Self-adjoint approximations of degenerate Schrodinger operator

TL;DR

This paper develops a method to construct a limiting quantum evolution for degenerate Hamiltonians lacking self-adjoint extensions, using regularization and algebraic approaches, revealing that pure states can evolve into mixed states.

Contribution

It introduces a novel approach to define quantum evolution for degenerate Hamiltonians without self-adjoint extensions via C*-algebra and Kraus decomposition.

Findings

Limiting evolution can be represented by Kraus decomposition with two terms.

Pure states can evolve into mixed states under the constructed evolution.

Properties of the evolution on C*-algebras are characterized.

Abstract

The problem of construction a quantum mechanical evolution for the Schrodinger equation with a degenerate Hamiltonian which is a symmetric operator that does not have self-adjoint extensions is considered. Self-adjoint regularization of the Hamiltonian does not lead to a preserving probability limiting evolution for vectors from the Hilbert space but it is used to construct a limiting evolution of states on a C*-algebra of compact operators and on an abelian subalgebra of operators in the Hilbert space. The limiting evolution of the states on 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 of isometric semigroup generated by the degenerate Hamiltonian. Properties of the limiting evolution of the states on the C*-algebras are investigated and it is shown that pure states could evolve into mixed states.

Equations

$$i\frac{\partial u}{\partial t}=\mathbf{H}u$$

$${\mathbf{H}}_{ϵ}=-ϵ\Delta +\mathbf{H}$$

$$i\frac{\partial u_{ϵ}}{\partial t}={\mathbf{H}}_{ϵ}{u}_{ϵ}$$

$$u_{ϵ}(0)=\phi ,\phi \in L^2(G)$$

$${\sigma }_{\phi }^t(A)=\underset{ϵ\to 0}{\lim }⟨u_{ϵ}(t),A{u}_{ϵ}(t)⟩=\underset{ϵ\to 0}{\lim }⟨\phi ,U_{ϵ}^{\ast }(t)A{U}_{ϵ}(t)\phi ⟩$$

$$\mathbf{H}=-b\hat{p}=ib\frac{\partial }{\partial x},b\in \mathbb{R}$$

$$\frac{\partial u}{\partial t}=b\frac{\partial u}{\partial x},t>0,x>0,$$

$$u(+0,x)=\phi (x),x>0,$$

$$u(t,+0)=0,t>0.$$

$${\mathbf{H}}_{ϵ}=-ϵ\frac{{\partial }^2}{\partial x^2}+ib\frac{\partial }{\partial x}$$

$$i\frac{\partial u_{ϵ}}{\partial t}=-ϵ\frac{{\partial }^2{u}_{ϵ}}{\partial x^2}+ib\frac{\partial u_{ϵ}}{\partial x},t>0,x>0,$$

$$u_{ϵ}(+0,x)=\phi (x),x>0,\phi \in H$$

$$u_{ϵ}(t,+0)=0,t>0.$$

$$(v(t),\psi )=(\phi ,\psi )+i\underset{0}{\overset{t}{\int }}(v(s),{\mathbf{H}}_{ϵ}\psi )ds,t\in R_{+},$$

$$u_{ϵ}(t,x)\sim {\Psi }_{ϵ}(t,x)=\phi (bt+x)-e^{i\frac{b}{ϵ}x}\phi (bt-x)$$

$${\Psi }_{ϵ}(t,0)=0,t>0$$

$$u_{ϵ}(t,x)=\frac{e^{-i\pi /4}}{\sqrt{4\pi ϵt}}{\int }_0^{\infty }[e^{\frac{i}{4ϵt}(x-y+bt{)}^2}-e^{\frac{i}{4ϵt}(x+y-bt{)}^2}{e}^{\frac{ib}{ϵ}x}]\phi (y)dy.$$

$$v(t,x)=e^{-\frac{i{b}^2}{4ϵ}t-\frac{ib}{2ϵ}x}{u}_{ϵ}(t,x)$$

$$i\frac{\partial v}{\partial t}=-ϵ\frac{{\partial }^{2}v}{\partial x^2},t>0,x>0.$$

