Hamiltonian treatment of time dispersive and dissipative media within the linear response theory

0.19 MB·English

by  A. Figotin

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a Hamiltonian treatment of time dispersive and dissipative media within
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the linear response theory
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Alexander Figotina and Jeffrey Schenker b
∗
]
h
aDepartment of Mathematics, University of California at Irvine, Irvine, CA 92697
p
-
s bInstitut fu¨r Theoretische Physik, ETH Zu¨rich, CH-8093 Zu¨rich, Switzerland
s
a
l
c WedevelopaHamiltoniantheoryforatimedispersiveanddissipative(TDD)inhomoge-
.
s neous medium, as described by a linear response equation respecting causality and power
c
i dissipation. The canonical Hamiltonian constructed here exactly reproduces the original
s
y dissipative evolution after integrating out auxiliary fields. In particular, for a dielectric
h
medium we obtain a simple formula for the Hamiltonian and closed form expressions for
p
[ the energy density and energy flux involving the auxiliary fields. The developed approach

also allows to treat a long standing problem of scattering from a lossy non-spherical
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v obstacle and, more generally, wave propagation in TDD media.
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1 1. Introduction
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1
There is an intrinsic ambiguity in identifying the field energy densities for radiation
4
0 in a time dispersive and dissipative (TDD) medium as described by the linear response
/
s theory,e.g.,inadielectricmediumdescribedbytheclassicallinearMaxwellequationswith
c
complex valued frequency dependent electric permittivity ε(ω) andmagnetic permeability
i
s
µ(ω). Consequently, there are problems with the interpretation of the energy balance
y
h equation [12, 77], [1, 1.5a], [5, 6.8], [17]. There were a number of efforts [17], [13],
p § § §
[20] to construct a consistent macroscopic theory of dielectric media that accounts for
:
v
dispersion and dissipation, based on more fundamental microscopic theories. At first
i
X
sight, it seems that the introduction of a realistic material medium in an explicit form
r
similar to [17], [13], [20] is the only way to model a TDD medium. In fact, that is not so
a
andinthis paper wedescribe aconsistent macroscopic approachwithin the linear response
theory. Full proofs of the statements outlined here will appear in a forthcoming paper [3].
A linear response TDD medium is an essentially open dissipative system, which in prin-
ciple can be obtained by i.) eliminating some degrees of freedom from a more involved
microscopic theory and ii.) making the approximation of linear response. Stopping short
of introducing a microscopic theory we ask, is there a conservative extended system which
exactly reproduces the given linear TDD system after reduction? In [2] we showed that
indeed such an extension is i.) possible and ii.) essentially uniquely determined, under
general conditions of causality, power dissipation, and minimality of the extension. Here
we go further and construct a canonical Hamiltonian for such a conservative extension
∗current address: School of Mathematics, Institute for Advanced Study, Princeton, NJ 08540
2
based only on the given TDD equations — without assumption on the underlying mi-
crostructure. In particular, we construct such a Hamiltonian for a dielectric medium
as defined by complex ε(ω) and µ(ω). The construction given here is not restricted to
a dielectric medium, however, but holds for TDD systems with a certain mathematical
structure — eqs. (8, 9, 10) below — including, in particular, elastic and acoustic media,
and it can be extended to space dispersive dissipative systems. A somewhat related con-
struction of the evolution equations for linear absorptive dielectrics was given by Tip [27].
The range of validity of the proposed theory is the same as for the linear response, though
nonlinear generalizations are clearly possible.
Other important benefits of the approach developed here are: i.) The constructed
Hamiltonian is an integral of a local energy density, which in the absence of TDD terms
reduces to the local field energy. This permits us to derive an expression for the energy
transport for TDD media. ii.) The present formulation allows to treat, in particular, a
long standing problem of scattering from a lossy nonspherical scatter - analyzed by other
methods with limited success [18] - by applying the well developed scattering theory
or conservative systems, see [22,26] and references therein. These applications will be
discussed in detail in forthcoming work [3], [4].
