Self-adjoint approximations of degenerate Schrodinger operator

0.19 MB·

by  V. Zh. Sakbaev

Self-adjoint approximations of
degenerate Schro¨dinger operator
V.Zh. Sakbaev and I.V. Volovich
∗
7
1
Steklov Mathematical Institute, RAS, Moscow
0
2

n
a
J
Abstract
0
1
TheproblemofconstructionaquantummechanicalevolutionfortheSchro¨dingerequation

] with a degenerate Hamiltonian which is a symmetric operator that does not have self-adjoint
h
extensions is considered. Self-adjoint regularization of the Hamiltonian does not lead to a
p
- preserving probability limiting evolution for vectors from the Hilbert space but it is used
h
to construct a limiting evolution of states on a C∗-algebra of compact operators and on an
t
a
abelian subalgebra of operators in the Hilbert space. The limiting evolution of the states on
m
the abelian algebra can be presented by the Kraus decomposition with two terms. Both of
[
this terms are corresponded to the unitary and shift components of Wold’s decomposition

1
of isometric semigroup generated by the degenerate Hamiltonian. Properties of the limiting
v
7 evolution of the states on the C*-algebras are investigated and it is shown that pure states
7 could evolve into mixed states.
7
2
0
1 Introduction
.
1
0
7 Degenerate elliptic and parabolic equations are widely studied, see for example [1, 2, 3].
1 Universal boundary conditions for various types of PDE were considered in [4].
:
v It was shown [2, 3] that the solvability of the Cauchy problem for the degenerate differen-
i
X tialequation with smoothcoefficients inthewholespaceholdsbutthedegeneratedifferential
r equation without smoothness of coefficients can have no solution. We will study an example
a
ofthesimplestfirstorderdifferentialoperatoronthesemi-line. Notethattheboundednessof
the domain in Euclidian coordinate space can arise as the result of singularity of coefficients
in the complement of this domain (see [5]).
In this paper we concern with the degenerate Schro¨dinger equation, see [6, 7, 8] of the
form
∂u
i = Hu (1)
∂t
where H is a second order differential operator with nonnegative characteristic form in a
−
region G Rn. It is assumed that under some boundary conditions H defines a symmetric
⊂
operator in the Hilbert space H = L2(G). We are interested in the case when H does not
have self-adjoint extensions.
∗[email protected]
1
The initial-boundary value problem for the Schrodinger equation 1 does not necessary
has a solution. In such a case we will use an elliptic regularization and proceed as follows.
Consider a family of operators H depending on the parameter ǫ > 0,
ǫ
H = ǫ∆+H (2)
ǫ
−
where∆is theLaplace operator. SupposethatH admitsaself-adjointextension (wedenote
ǫ
it by the same letter). Let u (t) = U (t)ϕ be a solution of the Cauchy problem
ǫ ǫ
∂u
i ǫ = H u (3)
ǫ ǫ
∂t
u (0) = ϕ, ϕ L2(G) (4)
ǫ
∈
Here U (t) = e−itHǫ is the unitary evolution operator.
ǫ
Shchrodinger equation with small paremeter arise in the study of limit behavior of a
class of a such quantum systems as electrons and holes in the nonhomogenious semiconducte
materials in the case of large value of effective mass of a particles in one part of materials
and small value of effective mass in another (see [9, 10, 8]).
The limit of u (t),ǫ 0 in the Hilbert space might be not exist however we will show
ǫ
→
that in some cases there exists the limit
σt(A) = lim u (t),Au (t) = lim ϕ,U∗(t)AU (t)ϕ (5)
ϕ ǫ→0h ǫ ǫ i ǫ→0h ǫ ǫ i
for any ϕ H,t 0 where A is an operator from a subalgebra of the algebra B(H) of
∈ ≥
bounded operators in H.
We will show that σt defines a state (a linear positive functional of norm 1) on a C∗-
ϕ
algebra and therefore it can be interpreted as quantum mechanical evolution generated by
the Schrodinger operator H.
Degenerate Hamiltonians arise in different problems.
1. Thetheory of semi-classical limit is the theory of the passage to the limit in the family
ofsecondorderdifferentialoperatorswiththesmallparemeter atthesecondderivatives. The
vanishing of the coefficients with second derivative in the case is the degeneration of Hamil-
tonians; the limit operator is the firstorder differential operator which can besymmetric but
not self-adjoint operator.