$$v(+0,x)=e^{-\frac{ib}{2ϵ}x}\phi (x),x\in {\mathbb{R}}_{+},$$

$$v(t,+0)=0,t\in {\mathbb{R}}_{+}$$

$$v(t,x)=\frac{e^{-i\pi /4}}{\sqrt{4\pi ϵt}}{\int }_0^{\infty }[e^{\frac{i}{4ϵt}(x-y{)}^2}-e^{\frac{i}{4ϵt}(x+y{)}^2}]e^{\frac{ib}{2ϵ}y}\phi (y)dy$$

$$u_{ϵ}(t,x)=\frac{e^{-i\pi /4}}{\sqrt{4\pi ϵt}}{e}^{\frac{i{b}^2}{4ϵ}t+\frac{ib}{2ϵ}x}{\int }_0^{\infty }[e^{\frac{i}{4ϵt}(x-y{)}^2}-e^{\frac{i}{4ϵt}(x+y{)}^2}]e^{\frac{ib}{2ϵ}y}\phi (y)dy$$

$$=\frac{e^{-i\pi /4}}{\sqrt{4\pi ϵt}}{\int }_0^{\infty }[e^{\frac{i}{4ϵt}(x-y+bt{)}^2}-e^{\frac{i}{4ϵt}(x+y-bt{)}^2}{e}^{\frac{ib}{ϵ}x}]\phi (y)dy.$$

$$u_{ϵ}\in C({\mathbb{R}}_{+},W_{2,0}^3({\mathbb{R}}_{+}))\bigcap C^1({\mathbb{R}}_{+},W_{2,0}^1({\mathbb{R}}_{+})).$$

$$W_{2,0}^3({\mathbb{R}}_{+})\ni \phi (\cdot )\to u_{ϵ}(t,\cdot )$$

$${\mathbf{U}}_{ϵ}(t),t\ge 0,$$

$$u_{ϵ}=v_{ϵ}+r_{ϵ},ϵ\in (0,1),$$

$$v_{ϵ}(t,x)=\phi (x+bt)-\theta (bt-x)\phi (bt-x)e^{\frac{ib}{ϵ}x},(t,x)\in {\mathbb{R}}_{+}\times {\mathbb{R}}_{+},$$

$$\underset{ϵ\to +0}{\lim }(\underset{t\in [0,T]}{\sup }\Vert r_{ϵ}(t,\cdot ){\Vert }_H)=0$$

Full text

1 Introduction

Degenerate elliptic and parabolic equations are widely studied, see for example [1, 2, 3]. Universal boundary conditions for various types of PDE were considered in [4].

It was shown [2, 3] that the solvability of the Cauchy problem for the degenerate differential equation with smooth coefficients in the whole space holds but the degenerate differential equation without smoothness of coefficients can have no solution. We will study an example of the simplest first order differential operator on the semi-line. Note that the boundedness of 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 Schrödinger equation, see [6, 7, 8] of the form

[TABLE]

where $-{\bf H}$ is a second order differential operator with nonnegative characteristic form in a region $G\subset\mathbb{R}^{n}$. It is assumed that under some boundary conditions ${\bf H}$ defines a symmetric operator in the Hilbert space $H=L^{2}(G)$. We are interested in the case when ${\bf H}$ does not have self-adjoint extensions.

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 ${\bf H}_{\epsilon}$ depending on the parameter $\epsilon>0$,

[TABLE]

where $\Delta$ is the Laplace operator. Suppose that ${\bf H}_{\epsilon}$ admits a self-adjoint extension (we denote it by the same letter). Let $u_{\epsilon}(t)=U_{\epsilon}(t)\varphi$ be a solution of the Cauchy problem

[TABLE]

[TABLE]

Here $U_{\epsilon}(t)=e^{-it{\bf H}_{\epsilon}}$ is the unitary evolution operator.

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 $u_{\epsilon}(t),\epsilon\to 0$ in the Hilbert space might be not exist however we will show that in some cases there exists the limit

[TABLE]

for any $\varphi\in H,t\geq 0$ where $A$ is an operator from a subalgebra of the algebra $B(H)$ of bounded operators in $H$.

We will show that $\sigma_{\varphi}^{t}$ defines a state (a linear positive functional of norm 1) on a $C^{*}$-algebra and therefore it can be interpreted as quantum mechanical evolution generated by the Schrodinger operator ${\bf H}$.