2. Construction of the Hamiltonian
We consider a system described by two canonical vector coordinates p,q H, with H a
∈
real Hilbert space. In the absence of TDD terms, the evolution is supposed to be induced
by a Hamiltonian A(p,q) of the form
1 1
A(p,q) = K p, K p + K q, K q , (1)
p p q q
2 h i 2 h i
with closed linear operators K , K from H into auxiliary spaces H , H respectively. To
p q p q
manifest the conservation of energy, it is convenient to consider the evolution of
f := K p H , f := K q H , (2)
p p p q q q
∈ ∈
in place of p,q. In the absence of dissipation, these quantities evolve according to
f 0 K f
∂ p = − p , zero dissipation, (3)
t(cid:18)fq(cid:19) (cid:18)K† 0 (cid:19)(cid:18)fq(cid:19)
with K := K K a closed linear map from H to H . Note that
p q† q p
1
A(p,q) = ( f 2 + f 2) (4)
p q
2 k k k k
is conserved due to the antisymmetry of the generator in (3).
The electromagnetic field in a non-dispersive inhomogeneous medium may be described
in this framework, with p = (4π) 1A (magnetic potential), q = D (electric displacement),
−
f = (2√π) 1H (magnetic field), and f = (2√π) 1E (electric field). Identifying (2) with
p − q −
the material relations, we determine the action of the operators K :
w
A A 1
K (~r) = (2√π)µ 1(~r) (~r) , K D(~r) = ǫ 1(~r) D(~r), (5)
p − q −
4π ·(cid:26)∇× 4π (cid:27) 2√π ·
3
where µ, ǫ are the static permeability and dielectric tensors, assumed real and symmetric.
We take (4π) 1A and D in the space H = H of divergence free vector fields — which
− curl
amounts to a choice of gauge and an assumption of no free charges. To complete the
picture we define H , H to be weighted L2 spaces with scalar products
p q
H, H = d3~r H(~r) µ(~r) H(~r), E, E = d3~r E(~r) ǫ(~r) E(~r). (6)
h iHp Z · · h iHq Z · ·
As a result
1
K = (2√π) , K = P , K = µ 1(~r) , K = ǫ 1(~r) , (7)
p† ∇× q† 2√π curl − ·∇× † − ·∇×
with P the orthogonal projection of (L2)3 onto H .
curl curl
An alternative formulation of the general system (1) is suggested by the example of the
EM field, namely to consider (2) as generalized material relations together with evolution
equations
p 0 K f
∂ = − q† p . (8)
t(cid:18)q(cid:19) (cid:18)K 0 (cid:19)(cid:18)f (cid:19)
p† q
In turn, this suggests a natural modification incorporating dispersion and dissipation by
replacing (2) with:
∞
f (t)+ dτ χ (τ)f (t τ) = K w(t), for w = p,q, w = p,q. (9)
w w w w
Z −
0
The TDD character of (9) comes from the operator valued generalized susceptibilities χ ,
w
w = p,q, theintegralsofwhichexplicitly satisfythe causality condition: valuesofK w(t)
w
depend only on f (t) for times t t.
w ′ ′
≤
Our main result is the following: Assume the susceptibilities χ satisfy the following
w
power dissipation condition (PDC),
1
Im{ζχˆw(ζ)} = 2i ζχˆw(ζ)−ζ∗χˆw(ζ)† ≥ 0, w = p,q, for all ζ = ω +iη, η ≥ 0, (10)
n o
for all ζ = ω +iη, η 0, where χˆ is the Fourier-Laplace transform of χ :
w w
≥
1
χˆ (ζ) = ∞dteiζtχ (t), w = p,q. (11)
w √2π Z w
0
Then it is possible to construct a Hamiltonian extension to (1), which reduces to (1) in
the limit of zero susceptibility, such that the subsystem p,q evolves according to (8, 9).