2. Similar effects has the family of Hamiltonians with large mass parameter. In the
semiconductors theory (see [9, 10]) the mass of a system is the nonnegative function on
the coordinate spece wich values is the constant in some domain of coordinate space and is
the large parameterin the complement of this domain. Then the limit Hamiltonian is the
degenered symmetric but not self-adjoint differential operator of second order in the domain
and of first order in its complement (see [7, 8, 5]).
3. Symmetric differential operators arise in the theory of conductance in the geometrical
graphs in nano-semiconductors theory (see [5, 11]). The degeneration of the second orger
coefficients ofdiffferential operatoronagraphcanbethereason oftheabsenceofself-adjoint
extensions of this operators.
4. The both of degeneration and nonsmoothness of coefficients of a first order transport
equation on a line is the reason of nonexistence or nonuniqueness of its solutions. Hence it
is the reason of luck of self-adjointness of the first order operator (see [3, 12]).
5. Positive-operator valued measures (POVM) are used in the quantum measurement
theory. Neurmak’s dilation theorem states that a POVM can be lifted to a projection-valued
measure. It is used in theory of open quantum systems (see [13, 14, 15, 16, 17]).
In the next section we study a simple model of quantum dynamics on the half-line with
the symmetric Hamiltonian which does not admit a self-adjoint extension.
2
2 Schrodinger equation on the half-line
In quantum mechanics, according to von Neumann’s axioms, observables correspond to self-
adjoint operators. However, Hermitian (symmetric) operators which don’t admit self-adjoin
extensions are also discussed as possible observables (see for example [13]).
It is well known that the momentum operator on a half-line is not self-adjoint. Here we
consider the momentum operator on the half-line R = x R : x > 0 as an example
+
{ ∈ }
of a degenerate Hamiltonian for the Schro¨dinger equation and try to define an appropriate
quantum dynamics. Let the Hamiltonian will have the form
∂
H = bpˆ= ib , b R (6)
− ∂x ∈
Thedifferential operator H defines a symmetric operator (denoted by the same letter) in the
Hilbert space H = L (R ) with the domain W1 (R ) which is a completion of the space
2 + 2,0 +
C∞(R ) with respect to the norm of the Sobolev space W1(R ), in particular u(+0) = 0 if
0 + 2 +
u W1 (R ). The adjoint operator H∗ is defined on the domain W1(R ) by the relation
∈ 2,0 + 2 +
(6). The deficiency index of the operator H is (n ,n ) = (1,0) if b < 0 and (0,1) if b > 0,
+ −
so the operator H doesn’t have self-adjoint extensions.
The Schrodinger equation with the Hamiltonian (6) reads
∂u ∂u
= b , t > 0, x > 0, (7)
∂t ∂x
with the initial-boundary value data
u(+0,x) = ϕ(x), x > 0, (8)
u(t,+0) =0, t > 0. (9)
The problem (7) - (9) does not have a solution in the case b > 0 since the solution u(t,x) =
ϕ(bt+x) of (7) - (8) does not satisfy the boundary condition (9) if b = 0, ϕ= 0.
6 6
But in the case b < 0 the problem (7) - (9) with the initial data ϕ D(H) has a unique
∈
solution which is given by the formula u(t,x) = ϕ(bt + x) if bt + x > 0; u(t,x) = 0 if
bt + x 0; (t,x) R R . The uniqueness of this solution is the consequence of the
+ +
≤ ∈ ×
symmetricity of Hamiltonian H.
To obtain a unified approach to this two cases we consider an approximate Hamiltonian
∂2 ∂
H = ǫ +ib (10)
ǫ − ∂x2 ∂x
where ǫ > 0. It defines a self-adjoint operator with the domain D(H ) = u W2(R ) :
ǫ { ∈ 2 +
u(+0) = 0 .
}
The corresponding Schro¨dinger equation reads
∂u ∂2u ∂u
ǫ ǫ ǫ
i = ǫ +ib , t > 0, x > 0, (11)
∂t − ∂x2 ∂x
with the initial-boundary data
u (+0,x) = ϕ(x), x > 0, ϕ H (12)
ǫ
∈
u (t,+0) = 0, t > 0. (13)
ǫ
3
A classical solution of the problem (11) - (13) on the interval I = (0,T), T > 0 is
a function v(t) C(I,C2(R ) D(H )) C1(I¯,C(R )) which satisfies Eq.(11) and the
+ ǫ +
∈ ∩ ∩
boundary conditions (8) - (9).