Degenerate Hamiltonians arise in different problems.

1. The theory of semi-classical limit is the theory of the passage to the limit in the family of second order differential operators with the small paremeter at the second derivatives. The vanishing of the coefficients with second derivative in the case is the degeneration of Hamiltonians; the limit operator is the first order differential operator which can be symmetric 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 of diffferential operator on a graph can be the reason of the absence of self-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 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 $\mathbb{R}_{+}=\{x\in\mathbb{R}:x>0\}$ as an example of a degenerate Hamiltonian for the Schrödinger equation and try to define an appropriate quantum dynamics. Let the Hamiltonian will have the form

[TABLE]

The differential operator $H$ defines a symmetric operator (denoted by the same letter) in the Hilbert space $H=L_{2}(\mathbb{R}_{+})$ with the domain $W_{2,0}^{1}(\mathbb{R}_{+})$ which is a completion of the space $C_{0}^{\infty}(\mathbb{R}_{+})$ with respect to the norm of the Sobolev space $W_{2}^{1}(\mathbb{R}_{+})$, in particular $u(+0)=0$ if $u\in W_{2,0}^{1}(\mathbb{R}_{+})$. The adjoint operator $H^{*}$ is defined on the domain $W_{2}^{1}(\mathbb{R}_{+})$ by the relation (6). The deficiency index of the operator ${\bf H}$ 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

[TABLE]

with the initial-boundary value data

[TABLE]

[TABLE]

The problem (7) - (9) does not have a solution in the case $b>0$ since the solution $u(t,x)=\varphi(bt+x)$ of (7) - (8) does not satisfy the boundary condition (9) if $b\neq 0,\,\varphi\neq 0$.

But in the case $b<0$ the problem (7) - (9) with the initial data $\varphi\in D({\bf H})$ has a unique solution which is given by the formula $u(t,x)=\varphi(bt+x)\,$ if $\,bt+x>0;\quad u(t,x)=0\,$ if $\,bt+x\leq 0;\ (t,x)\in\mathbb{R}_{+}\times\mathbb{R}_{+}.$ The uniqueness of this solution is the consequence of the symmetricity of Hamiltonian ${\bf H}$.

To obtain a unified approach to this two cases we consider an approximate Hamiltonian

[TABLE]

where $\epsilon>0$. It defines a self-adjoint operator with the domain $D({\bf H}_{\epsilon})=\{u\in W_{2}^{2}(\mathbb{R}_{+}):u(+0)=0\}$.

The corresponding Schrödinger equation reads

[TABLE]

with the initial-boundary data

[TABLE]

[TABLE]

A classical solution of the problem (11) - (13) on the interval $I=(0,T),\,T>0$ is a function $v(t)\in C(I,C^{2}(\mathbb{R}_{+})\cap D(H_{\epsilon}))\cap C^{1}(\bar{I},C(\mathbb{R}_{+}))$ which satisfies Eq.(11) and the boundary conditions (8) - (9).

A generalized solution of the problem (11) - (13) on the interval $I=(0,T),\,T>0$ is a function $v(t)\in C(I,H)$ which satisfies the integral equality

[TABLE]

for arbitrary element $\psi\in D({\bf H}_{\epsilon})$.

The uniqueness of solution of the Cauchy problem (11) - (13) is the consequence of the self-adjointness of the Hamiltonian ${\bf H}_{\epsilon}$.

We will show that the asymptotic behavior of the solution $u_{\epsilon}(t,x)$ of the problem (11) - (13) as $\epsilon\to 0$ has the form

[TABLE]

Note that

[TABLE]

Firstly we obtaine the representation of the solution of the problem (11) - (13) for the sufficiently smooth initaial data $\varphi$ and after that we extend this result to arbitrary initial data.