The extended Hamiltonian (P,Q) is a function of extended momentum P and co-
A
ordinate Q variables, each taking values in a Hilbert space H, and has the same
H ⊃
structure as (1), i.e.
1 1
(P,Q) = P , P + Q, Q , (12)
p p q q
A 2 hK K i 2 hK K i
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with , closed operators from to H , H , which extend K and K
p q p p q q p q
K K H H ⊃ H ⊃
respectively (see (48) below).
Before presenting the general construction, let us illustrate it with the example of a
linear TDD dielectric medium, described by the macroscopic Maxwell equations without
external charges and currents
∂ D = H, ∂ B = E, B = 0, D = 0, (13)
t t
∇× −∇× ∇· ∇·
in units with c,ε ,µ = 1. Here
0 0
D = E+4πP, B = H+4πM, (14)
with the polarization P and magnetization M given by linear response
∞ ∞
P(~r,t) = dτ χ (~r,τ)E(~r,t τ), M(~r,t) = dτ χ (~r,τ)H(~r,t τ). (15)
E H
Z − Z −
0 0
The electric and magnetic susceptibilities should satisfy the PDC (10) for each ~r, and for
simplicity we take them to be real valued scalars. (The frequency domain susceptibilities
χ (ω) may nonetheless be complex.)
F
Motivated by [2] and the Lamb model (see Fig. 1 below), we introduce canonical vari-
b
ables
P = (4π) 1A(~r), θ (~r,s), ϕ (~r,s) , Q = (D(~r), ϕ (~r,s), θ (~r,s)), (16)
− E H E H
(cid:0) (cid:1)
with A,D H and auxiliary vector fields ϕ ,θ , F = E,H, which are functions of ~r
curl F F
∈
and an auxiliary coordinate < s < . For these variables we define a Hamiltonian
−∞ ∞
(P,Q) = (P)+ (Q), (17)
A T U
with,
1 A(~r) 2
(P) = d3~r 2√π ∞ dsς (~r,s)ϕ (~r,s) (18)
T 2 Z (cid:12) ∇× 4π −Z H H (cid:12)
(cid:12) (cid:12)
(cid:12)1 −∞ (cid:12)
+(cid:12) d3~r ∞ ds θ (~r,s) 2 + ∂ ϕ (~r,s) 2(cid:12) ,
2 Z Z | E | | s H |
(cid:2) (cid:3)
−∞
2
1 1
(Q) = d3~r D(~r) ∞ dsς (~r,s)ϕ (~r,s) (19)
U 2 Z (cid:12)2√π −Z E E (cid:12)
(cid:12) (cid:12)
(cid:12)1 −∞ (cid:12)
+(cid:12) d3~r ∞ ds ∂ ϕ (~r,s) 2 + θ ((cid:12)~r,s) 2 ,
2 Z Z | s E | | H |
(cid:2) (cid:3)
−∞
where ς , F = E,H, are scalar functions to be specified below.
F
5
The resulting Hamilton equations of motion for the extended Maxwell system are:
A(r,t) 1
∂ = D(r,t) 2√π ς ,ϕ (r,t) , (20)
t 4π −4π − h E Eis
(cid:2) (cid:3)
∂ ϕ (r,s) = θ (r,s),
t H H
−
1
∂ θ (r,s) = ς (r,s) D(r,t) 2√π ς ,ϕ (r,t) +∂2ϕ (r,s),
t E 2√π E − h E Eis s E
(cid:2) (cid:3)
∂ D(r,t) = A(r,t) 2√π ς ,ϕ (r,t) , (21)
t ∇× ∇× − h H His
(cid:2)1 (cid:3)
∂ θ (r,s) = ς (r,s) A(r,t) 2√π ς ,ϕ (r,t) ∂2ϕ (r,s).