A generalized solution of the problem (11) - (13) on the interval I = (0,T), T > 0 is a
function v(t) C(I,H) which satisfies the integral equality
∈
t
(v(t),ψ) = (ϕ,ψ)+i (v(s),H ψ)ds, t R ,
Z ǫ ∈ +
0
for arbitrary element ψ D(H ).
ǫ
∈
The uniqueness of solution of the Cauchy problem (11) - (13) is the consequence of the
self-adjointness of the Hamiltonian H .
ǫ
We will show that the asymptotic behavior of the solution u (t,x) of the problem (11) -
ǫ
(13) as ǫ 0 has the form
→
b
uǫ(t,x) Ψǫ(t,x) = ϕ(bt+x) eiǫxϕ(bt x) (14)
∼ − −
Note that
Ψ (t,0) = 0, t > 0 (15)
ǫ
Firstly we obtaine the representation of the solution of the problem (11) - (13) for the
sufficiently smooth initaial data ϕ and after that we extend this result to arbitrary initial
data.
Lemma 1. Let b > 0 and ϕ W3 (R ). Then the solution u of the problem (11) - (13)
∈ 2,0 + ǫ
can be represented in the form
uǫ(t,x) = √e−4iππ/ǫ4t Z ∞[e4iǫt(x−y+bt)2 −e4iǫt(x+y−bt)2eiǫbx]ϕ(y)dy. (16)
0
Proof. Let u is a solution of Eq.(11). Then the function
ǫ
ib2 ib
v(t,x) = e−4ǫt−2ǫxuǫ(t,x) (17)
satisfies the following Schro¨dinger equation for the free particle
∂v ∂2v
i = ǫ , t > 0, x > 0. (18)
∂t − ∂x2
Equivalent problem for Eq.(18) has the following initial-boundary value conditions:
v(+0,x) = e−2ibǫxϕ(x), x R+, (19)
∈
v(t,+0) = 0, t R (20)
+
∈
It is known that the solution of the problem (18), (19), (20) has the form
v(t,x) = e−iπ/4 ∞[e4iǫt(x−y)2 e4iǫt(x+y)2]e2ibǫyϕ(y)dy (21)
√4πǫt Z −
0
Therefore the solution of the problem (11) - (13) will have the form:
uǫ(t,x) = √e−4iππ/ǫ4tei4bǫ2t+2ibǫxZ ∞[e4iǫt(x−y)2 −e4iǫt(x+y)2]e2ibǫyϕ(y)dy (22)
0
4
= e−iπ/4 ∞[e4iǫt(x−y+bt)2 e4iǫt(x+y−bt)2eiǫbx]ϕ(y)dy.
√4πǫt Z −
0
Note that under the assumption ϕ W3 (R ) we have
∈ 2,0 +
u C(R ,W3 (R )) C1(R ,W1 (R )).
ǫ ∈ + 2,0 + + 2,0 +
\
Hence the function u is the classical solution for the problem (11) - (13). Since the space
ǫ
W3 (R ) is dense in the space H = L (R ) and since the map U (t)
2,0 + 2 + ǫ
W3 (R ) ϕ() u (t, )
2,0 + ∋ · → ǫ ·
foranyt 0preservestheL -norm: u (t, ) = ϕ thenthefamilyofmapsU (t), t 0,
2 ǫ H H ǫ
≥ k · k k k ≥
can be redefined by the continuity onto the unitary semigroup
U (t), t 0, (23)
ǫ
≥
in the space H. Then for any ϕ H the equality (22) gives the function U (t)ϕ and this
ǫ
∈
function is the generalised solution of the problem (11) - (13).
Theorem 1. Let b > 0 and ϕ H. Then for the solution u (16) of the problem (11) -
ǫ
∈
(13) the following asymptotic expansion holds as ǫ 0
→
u = v +r , ǫ (0,1),
ǫ ǫ ǫ
∈
where
ib
v (t,x) = ϕ(x+bt) θ(bt x)ϕ(bt x)eǫx, (t,x) R R , (24)
ǫ + +
− − − ∈ ×
and
lim ( sup r (t, ) )= 0 (25)
ǫ H
ǫ→+0 t∈[0,T]k · k
for any T > 0.