Lemma 1. Let $b>0$ and $\varphi\in W^{3}_{2,0}(\mathbb{R}_{+})$. Then the solution $u_{\epsilon}$ of the problem (11) - (13) can be represented in the form

[TABLE]

Proof. Let $u_{\epsilon}$ is a solution of Eq.(11). Then the function

[TABLE]

satisfies the following Schrödinger equation for the free particle

[TABLE]

Equivalent problem for Eq.(18) has the following initial-boundary value conditions:

[TABLE]

[TABLE]

It is known that the solution of the problem (18), (19), (20) has the form

[TABLE]

Therefore the solution of the problem (11) - (13) will have the form:

[TABLE]

[TABLE]

Note that under the assumption $\varphi\in W^{3}_{2,0}(R_{+})$ we have

[TABLE]

Hence the function $u_{\epsilon}$ is the classical solution for the problem (11) - (13). Since the space $W^{3}_{2,0}(\mathbb{R}_{+})$ is dense in the space $H=L_{2}(\mathbb{R}_{+})$ and since the map ${\bf U}_{\epsilon}(t)$

[TABLE]

for any $t\geq 0$ preserves the $L_{2}$-norm: $\|u_{\epsilon}(t,\cdot)\|_{H}=\|\varphi\|_{H}$ then the family of maps ${\bf U}_{\epsilon}(t),\,t\geq 0,$ can be redefined by the continuity onto the unitary semigroup

[TABLE]

in the space $H$. Then for any $\varphi\in H$ the equality (22) gives the function ${\bf U}_{\epsilon}(t)\varphi$ and this function is the generalised solution of the problem (11) - (13).

Theorem 1. Let $b>0$ and $\varphi\in H$. Then for the solution $u_{\epsilon}$ (16) of the problem (11) - (13) the following asymptotic expansion holds as $\epsilon\to 0$

[TABLE]

where

[TABLE]

and

[TABLE]

for any $T>0$.

Proof.

The proof of the theorem 1 consists of a several lemmas.

Lemma 2. Let $b>0$ and $\varphi\in H$. Then for any $T>0$

[TABLE]

Let $v_{0}\in L_{2}(\mathbb{R})$ be the function which is defined by the conditions: $v_{0}|_{\mathbb{R}_{-}}=0$ and $v_{0}|_{\mathbb{R}_{+}}=\varphi$. Then for any $t>0$ the following function $v_{\epsilon}(t,\cdot):\ mathbb{R}\to\mathbb{C}$ is defined by the equality

[TABLE]

Hence Foirier image $\hat{v}(t,\xi)={\cal F}(v(t,\cdot))(\xi)$ satisfies the equality

[TABLE]

Since $u_{0}\in L_{2}(\mathbb{R}_{+})$, then $v_{0}\in L_{2}(\mathbb{R})$ and $\hat{v}_{0}\in L_{2}(\mathbb{R})$. Therefore for any $\sigma>0$ there is a number $L>0$ such that $\|\hat{u}_{0}-{\bf P}_{L}\hat{u}_{0}\|_{L_{2}(\mathbb{R})}<{{\sigma}\over 4}$ where ${\bf P}_{L}$ is the operator of multiplication onto the indicator function $\chi_{[-L,L]}$ of the segment $[-L,L]$. Since the functions $\{i\epsilon t\xi^{2},\,\epsilon>0$, tends to zero as $\epsilon\to 0$ uniformly on any rectangular $[0,T]\times[-L,L]$, then there is the number $\epsilon_{0}>0$ such that the estimate $\sup\limits_{t\in[0,T]}\int\limits_{-L}^{L}|(e^{i\epsilon t\xi^{2}}-1)e^{ibt\xi}\hat{u}_{0}(\xi)|^{2}d\xi\leq{{{\sigma}^{2}}\over 2}$ holds for any $\epsilon\in(0,\epsilon_{0})$.

Thus there is the number $\epsilon_{0}>0$ such that the estimates $\sup\limits_{t\in[0,T]}\|\hat{v}_{\epsilon}(t,\xi)-e^{ibt\xi}\hat{v}_{0}(\xi)\|_{L_{2}(\mathbb{R})}^{2}=\sup\limits_{t\in[0,T]}(\int\limits_{-\infty}^{-L}+\int\limits_{-L}^{L}+\int\limits_{L}^{\infty})|(e^{i\epsilon t\xi^{2}}-1)e^{ibt\xi}\hat{u}_{0}(\xi)|^{2}d\xi\leq\|2(u_{0}-{\bf P}_{L}u_{0})\|_{L_{2}(R)}^{2}+\sup\limits_{t\in[0,T]}\int\limits_{-L}^{L}|(e^{i\epsilon t\xi^{2}}-1)e^{ibt\xi}\hat{u}_{0}(\xi)|^{2}d\xi\leq\sigma^{2}$ hold for any $\epsilon\in(0,\epsilon_{0})$.