t H −2√π H ∇× − h H His − s H
(cid:2) (cid:3)
∂ ϕ (r,s) = θ (r,s),
t E E
where
, = ds. (22)
h• •is Z ••
We make the identifications:
B(r,t) = A(r,t) (23)
∇×
H(r,t) = B(r,t) 2√π ς ,ϕ (r,t) = B(r,t) 4πM(r,t), (24)
− h H His −
E(r,t) = D(r,t) 2√π ς ,ϕ (r,t) = D(r,t) 4πP(r,t),
− h E Eis −
with
ς ,ϕ (r,t) = 2√πM(r,t), ς ,ϕ (r,t) = 2√πP(r,t), (25)
h H His h E Eis
resulting in the following equivalent system of extended Maxwell equations
∂ H(r,t) = E(r,t)+2√π ς ,θ (r,t), (26)
t −∇× h H His
1
∂ ϕ (r,s) = θ (r,s), ∂ θ (r,s) = ς (r,s)E(r,t)+∂2ϕ (r,s),
t H − H t E 2√π E s E
∂ E(r,t) = H(r,t) 2√π ς ,θ (r,t), (27)
t ∇× − h E Eis
1
∂ θ (r,s) = ς (r,s)H(r,t) ∂2ϕ (r,s), ∂ ϕ (r,s) = θ (r,s).
t H −2√π H − s H t E E
Combining the first order Hamilton equations of motion for ϕ and θ into a single
F F
second order equation for ϕ , F = E,H, we obtain a driven wave equation
F
1
∂2 ∂2 ϕ (~r,s,t) = ς (~r,s)F(~r,t), F = E,H. (28)
t − s F 2√π F
(cid:8) (cid:9)
Assuming ϕ to be at rest (ϕ = ∂ ϕ = 0) in the distant past, the solution to (28) is
F F t F
given by
1 s+τ
∞
ϕ (~r,s,t) = dτ dσς (~r,σ)F(~r,t τ), F = E,H, (29)
F 4√π Z Z F −
0 s τ
−
6
implying with (15) and (25) the following expression for the susceptibilities
1 s+t
∞
χ (~r,t) = ds dσς (~r,s)ς (~r,σ), F = E,H. (30)
F 8π Z Z F F
s t
−∞ −
The key fact is that, due to the power dissipation condition (10), it is possible to invert
(30) and write ς as a function of χ . An explicit solution is
F F
2
ς (~r,s) = ∞ dωe iωs ωImχˆ (~r,ω +i0), F = E,H. (31)
F √4 2π Z − F
p
−∞
Note that ς is real and invariant under s s.
F
7→ −
The above discussion implies the following result on the extended system: Let the
Hamiltonian (17, 18, 19) be given with ς defined by (31) for χ which obey (10). Then
F F
for any solution to the Hamilton equations of motion with ϕ ,θ 0 as t , the
F F
→ → −∞
variables D(~r,t) and B(~r,t) = A(~r,t) evolve according to the macroscopic Maxwell
∇×
equations (13-15).
Based on the constructed TDD Hamiltonian (17), we obtain an expression for the
energy density of the EM field and the medium
1 1
(~r,t) = E 2 + H 2 (~r,t)+ ∂ ϕ 2 + θ 2 + ∂ ϕ 2 + θ 2 (~r,t), (32)
E 8π | | | | 2 k s Hks k Hks k s Eks k Eks
(cid:8) (cid:9) (cid:8) (cid:9)
where
2 2
= ds. (33)
k•ks Z |•|
This results in the conservation law
∂ + S = 0, (34)
t
E ∇·
with the familiar Poynting vector for the energy flux
1
S(~r,t) = E(~r,t) H(~r,t). (35)
4π ×
These identities follow from (17-19) and the general theory of Hamiltonian fields [11].
