_j

− g_k R^1/2_k

− g_l R^1/2_l



.

It is easy to see that the matrix L is symmetric and positive semi-definite, i.e.

hg, L f i = hLg, f i and h f , L f i ≥ 0, for all functions g = g(ξ ) and f = f (ξ ).

Furthermore, h f , L f i = 0 if and only if f_i

R^1/2_i

= f_j R^1/2_j

+ f_k R^1/2_k

(27)

for all indices satisfying Γ_jkⁱ 6= 0 if Γ = 0.

If Γ 6= 0, h f , L f i = 0 if and only if also, additionally to Eq.(27), f_i

R^1/2_i + f_j

R^1/2_j

= f_k R^1/2_k

+ f_l R^1/2_l

(28)

for all indices satisfying Γ_{i j}^kl 6= 0. We denote f = R^1/2φ in Eq.(27) and Eq.(28) and obtain Eq.(9) and Eq.(10) respectively. Hence, since L is semi-positive,

L f = 0 if and only if f = R^1/2φ , where φ is a collision invariant (12).

Then also D

S( f ) , R^1/2φ E

= hC12(F) + Γ C22(F) , φ i +D

F, LR^1/2φ E

= 0 for all collision invariants φ .

The system (6) transforms in Bd f

dx+ L f = S( f ). (29)

The diagonal matrix B (6) (under our assumptions) has no zero diagonal elements and is non-singular. We denote f (0) = f0. Then the formal solution of Eq.(29) reads

f(x) = e^−xB−1Lf₀+

x Z

0

e^{(σ −x)B−1L}B⁻¹[S ( f )] (σ ) dσ .

As in the case of DVMs for the Boltzmann equation, the linearized operator Lis symmetric, positive semi-definite, and have a non-trivial null-space, and by assumption, the matrix B is non-singular. Therefore, we can apply a result obtained by Bobylev and Bernhoff in [9] (see also [5]), that we will present below.

We denote by n^±, where n⁺+ n⁻= N, and m^±, with m⁺+ m⁻= q, the num- bers of positive and negative eigenvalues (counted with multiplicity) of the matri- ces B and B⁻¹Lrespectively, and by m⁰the number of zero eigenvalues of B⁻¹L.

Moreover, we denote by k⁺, k⁻, and l the numbers of positive, negative, and zero eigenvalues of the ρ × ρ matrix K (ρ = d + 1 for normal discrete models), with entries

k_{i j}=y_i, yj

B=y_i, Byj ,

such thaty₁, ..., y_ρ is a basis of the null-space of L, i.e. in our case span y1, ..., yρ
= N(L) = span

R^1/2p¹, ..., R^1/2p^d, R^1/2(|p|²+ n) . Here and below, we denote h·, ·i_B= h·, B·i and by N(L) the null-space of L.

In applications, the number ρ of collision invariants is usually relatively small compared to N (note that formally N = ∞ for the continuous equation whereas ρ ≤ 4). Also, the matrix B is diagonal and therefore all its eigenvalues are known.

This explains the importance of the following result [9]. The theorem is valid for any real symmetric matrices L and B, such that L is semi-positive, B is invertible, and dim(N(L)) = ρ ≥ 1.

Theorem 1 The numbers of positive, negative and zero eigenvalues of B⁻¹L are given by







m⁺= n⁺− k⁺− l m⁻= n⁻− k⁻− l m⁰= ρ + l.

In the proof of Theorem 1 a basis

u1, ..., uq, y1, ..., y_k, z1, ..., z_l, w1, ..., w_l

(30) of R^N, such that

y_i, z_r∈ N(L), B⁻¹Lw_r= z_rand B⁻¹Luα= λαuα, (31) and

uα, u_β

B= λαδ_{α β}, with λ1, ..., λ_m⁺> 0 and λ_m⁺₊₁, ..., λq< 0, yi, yj

B= γiδi j, with γ1, ..., γ_k⁺> 0 and γ_k⁺₊₁, ..., γ_k< 0, hu_α, zri_B= huα, wri_B= huα, yii_B= hwr, yii_B= hzr, yii_B= 0,

hw_r, w_si_B= hz_r, z_si_B= 0 and hw_r, z_si_B= δ_rs, (32) is constructed.