Since $\sigma>0$ and $T>0$ are arbitrary numbers then the sequence of functions $\hat{v}_{\epsilon},\,\epsilon\to 0,$ converges in the space $L_{2}(\mathbb{R})$ to the function $\hat{w}(t,\xi)=e^{ibt\xi}\hat{v}_{0}(\xi)$ uniformly on the seqment $[0,T]$ with respect to parameter $t$. According to the unitarity of Fourier transformation the sequence of functions $v_{\epsilon},\,\epsilon\to 0,$ converges in the space $L_{2}(\mathbb{R})$ to the function $w(t,x)=v_{0}(x+bt),\,x\in\mathbb{R},$ uniformly on the segment $[0,T]$ with respect to the parameter $t$. Since the functions $u_{\epsilon}$ are the restrictions of the functions $v_{\epsilon}$ on the quadrant $K=\{t>0,\,x>0\}$ then the sequence of functions $u_{\epsilon},\,\epsilon\to 0,$ converges in the space $L_{2}(\mathbb{R}_{+})$ to the function $u(t,x)=v_{0}(x+bt)|_{K}$ uniformly on the segment $[0,T]$ with respect to parameter $t$ (note that if $b>0$ then $v_{0}(x+bt)|_{K}=\varphi(x+bt)|_{K}$). Lemma 2 is proved.

Lemma 3. Let $b>0$ and $\varphi\in H$. Then for any $T>0$

[TABLE]

To investigate the limit behavior of the second term in the expression (16) we should introduced the following sequence of functions $\{\psi_{\epsilon}:\ (0,+\infty)\times\mathbb{R}\,\to\,\mathbb{R},\ \epsilon\in(0,1)\}$, where for any $t>0$ the function $\psi_{\epsilon}(t,cdot)$ is defined on the line $\mathbb{R}$ by the equality

[TABLE]

By means of the convergence of the sequence (27) we can prove that the sequence of functions $\psi_{\epsilon},\,\epsilon\to 0,$ converges in the space $L_{2}(\mathbb{R})$ to the function $\psi(t,x)=v_{0}(bt-x)$ uniformly on 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 $\psi(t,x)=v_{0}(bt-x)$ (note that the support of the last function is ste subset of the angle $x>0,\,bt-x>0$).

Hence the equality

[TABLE]

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 $\{{\bf U}_{\epsilon}\}$ acting in the space $H$

Let the operator $\bf H$ in the space $H$ is defined by the relation (6) and the sequence of regularized Hamiltonians ${\bf H}_{\epsilon},\ \epsilon>0,\,\epsilon\to 0$ is given by the expression (10).

Then according to the theorem 1 we obtain the following results about the behavior of the sequence $\{{\bf U}_{\epsilon}\}$ as $\epsilon\to 0$, where ${\bf U}_{\epsilon}(t)=e^{-it{\bf H}_{\epsilon}},\,t\in\mathbb{R}$.

Proposition 1 ([8]). Let $\bf H$ is maximal symmetric operator in the space $H$ with deficient indexes $(n_{-},n_{+})$. Then

1. If $n_{+}=0$ then the operator $-i{\bf H}$ is the generator of isometric semigroup ${\bf U}_{\bf L}(t)=e^{-it{\bf L}},\,t\geq 0,$ and the operator $-i{\bf H}^{*}$ is the generator of contractive semigroup ${\bf U}_{{\bf L}^{*}}(t)=e^{-it{\bf H}*},\,t\leq 0$.