When the interaction ς is set to zero, the EM and auxiliary fields decouple and (32)
reduces to
(~r,t) = (~r,t)+ (~r,t) (36)
0 EM S
E E E
with the energy density of the EM field
1 1
(~r,t) = D 2(~r,t)+ A 2(~r,t), (37)
EM
E 8π | | 8π |∇× |
and the energy density of the auxiliary fields
1
(~r,t) = ∂ ϕ 2 + θ + ∂ ϕ 2 + θ 2 (~r,t). (38)
ES 2 k s Hks k Hks k s Eks k Eks
(cid:8) (cid:9)
7
Subtracting (36) from (32) gives the energy shift due to the interaction of the EM field
and the matter
1
δ (~r,t) = E(r,t) 2 D(r,t) 2 + H(r,t) 2 B(r,t) 2 . (39)
E 8π | | −| | | | −| |
(cid:8) (cid:9)
In general, it is not possible to give an expression for the energy density (~r,t) of the
S
E
medium in terms of the instantaneous EM fields E(~r,t) and H(~r,t). However, using (29,
30) and the equations of motion we have calculated that
1 1
∂ ∂ ϕ 2 + θ 2 (~r,t) = [∂ ς ,ϕ (~r,t)] F(~r,t) (40)
t 2 k s Fks k Fks 2√π th F Fi ·
(cid:8) (cid:9)
= [∂ P(~r,t)] E(~r,t), F = E, (41)
t
·
with a similar expression for F = H.
The result (41) is the usual expression for the rate of change the density of EM energy
stored in a dielectric. For a wave packet E(~r,t) = Re e iω0tE (~r,t) with E (~r,t) a
− 0 0
slowly varying function of t, we have (cid:8) (cid:9)
dω χ (~r,ω )
∂ P(~r,t) √2πRe iω χ (~r,ω )e iω0tE (~r,t)+ 0 E 0 e iω0t∂ E (~r,t) , (42)
t ≈ (cid:26)− 0 E 0 − 0 dω − t 0 (cid:27)
0
b
b
and thus
[∂ P(~r,t)] E(~r,t)
t
·
√2π dω χ (~r,ω )
Re iω χ (~r,ω ) E (~r,t) 2 + 0 E 0 [∂ E (~r,t)] E (~r,t)
≈ 2 (cid:26)− 0 E 0 | 0 | dω t 0 · ∗0 (cid:27)
0
b
b + terms with a factor of e iω0t or eiω0t. (43)
−
If we consider the time averaged power density, averaged over a time scale longer than
1/ω , the terms with oscillatory factors are very small and we have (with denoting time
0
•
averaging):
[∂ P(~r,t)] E(~r,t)
t
·
π 1 d
ω Reχ (~r,ω ) ∂ E (~r,t) 2 +ω Imχ (~r,ω ) E (~r,t) 2 , (44)
≈ r2 (cid:26)2 (cid:20)dω 0 E 0 (cid:21) t| 0 | 0 E 0 | 0 | (cid:27)
0
b b
where we have assumed for simplicity that the slowly varying function E (~r,t) is real.
0
The first term on the r.h.s. of (44) is a total derivative, which when integrated gives the
Brillouin formula for the time averaged energy density in a material medium (see [12,
Section 80]):
√2π d
ω Reχ (~r,ω ) E (~r,t) 2 , (45)
4 (cid:20)dω 0 E 0 (cid:21) | 0 |
0
b
which neglects losses entirely, and hence is useful only if Imχ (~r,ω ) is zero or so small as
E 0
to be irrelevant. The second term of (44) is strictly positive and gives the contribution
b
π t
ω Imχ (~r,ω ) E (~r,τ) 2dτ (46)
r2 0 E 0 Z | 0 |
−∞
b
8
to the time averaged material energy density, which is non-decreasing in time and incor-
porates losses in an approximate way.