Proof.
The proof of the theorem 1 consists of a several lemmas.
Lemma 2. Let b > 0 and ϕ H. Then for any T > 0
∈
ǫ→lim+0(t∈s[u0,pT]kϕ(x+bt)− √e−4iππ/ǫ4t Z0∞e4iǫt(x−y+bt)2ϕ(y)dykH) = 0. (26)
Let v0 ∈ L2(R) be the function which is defined by the conditions: v0|R− = 0 and
v0|R+ = ϕ. Then for any t > 0 the following function vǫ(t,·) : mathbbR → C is defined by
the equality
vǫ(t,x) = √ei4ππ/4ǫt Z ∞ e−4iǫt(x−y+bt)2v0(y)dy, x ∈R, t > 0. (27)
−∞
Hence Foirier image vˆ(t,ξ) = (v(t, ))(ξ) satisfies the equality
F ·
vˆ (t,ξ) =e−2iǫtξ2−ibtξvˆ (ξ).
ǫ 0
Since u L (R ), then v L (R) and vˆ L (R). Therefore for any σ > 0 there is a
0 2 + 0 2 0 2
∈ ∈ ∈
number L > 0 such that kuˆ0 −PLuˆ0kL2(R) < σ4 where PL is the operator of multiplication
onto the indicator function χ of the segment [ L,L]. Since the functions iǫtξ2, ǫ > 0,
[−L,L]
− {
5
tendstozero as ǫ 0uniformlyon any rectangular [0,T] [ L,L], then thereis thenumber
→ × −
L
ǫ > 0 suchthat theestimate sup (eiǫtξ2 1)eibtξuˆ (ξ)2dξ σ2 holds forany ǫ (0,ǫ ).
0 | − 0 | ≤ 2 ∈ 0
t∈[0,T]−RL
Thusthereisthenumberǫ > 0suchthattheestimates sup vˆ (t,ξ) eibtξvˆ (ξ) 2 =
0 k ǫ − 0 kL2(R)
t∈[0,T]
−L L ∞ L
sup ( + + )(eiǫtξ2 1)eibtξuˆ (ξ)2dξ 2(u P u ) 2 + sup (eiǫtξ2
| − 0 | ≤ k 0 − L 0 kL2(R) | −
t∈[0,T] −R∞ −RL LR t∈[0,T]−RL
1)eibtξuˆ (ξ)2dξ σ2 hold for any ǫ (0,ǫ ).
0 0
| ≤ ∈
Since σ > 0 and T > 0 are arbitrary numbers then the sequence of functions vˆ , ǫ 0,
ǫ
→
converges in the space L (R) to the function wˆ(t,ξ) = eibtξvˆ (ξ) uniformly on the seqment
2 0
[0,T] with respect to parameter t. According to the unitarity of Fourier transformation the
sequence of functions v , ǫ 0, converges in the space L (R) to the function w(t,x) =
ǫ 2
→
v (x +bt), x R, uniformly on the segment [0,T] with respect to the parameter t. Since
0
∈
the functions u are the restrictions of the functions v on the quadrant K = t > 0, x > 0
ǫ ǫ
{ }
then the sequence of functions u , ǫ 0, converges in the space L (R ) to the function
ǫ 2 +
→
u(t,x) = v (x+bt) uniformly on the segment [0,T] with respect to parameter t (note that
0 K
|
if b > 0 then v (x+bt) = ϕ(x+bt) ). Lemma 2 is proved.