*2. If $n_{-}=0$ then the operator $-i{\bf H}$ is the generator of isometric semigroup ${\bf U}_{\bf L}(t)=e^{-it{\bf L}},\,t\leq 0,$ and the operator $-i{\bf H}^{*}$ is the generator of contractive semigroup ${\bf U}_{{\bf L}^{*}}(t)=e^{-it{\bf H}*},\,t\geq 0,$. *

For any $b\in R$ we define the operator-valued function ${\bf V}_{b}$ by the following conditions:

1. If $b\leq 0$ then ${\bf V}_{b}(t)=e^{-it{\bf H}},\,t\geq 0$ and ${\bf V}_{b}(t)=e^{-it{\bf H}^{*}},\,t\leq 0$.

2. If $b\geq 0$ then ${\bf V}_{b}(t)=e^{-it{\bf H}},\,t\leq 0$ and ${\bf V}_{b}(t)=e^{-it{\bf H}^{*}},\,t\geq 0$.

Proposition 2 ([8]). Let the operator $\bf H$ in the space $H$ is defined by the relation (6) and the sequence of regularized Hamiltonians ${\bf H}_{\epsilon},\ \epsilon>0,\,\epsilon\to 0$ is given by the expression (10). Then the sequence of unitary group $\{{\bf U}_{\epsilon}\}$ converges in the weak operator topology uniformly on any segment $[c,d]\in\mathbb{R}$ to the operator-valued function ${\bf V}_{b}$:

[TABLE]

Moreover, if $b\leq 0$ then

[TABLE]

or if $b\geq 0$ then

[TABLE]

3 Quantum dynamics on the half-line

It is evident from Theorem 1 that the limit of the solution $u_{\epsilon}$ as $\epsilon\to 0$ doesn’t exist in ${\cal H}$ due to the oscillation factor $e^{ibx/\epsilon}$. 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 ${\cal A}$ with identity. Observables correspond to self-adjoint elements of ${\cal A}$. The states are normalized positive linear functionals on ${\cal A}$. The time dynamics is given by a (semi)group of $\ast$-automorphism of ${\cal A}$.

The quantum theory can be presented by the following data $\{{\cal A},S,\alpha^{t}\}$, where ${\cal A}$ is the $C^{*}$-algebra (or von Neyman algebra), $S$ is the set of states on the algebra ${\cal A}$, $\alpha^{t}$ is the (semi)group of authomorphisms of the algebra ${\cal A}$. The (semi)group $T^{t}$ of dual authomorphisms of the set of states is defined by the formula

[TABLE]

In both cases the group property takes place: $\alpha^{t+\tau}=\alpha^{\tau}\alpha^{t}$ and $T^{t+\tau}=T^{t}T^{\tau}$.

Any group $U_{t}$ of unitary operators in the Hilbert space $H$ generates the group of authomorphisms $\alpha^{t}$ of the algebra ${\cal A}=B(H)$ of all bounded linear operators in the space $H$ which is given by the formular $\alpha^{t}(A)=U^{*}_{t}AU_{t}$,   $A\in{\cal A}$.

For investigation of our model we consider some different choises of a $C^{*}$-algebra $\cal A$ of observables: as the $C^{*}$-algebra $\cal A$ the following cases will be studied:

1. the algebra $B(H)$ of all bounded linear operators,

2. the algebra ${\cal A}_{comp}={\cal K}\bigcup\{{\bf I}\}$ consists of the ring $\cal K$ of compact operators and the unity operator,

3. the algebra ${\cal A}_{mult}$ of operators of multiplication on the measurable essentially bounded function which can be presented by the abelian algebra $L_{\infty}(R_{+})$.

4. the algebra ${\cal A}_{sum}={\cal A}_{comp}\oplus{\cal A}_{mult}$ of operators which can be presented as the summ of two terms ${\bf A}_{1}\in{\cal A}_{comp}$ and ${\bf A}_{2}\in{\cal A}_{mult}$.