4 Half-space problems

We consider the inhomogeneous (or homogeneous if g = 0) linearized problem Bd f

dx + L f = g, (33)

where g = g(x) ∈ L¹(R+, Rⁿ), with one of the boundary conditions (O) the solution tends to zero at infinity, i.e.

f(x) → 0 as x → ∞;

(P) the solution is bounded, i.e.

| f (x)| < ∞ for all x ∈ R+; (Q) the solution can be slowly increasing at infinity, i.e.

| f (x)| e^−εx→ 0 as x → ∞, for all ε > 0.

In case of boundary condition (O) we additionally assume that

g(x) ∈ N(L)^⊥for all x ∈ R+. (34) Remark 2 The boundary condition (O) corresponds to the case when we have made the expansion (20) around a Planckian P, such that F → P as x → ∞. The boundary conditions (P) and (Q) are the boundary conditions in the Milne and Kramers problem respectively.

We can (without loss of generality) assume that B= B₊ 0

0 −B−



, (35)

where

B₊= diag p¹₁, ..., p¹_n+
and B₋= −diag p¹_n++1, ..., p¹_N
, with p¹₁, ..., p¹_n+> 0 and p¹_n++1, ..., p¹_N< 0.

We also define the projections R+: R^N→ Rⁿ⁺and R−: R^N→ Rⁿ⁻, by R+s= s⁺= (s1, ..., s_n⁺) and R₋s= s⁻= (s_n⁺₊₁, ..., sN) for s = (s1, ..., sN).

At x = 0 we assume the boundary condition

f⁺(0) = h0, (36)

where h0∈ Rⁿ⁺.

The solutions of the system (33) with one of the boundary conditions (O) (together with condition (34)), (P), and (Q) reads

f(x) = Ψ⁺(x) +Ψ⁻(x) + Φ⁺(x) + Φ⁻(x),

where (in the notations of (30)-(32))

Ψ⁺(x) =

m⁺ r=1∑

u_r



βr(0)e^−λrx+

x Z

0

e^{(σ −x)λr}hg (σ ) , uri λ_r dσ



,

Ψ⁻(x) = −

q

∑

r=m⁺+1

u_r

∞ Z

x

e^{(σ −x)λr}hg (σ ) , u_ri λr

dσ ,

Φ⁺(x) =

k⁺ i=1∑

y_i



µi(0) +

x Z

0

hg (σ ) , yii hy_i, y_ii_B dσ





+

l

∑j=1

z_j



ηj(0) +

x Z

0

g (σ) , wj dσ − x



αj(0) +

x Z

0

g (σ) , zj dσ







,

Φ⁻(x) =

k

∑

i=k⁺+1

y_i



µi(0) +

x Z

0

hg (σ ) , y_ii hy_i, yii_B dσ





+

l

∑

j=1

w_j



α_j(0) +

x Z

0

g (σ) , zj dσ



,

with for the case with boundary condition (O) (note that hg (x) , y_ii =g (σ) , zj = 0 for i = 1, ..., k and j = 1, ..., l, by condition (34))

µi(0) = αj(0) = 0 for i = 1, ..., k and j = 1, ..., l, and η_j(0) = −

∞ Z

0

g (σ) , wj dσ for j = 1, ..., l,

and for the case with boundary condition (P)

αj(0) = −

∞ Z

0

g (σ) , zj dσ for j = 1, ..., l.

By the boundary condition (36), or equivalently

R₊Ψ⁺(0) + R₊Φ⁺(0) = h₀− R₊Ψ⁻(0) − R₊Φ⁻(0), and that

{R₊u1, ..., R+u_m⁺, R+y1, ..., R+y_k⁺, R+z1, ..., R+z_l}