0 K K
| |
Lemma 3. Let b > 0 and ϕ H. Then for any T > 0
∈
ǫ→lim+0(t∈s[u0,pT]kθ(bt−x)ϕ(bt−x)− √e−4iππ/ǫ4t Z0∞e4iǫt(x+y−bt)2ϕ(y)dykH)= 0. (28)
To investigate the limit behavior of the second term in the expression (16) we should
introduced the following sequence of functions ψ : (0,+ ) R R, ǫ (0,1) , where
ǫ
{ ∞ × → ∈ }
for any t > 0 the function ψ (t,cdot) is defined on the line R by the equality
ǫ
ψǫ(t,x) = √ei4ππ/4ǫt Z ∞e−4iǫt(x+y−bt)2v0(y)dy, x ∈ R, t > 0. (29)
0
Bymeansoftheconvergence ofthesequence(27)wecanprovethatthesequenceoffunctions
ψ , ǫ 0, converges in the space L (R) to the function ψ(t,x) = v (bt x) uniformly on
ǫ 2 0
→ −
the segment [0,T] with respect to parameter t. Hence the sequence of its restrictions on the
quadrant K converges to the restriction on the quadrant K of a function ψ(t,x) = v (bt x)
0
−
(note that the support of the last function is ste subset of the angle x > 0, bt x > 0).
−
Hence the equality
ǫl→im0t∈s[u0,pT]k√ei4ππ/4ǫte−ibǫxZ0∞e−4ǫit(x+y−bt)2u0(y)dy−e−ibǫxψ(t,x)kL2(R+) = 0
holds for any T >0.
Therefore the equalities (24)-(25) hold as the consequence of Lemmas 2 and 3.
The convergence of the sequence of regularized unitary groups U acting in
ǫ
{ }
the space H
Let the operator H in the space H is defined by the relation (6) and the sequence of
regularized Hamiltonians H , ǫ > 0, ǫ 0 is given by the expression (10).
ǫ
→
Then according to the theorem 1 we obtain the following results about the behavior of
the sequence U as ǫ 0, where U (t) = e−itHǫ, t R.
ǫ ǫ
{ } → ∈
Proposition 1 ([8]). Let H is maximal symmetric operator in the space H with deficient
indexes (n ,n ). Then
− +
6
1. If n+ = 0 then the operator iH is the generator of isometric semigroup UL(t) =
e−itL, t 0, and the operator iH−∗ is the generator of contractive semigroup UL∗(t) =
e−itH∗, t≥ 0. −
≤
2. If n− = 0 then the operator iH is the generator of isometric semigroup UL(t) =
e−itL, t 0, and the operator iH−∗ is the generator of contractive semigroup UL∗(t) =
e−itH∗, t≤ 0,. −
≥
For any b R we define the operator-valued function V by the following conditions:
b
1. If b 0∈then V (t)= e−itH, t 0 and V (t) = e−itH∗, t 0.
b b
2. If b≤ 0 then V (t)= e−itH, t ≥ 0 and V (t) = e−itH∗, t ≤ 0.
b b
≥ ≤ ≥
Proposition 2 ([8]). Let the operator H in the space H is defined by the relation (6) and
the sequence of regularized Hamiltonians H , ǫ > 0, ǫ 0 is given by the expression (10).
ǫ
→
Then the sequence of unitary group U converges in the weak operator topology uniformly
ǫ
{ }
on any segment [c,d] R to the operator-valued function V :
b
∈
lim sup (φ,(U (t) V (t))ψ) = 0 c,d R; φ,ψ H.
ǫ b
ǫ→0t∈[c,d]| − | ∀ ∈ ∈
Moreover, if b 0 then
≤
lim sup (U (t) V (t))ψ) = 0 c,d [0,+ ); ψ H,
ǫ b
ǫ→0t∈[c,d]k − k ∀ ∈ ∞ ∈
or if b 0 then
≥
lim sup (U (t) V (t))ψ) = 0 c,d ( ,0]; ψ H.
ǫ b
ǫ→0t∈[c,d]k − k ∀ ∈ −∞ ∈
3 Quantum dynamics on the half-line
It is evident from Theorem 1 that the limit of the solution u as ǫ 0 doesn’t exist in
ǫ
→ H
due to the oscillation factor eibx/ǫ. However there exists a weak limit and moreover we prove
now that there exists the limit (5) for mean values of operators from a C∗-subalgebra of the
C∗-algebra B(H) of bounded operators in H.
In an algebraic approach to quantum mechanics [18] a physical system is defined by its
C∗-algebra withidentity. Observablescorrespondto self-adjointelements of . Thestates
A A
arenormalizedpositivelinearfunctionalson . Thetimedynamicsisgiven bya(semi)group
A
of -automorphism of .