We study the limit behavior as $\epsilon\to 0$ of the family of the groups of automorphisms $\{\alpha^{t}_{\epsilon},\,t\in R\}$ of $C^{*}$-algebra $\cal A$ and the family of the groups of conjugate automorphisms $T^{t}_{\epsilon}$ of the set of states $\Sigma({\cal A})$. For any $\epsilon>0$ and $t\in R$ the map $\alpha^{t}_{\epsilon}:\ {\cal A}\,\to\,{\cal A}$ is defined by the equalities $\alpha^{t}_{\epsilon}({\bf A})=U_{\epsilon}^{*}(t){\bf A}U_{\epsilon}(t),\ {\bf A}\in{\cal A}.$ The unitary groups ${\bf U}_{\epsilon}$ is generated by the regularized Hamiltonian (10) according to the equality

[TABLE]

Firstly we consider the case ${\cal A}=B({\cal H}),\,\,\alpha^{t}_{\epsilon}(A)=U_{\epsilon}^{*}(t)AU_{\epsilon}(t).$ The regularized dynamic $T_{\epsilon}^{t}$ of the set of states is defined by the action of the map $T_{\epsilon}^{t}$ on any pure state $\rho_{\varphi}$, where $\varphi$ is a unit vector from the space ${\cal H}$:

[TABLE]

Therefore the group property for the regularized dynamics holds: $\alpha^{t+\tau}_{\epsilon}=\alpha^{\tau}_{\epsilon}\alpha^{t}_{\epsilon},\,\,T^{t+\tau}_{\epsilon}=T^{\tau}_{\epsilon}T^{t}_{\epsilon}$.

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 $\Sigma(H)$ is the set of linear continuous nonnegative normalized functionals on the space $B(H)$: the set $\Sigma(H)$ is the intersection of the unique sphere with the positive cone of the space $(B(H))^{*}$ (see [19]).

Let $\Sigma_{n}(H)$ be the set of all quantum states which is continuous not only in the norm 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 $\rho\in\Sigma_{n}(H)$ can be uniquely represented by some nonnegative trace-class operator $r\in T(H)$ with unique trace-norm. Let $\Sigma_{p}(H)$ be the set of pure vector states. Any pure state can be uniquely represented by one dimensional orthogonal projector. The pure state $\rho_{u}$ which is given by the orthogonal projector on the unique vector $u\in H$ defines the linear continuous functional $\rho_{u}:\quad\langle\rho_{u},{\bf A}\rangle=(u,{\bf A}u)_{H},\,{\bf A}\in B(H)$, on the space $B(H)$. According to [18] the set $\Sigma_{p}(H)$ is the set of extreme points of the set $\Sigma_{n}(H)$.

The state $\rho\in\Sigma(H)$ is called singular state iff the following equality holds

[TABLE]

(where ${\bf P}_{u}$ is orthogonal projector on the unit vector $u\in H$). The set of singular states is denoted by $\Sigma_{s}(H)$. The following analogue of the statement of Iosida-Hewitt theorem for measures takes place for the states.

Lemma 4. (See [20, 21]) For any states $\rho\in\Sigma(H)$ there is the unique two functionals $r_{n}\in T(H),\,r_{b}\in B^{*}(H)\backslash T(H)$ such that $\rho=r_{n}+r_{b}$.

To investigate the asymptotics of the regularized trajectories $\rho_{u_{\epsilon}(t)},\,t\geq 0,$ in the space of quantum states we use the decomposition of the solution of regularized Cauchy problem (11) - (13) in the form (see Lemma 1):

[TABLE]

where $\phi_{\epsilon}(t,x)=u_{\epsilon}(t,x)-\psi_{\epsilon}(t,x)$ and $\psi_{\epsilon}(t,x)=\theta(bt-x)\varphi(bt-x)e^{i{b\over{\epsilon}}x}$ (see theorem 1). Then according to the theorem 1 we have that the following two statements hold:

Remark 1. The sequence of functions $\{\phi_{\epsilon}(t,x)\}$ converges in the space $C(R_{+},H)$ uniformly on any segment $[0,T]$ to the limit function $v(t,x)=\varphi(x+bt)$ i.e. for any number $T>0$ the equality holds

[TABLE]

Remark 2. the sequence of functions $\{\psi_{\epsilon}(t,x)\}$ converges weakly in the space $C(R_{+},H)$ to zero uniformly on any segment $[0,T]$ i.e. for any function $g\in H$ and any number $T>0$ the equality holds

[TABLE]

Lemma 5. For any operator ${\bf A}\in B(H)$ the equality $\lim\limits_{\epsilon\to 0}\langle\rho_{{u_{\epsilon}}},{\bf A}\rangle=(v,{\bf A}v)+\lim\limits_{\epsilon\to 0}(\psi_{\epsilon},{\bf A}\psi_{\epsilon})$ holds.