For a general system of the form (8, 9, 10), a TDD Hamiltonian can be constructed in
the same way. We define canonical variables
P = p, θ , ϕ , Q = q, ϕ , θ , (47)
q p q p
(cid:0) (cid:1) (cid:0) (cid:1)
with θ (s),ϕ (s) functions of an auxiliary coordinate < s < , taking values in the
w w
−∞ ∞
Hilbert spaces H , w = p,q. If χ (t) = χ (t) for all t 0, the Hamiltonian is of the
w w w †
≥
form (12), with
∞
P = K p dsς (s)ϕ (s), θ , ∂ ϕ , (48)
p p p p q s p
K (cid:18) −Z (cid:19)
−∞
∞
Q = K q dsς (s)ϕ (s), ∂ ϕ , θ ,
q q q q s q p
K (cid:18) −Z (cid:19)
−∞
where
ˆς (ω) = (2π) 1/4 2ωImχ (ω +i0), w = p,q, (49)
w − w
p
with √ the operator square root.
•
In particular, we can handle it this way: i.) Non-isotropic media, provided the tensors
χ are real symmetric. (Gyrotropy could in principle be handled by a more involved
w
construction with terms mixing momenta and coordinates.) ii.) Space dispersion, in
which case terms depending on ϕ and θ appear in the Hamiltonian. Details of the
∇ ∇
abstract construction and further examples will be given in forthcoming work [3].
3. Discussion and comparison with prior work
The need for a Hamiltonian description of a dissipative system has long been known,
having been emphasized by Morse and Feshbach [19, Ch. 3.2.] forty years ago. They
constructed, for a damped oscillator, an artificial Hamiltonian based on a “mirror-image”
trick, incorporating a second oscillator with negative friction. The resulting Hamiltonian
is quite un-physical: it is unbounded from below and under time reversal the oscillator
is transformed into its “mirror-image.” The artificial nature of this construction was
described in [19, Ch. 3.2.]: “By this arbitrary trick we are able to handle dissipative
systems as though they were conservative. This is not very satisfactory if an alternate
method of solution is known...”
The Hamiltonian we construct for TDD media can be viewed as a quite general “satis-
factory solution” to the problem posed in [19, Ch. 3.2.] since we do not introduce negative
friction and, in particular, we do not make use of “mirror-images.” Instead we couple a
given TDD system to an effective model for the normal modes of the underlying medium.
For the combined system we give a non-negative Hamiltonian with a transparent inter-
pretation as the system energy. As regards the underlying microscopic theory, this is an
effective Hamiltonian for those modes well approximated by linear response.
The evolution equations of the proposed theory come from a Hamiltonian and are thus
time reversible. Nonetheless, an irreversible motion of the TDD system stems from the
9
3
2
1
0
–1
–2
–3
–10 –8 –6 –4 –2 0
s
Figure 1. The Lamb model, introduced in [10] to describe radiation damping, is a point
massattached toaninfinite elastic string anda Hook’slawspring. The point massevolves
as a classical linearly damped oscillator.
infinite heat capacity of the auxiliary system. This is demonstrated in its simplest form
by the damped harmonic oscillator
m∂2q(t)+γ∂ q(t)+kq(t) = 0, (50)
t t
which results from the TDD system
∂ p = −fq , fp(t)+γ 0∞fp(t−τ)dτ = √1mp(t) , (51)
t(cid:18)q(cid:19) (cid:18) f (cid:19) (cid:18) Rf (t) (cid:19) (cid:18)√kq(t)(cid:19)
p q
with m,k,γ > 0. The construction presented above reproduces in this simple case a
model due to Lamb in 1900 [10] — see Fig. 1 — in which the energy of an oscillator
escapes to infinity along an attached flexible string. The theory proposed here illustrates
that, from the standpoint of thermodynamics, dissipation in classical linear response is
an idealization which assumes infinite heat capacity of (hidden) degrees of freedom.