∗ A
Thequantumtheorycanbepresentedbythefollowingdata ,S,αt ,where istheC∗-
{A } A
algebra(orvonNeymanalgebra),S isthesetofstatesonthealgebra ,αt isthe(semi)group
A
of authomorphisms of the algebra . The (semi)group Tt of dual authomorphisms of the set
A
of states is defined by the formula
< Ttρ,A>=< ρ,αt(A) >, ρ S,A .
∈ ∈ A
In both cases the group property takes place: αt+τ = αταt and Tt+τ = TtTτ.
Any group U of unitary operators in the Hilbert space H generates the group of autho-
t
morphismsαt of the algebra = B(H) of all boundedlinear operators in the space H which
A
is given by the formular αt(A) = U∗AU , A .
t t ∈ A
For investigation of our model we consider some different choises of a C∗-algebra of
A
observables: as the C∗-algebra the following cases will be studied:
A
7
1. the algebra B(H) of all bounded linear operators,
2. the algebra = I consists of the ring of compact operators and the unity
comp
A K { } K
operator, S
3. the algebra of operators of multiplication on themeasurableessentially bounded
mult
A
function which can be presented by the abelian algebra L (R ).
∞ +
4. the algebra = of operators which can be presented as the summ
sum comp mult
A A ⊕A
of two terms A and A .
1 comp 2 mult
∈ A ∈ A
We study the limit behavior as ǫ 0 of the family of the groups of automorphisms
→
αt, t R of C∗-algebra and the family of the groups of conjugate automorphisms Tt
{ ǫ ∈ } A ǫ
of the set of states Σ( ). For any ǫ > 0 and t R the map αt : is defined by
A ∈ ǫ A → A
the equalities αt(A) = U∗(t)AU (t), A . The unitary groups U is generated by the
ǫ ǫ ǫ ∈ A ǫ
regularized Hamiltonian (10) according to the equality
Ut = e−itHǫ, t R. (30)
ǫ ∈
Firstly we consider the case = B( ), αt(A) = U∗(t)AU (t). The regularized dynamic
A H ǫ ǫ ǫ
Tt of the set of states is defined by the action of the map Tt on any pure state ρ , where ϕ
ǫ ǫ ϕ
is a unit vector from the space :
H
< Ttρ ,A >=<U (t)ϕ,AU (t)ϕ > .
ǫ ϕ ǫ ǫ
Therefore the group property for the regularized dynamics holds: αt+τ = αταt, Tt+τ =
ǫ ǫ ǫ ǫ
TτTt.
ǫ ǫ
Now we give the description of the set of quantum states on the algebra B(H). Let
B(H) be the Banach space of linear bounded operators in the space H and B∗(H) is the
conjugate space. The set of quantum states Σ(H) is the set of linear continuous nonnegative
normalized functionals on the space B(H): the set Σ(H) is the intersection of the unique
sphere with the positive cone of the space (B(H))∗ (see [19]).
Let Σ (H) be the set of all quantum states which is continuous not only in the norm
n
topology but also in strong operator topology on the space B(H). Let T(H) be the Banach
space of trace-class operators endowing with trace norm. According to the results by [18] an
arbitrary normal state ρ Σ (H) can be uniquely represented by some nonnegative trace-
n
∈
class operator r T(H) with unique trace-norm. Let Σ (H) be the set of purevector states.
p
∈
Any pure state can be uniquely represented by one dimensional orthogonal projector. The
pure state ρ which is given by the orthogonal projector on the unique vector u H defines
u
∈
the linear continuous functional ρ : ρ ,A = (u,Au) , A B(H), on the space B(H).
u u H
h i ∈
According to [18] the set Σ (H) is the set of extreme points of the set Σ (H).
p n
The state ρ Σ(H) is called singular state iff the following equality holds
∈
sup ρ,P = 0
u
h i
kukH=1
(where P is orthogonal projector on the unit vector u H). The set of singular states is
u
∈
denoted by Σ (H). The following analogue of the statement of Iosida-Hewitt theorem for
s
measures takes place for the states.
Lemma 4. (See [20, 21]) For any states ρ Σ(H) there is the unique two functionals
∈
r T(H), r B∗(H) T(H) such that ρ= r +r .
n b n b
∈ ∈ \
To investigate the asymptotics of the regularized trajectories ρ , t 0, in the space
uǫ(t) ≥
of quantum states we use the decomposition of the solution of regularized Cauchy problem
(11) - (13) in the form (see Lemma 1):
u (t,x) = φ (t,x)+ψ (t,x),
ǫ ǫ ǫ
8
b
where φǫ(t,x) = uǫ(t,x) ψǫ(t,x) and ψǫ(t,x) = θ(bt x)ϕ(bt x)eiǫx (see theorem 1).
− − −
Then according to the theorem 1 we have that the following two statements hold:
Remark 1. The sequence of functions φ (t,x) converges in the space C(R ,H) uni-
ǫ +
{ }
formly on any segment [0,T] to the limit function v(t,x) = ϕ(x +bt) i.e. for any number
T > 0 the equality holds
lim sup v(t) φ (t) = 0;
ǫ H
ǫ→0t∈[0,T]k − k
Remark 2. the sequence of functions ψ (t,x) converges weakly in the space C(R ,H) to
ǫ +
{ }
zero uniformly on any segment [0,T] i.e. for any function g H and any number T > 0 the
∈
equality holds
lim sup (g,ψ (t)) = 0.
ǫ H
ǫ→0t∈[0,T]| |
Lemma5. ForanyoperatorA B(H)theequality lim ρ ,A = (v,Av)+lim(ψ ,Aψ )
∈ ǫ→0h uǫ i ǫ→0 ǫ ǫ
holds.
In fact, for any ǫ > 0 the following equality:
(u ,Au ) = (φ ,Aφ )+2Re((ψ ,Aφ ))+(ψ ,Aψ ) (31)
ǫ ǫ ǫ ǫ ǫ ǫ ǫ ǫ
holds. Then according to Remarks 1 and 2 the second term of the last expression tends to
zero as ǫ 0, therefore the statement of lemma B is proved.
→
The divergence of the sequence of regularized dynamics in the whole algebra
B(H)
The solution u (t,x) of regularized problem (11) - (13) can be presented by the regular-
ǫ
iszed group Ut of unitary operators in the Hilbert space :
ǫ H
u (t,x) = (Ut)ϕ(x),
ǫ ǫ
where Ut =e−itHǫ, t R (see (30)).
ǫ ∈
Theorem 2. If for some t > 0 and ϕ the sequence (Ut)ϕ, ǫ +0 converges
∈ H { ǫ → }
in weak topology of the space but diverges in the strong topology then the sequence
H
(Tt)ρ diverges in topologe of point-wice convergence on the space B(H). I.e. for any
{ ǫ ϕ}
infinitesimalsequence ǫ thereis the operator A B(H)such that thenumericalsequence
k
{ } ∈
(Tt)ρ ,A diverges.
{h ǫ ϕ i}
Theproofof thetheorem 2 follows from amore general theorem which is publishedin the
paper [8] (see theorem 14.2) or in the paper [7] (see theorem 7). Similar results is considered
in the work [22].
The convergence of the sequence regularized dynamics in the algebra
comp
A
Now we consider the dynamics of the states in the algebra of compact operators
comp
A
with the unity operator.
Let symbols Σ(H) and Σ (H) denote the set of state the set of pure states respectively
p
on the algebra B(H) of all bounded linear operators. Analogously, let symbols Σ( ) and
A
Σ ( ) note the set of state the set of pure states respectively on some subalgebra of
p
A A
algebra B(H). Let symbol ρ , u H, u = 1, note the state on the algebra acting by
u
∈ k k A
the equality ρ ,A = (u,Au), A B(H).
u
h i ∈
Let us define by means of degenerate Hamiltonian H = ibd/dx the following one-
parameter family Tt , t R, of maps of the subset Σ( ) into itself.
comp ∈ Acomp
1. Case b 0.
If t ≥ 0 t≤hen Tctompρϕ = ρe−itHϕ where ρe−itHϕ(A) = (e−itHϕ,Ae−itHϕ) for any A ∈
,ϕ .
A ∈ H
9
(e−iItfHt∗≤ϕ,0Ateh−eintHT∗ctϕom)pfJor=anJyaAndTctompρϕ a=ndρe−Ji(tHI)∗ϕ=+1(;1−Jk(Ae−)it=H∗0ϕkfo2r)Jalwlhceormepρae−ctitHo∗pϕe(rAat)or=s
comp
∈ A
A.