In general, the auxiliary system, described by the fields ϕ ,ϕ ,θ ,θ , is governed by
p q p q
a Hamiltonian H (ϕ,θ) of the following simple and univer(cid:8)sal form (cid:9)
hb
1
∞ 2 2
H (ϕ,θ) = θ(s) + ∂ ϕ(s) ds. (52)
hb 2 Z0 hk kHp⊕Hq k s kHp⊕Hqi
The Hamiltonian H (ϕ,θ) in (52) is a canonical heat bath (justifying the index hb) as
hb
described in [6, Section 2], [25, Section 2].
The physical concept of an ideal or canonical heat bath originates in thermodynam-
ics. Statistical mechanical models at the mathematical level of rigor were introduced,
motivated, and described rather recently (to our best knowledge), see [7, Section 1], [6,
10
Section 2], [25, Section 2] and references therein. The consensus among the references, on
the basis of statistical mechanics, is that the generator ofmotion for a canonical heat bath
must be i a self-adjoint operator with absolutely continuous spectrum with no gaps, i.e.
×
R
the spectrum must be the entire real line , and the spectrum must be of a uniform
multiplicity. These requirements lead to a system unitarily equivalent to the universal
form Hamiltonian (52), [6, Section 2].
General statistical mechanics considerations indicate that for a system to behave ac-
cording the thermodynamics it must be properly coupled to the heat bath. In particular,
the coupling should involve all modes of the heat bath, [7, Section 1], [6, Section 2], [25,
Section 2]. In our Hamiltonian setting the coupling is essentially K w,T ϕ , which is
w w w
h i
the dipole approximation, [25, Section 1,2], and the condition of coupling to all modes is
the following constraint on the dissipation:
RanImχ (ω) = H , w = p,q. (53)
w w
ω[R
∈ b
There is some relation in spirit between our theory and a recently proposed hydro-
dynamic theory (HT) [16], [8], [9]. Both theories are self contained, macroscopic, and
make no assumption on the underlying microstructure. However the theory proposed
here, unlike the HT, makes no use of parameters other than the susceptibilities of linear
response theory. Furthermore, the present theory is truly conservative, with dissipative
effects modeled by effectively irreversible energy transport to an auxiliary system, which
may be conceived of as constructed from flexible strings. In contrast, the HT makes use
of explicitly dissipative, nonconservative equations similar to those of Navier-Stokes.
Adeeper relationcanbefoundbetween theapproachdescribed hereandthewell known
dilation theory which, in certain cases, provides a treatment for dissipation and resonance
phenomena. We first give a brief account of the dilation theory, based on B. Pavlov’s
extensive review [23] as well as his more recent work [24]. For more detailed exposition
on the subject we refer the reader to [23], [24] and references therein.
The dilation theory was the first rather general approach to the construction of a
spectral theory for dissipative operators. It is based on an abstract version of the Lax-
Phillips scattering theory [15] which assumes that there are i.) a dynamical unitary
evolution group U = eiΩt in a Hilbert space H where Ω is a self-adjoint operator in H; ii.)
t
an “incoming” subspace D H invariant with respect to the semi-group U , t < 0, and
t
− ⊂
an “outgoing” subspace D H invariant with respect to the semi-group U , t > 0. The
+ t
⊂
invariant subspaces (also called scattering channels) D are assumed to be orthogonal.
±
Then oneintroduces the“observation”subspace K = H (D D )whichiscoinvariant
+
⊖ − ⊕
in the sense that the restriction of U , t > 0, to K is a well defined semigroup on its own,
t
namely for t > 0
Z = P U = eiBt, where P is the orthogonal projection on K. (54)
t K t|K K
Inmanyinteresting cases thegeneratorB ofthesemigroupZ isdissipative, i.e. ImB
t
≥
0 or, even, ImB > 0. So the relation (54) provides an interesting scenario within the Lax-
Phillips scattering theory for the rise of a dissipative operator. The dilation theory yields
the spectral theory through the construction of generalized eigenmodes (the scattering
theory) provided, of course, the conditions discussed above are satisfied.