1. Case b 0.
(e−iIItffHtt∗≥≤ϕ,00Attehh−e≥einntHTT∗ctϕcotom)mpfpJoρrϕ=a=nJyρaeAn−ditHTϕctomw. phρeϕre=ρeρ−ei−tHitHϕ∗(ϕA+)(=1−(ek−e−itHitHϕ∗,ϕAke2−)JitHwϕh)erfeorρea−niytHA∗ϕ(∈AA).=
∈ A
Let for any ǫ (0,1) the regularized group Tt, t R, of transformation of the set Σ ( )
∈ ǫ ∈ p H
is generated by the regularized Hamiltonian L by the equality Ttρ = ρt where
ǫ ǫ ϕ ϕ,ǫ
ρt (A) =< Utϕ,AUtϕ > Ttρ ,A , A B(H), (32)
ϕ,ǫ ǫ ǫ ≡ h ǫ ϕ i ∀ ∈
and Ut = e−itHǫ, t R (see ((30))).
ǫ ∈
Since the one-parameter families of maps are defined on the set of the states ρ , ϕ
ϕ
{ ∈ H
and this set of states is dense in the set Σ( ) in the topology of point-wice convergence
comp
A
(see [22, 8]) then it can be uniquely extended onto the whole set Σ( ).
comp
A
Theorem 3. The sequence of regularized groups Tt converges to the one-parameter
{ ǫ}
family of maps Tt in the following sense:
comp
for any A , (of the type A = λ 1+K) the equality
comp
∈ A ·
lim sup Ttρ ,A Tt ρ ,A = 0. (33)
ǫ→0t∈[−L,L]|h ǫ ϕ i−h comp ϕ i|
holds for any L > 0,ϕ H.
∈
The statement of the theorem 3 is the consequence of the lemma 5 and theorem 1.
Proof. Let K be a compact self-adjoint operator in the space H and let σ is a positive
number. Then there is the finite number of unique vectors φ ,...,φ and the finite number
1 m
m
of real numbers c ,...,c such that K c P < 1σ where P is the operator of
1 m k − k φkkB(H) 2 φk
kP=1
orthogonal projection on the one-dimensional subspace lin(φ ) of the space .
k
m m mH
Therefore ρt (K) ρt ( c P ) 1σ. Since ρt ( c P ) = c ρt (P ) =
| ϕ,ǫ − ϕ,ǫ k φk | ≤ 2 ϕ,ǫ k φk k ϕ,ǫ φk
kP=1 kP=1 kP=1
m
c (u (t),φ )2 then according to the theorem 1 and Rieman oscilation theorem
k ǫ k
| |
kP=1
m m m +∞
limρt ( c P ) = c lim[(u (t),φ )]2 = c (ϕ(x+bt)φ (x))dx 2.
ǫ→0 ϕ,ǫ k φk k|ǫ→0 ǫ k | k| k |
kP=1 kP=1 kP=1 R0
m
If u∗(t,x) = φ(x + bt), (t,x) R R , then limρt ( c P ) =
∈ + × + ǫ→0 ϕ,ǫ k φk
kP=1
m
(u∗(t, ),( c P )u∗(t, )).
· k φk ·
kP=1
m
Since (u∗(t, ),Ku∗(t, )) (u∗(t, ),( c P )u∗(t, )) < 1σ then
| · · − · k φk · | 2
kP=1
lim (u∗(t, ),Ku∗(t, )) ρt (K) σ.
ǫ→0| · · − ϕ,ǫ | ≤
Since σ > 0 is arbitrary number then limρt (K) = (u∗(t, ),Ku∗(t, )) for arbitrary t 0
ǫ→0 ϕ,ǫ · · ≥
and arbitrary compact operator K. The theorem 3 is proved.
Theorem 4. The family of maps Tt , t R, is not a group. But the restrictions
comp ∈
Tt (b), t R are both semigroups of maps of the set Σ( ) into itself: Tt Tτ =
comp ∈ ± Acomp comp comp
Tt+τ , t,τ R .
comp ∈ +
Proof. The function Tt , t R, is not a group since, in particular, the equalities
comp ∈
T−t Tt ρ = ρ = Tt T−t ρ can’t hold both for any t > 0. The semigroup properties
comp comp ϕ ϕ comp comp ϕ
10