In fact, for any $\epsilon>0$ the following equality:

[TABLE]

holds. Then according to Remarks 1 and 2 the second term of the last expression tends to zero as $\epsilon\to 0$, therefore the statement of lemma B is proved.

The divergence of the sequence of regularized dynamics in the whole algebra $B(H)$

The solution $u_{\epsilon}(t,x)$ of regularized problem (11) - (13) can be presented by the regulariszed group $U_{\epsilon}^{t}$ of unitary operators in the Hilbert space ${\cal H}$:

[TABLE]

where $U^{t}_{\epsilon}=e^{-it{\bf H}_{\epsilon}},\,t\in\mathbb{R}$ (see (30)).

Theorem 2. If for some $t>0$ and $\varphi\in\cal H$ the sequence $\{(U^{t}_{\epsilon})\varphi,\ {\epsilon\to+0}\}$ converges in weak topology of the space $\cal H$ but diverges in the strong topology then the sequence $\{(T^{t}_{\epsilon})\rho_{\varphi}\}$ diverges in topologe of point-wice convergence on the space $B(H)$. I.e. for any infinitesimal sequence $\{\epsilon_{k}\}$ there is the operator ${\bf A}\in B(H)$ such that the numerical sequence $\{\langle(T^{t}_{\epsilon})\rho_{\varphi},{\bf A}\rangle\}$ diverges.

The proof of the theorem 2 follows from a more general theorem which is published in 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 ${\cal A}_{comp}$

Now we consider the dynamics of the states in the algebra ${\cal A}_{comp}$ of compact operators with the unity operator.

Let symbols $\Sigma(H)$ and $\Sigma_{p}(H)$ denote the set of state the set of pure states respectively on the algebra $B(H)$ of all bounded linear operators. Analogously, let symbols $\Sigma({\cal A})$ and $\Sigma_{p}({\cal A})$ note the set of state the set of pure states respectively on some subalgebra $\cal A$ of algebra $B(H)$. Let symbol $\rho_{u},\,u\in H,\,\|u\|=1,$ note the state on the algebra $\cal A$ acting by the equality $\langle\rho_{u},{\bf A}\rangle=(u,{\bf A}u),\,{\bf A}\in B(H)$.

Let us define by means of degenerate Hamiltonian ${\bf H}=ibd/dx$ the following one-parameter family $T^{t}_{comp},\,t\in\mathbb{R},$ of maps of the subset $\Sigma({\cal A}_{comp})$ into itself.

1. Case $b\leq 0$.

If $t\geq 0$ then $T_{comp}^{t}\rho_{\varphi}=\rho_{e^{-it{\bf H}}\varphi}$ where $\rho_{e^{-it{\bf H}}\varphi}({\bf A})=(e^{-it{\bf H}}\varphi,{\bf A}e^{-it{\bf H}}\varphi)$ for any ${\bf A}\in\cal A,\varphi\in H$.

If $t\leq 0$ then $T_{comp}^{t}J=J$ and $T_{comp}^{t}\rho_{\varphi}=\rho_{e^{-it{\bf H}^{*}}\varphi}+(1-\|e^{-it{\bf H}^{*}}\varphi\|^{2})J$ where $\rho_{e^{-it{\bf H}^{*}}\varphi}({\bf A})=(e^{-it{\bf H}^{*}}\varphi,{\bf A}e^{-it{\bf H}^{*}}\varphi)$ for any ${\bf A}\in{\cal A}_{comp}$ and $J({\bf I})=1;\ J({\bf A})=0$ for all compact operators $\bf A$.

1. Case $b\geq 0$.

If $t\leq 0$ then $T_{comp}^{t}\rho_{\varphi}=\rho_{e^{-it{\bf H}}\varphi}$ where $\rho_{e^{-it{\bf H}}\varphi}({\bf A})=(e^{-it{\bf H}}\varphi,{\bf A}e^{-it{\bf H}}\varphi)$ for any ${\bf A}\in\cal A$.

If $t\geq 0$ then $T_{comp}^{t}J=J$ and $T_{comp}^{t}\rho_{\varphi}=\rho_{e^{-it{\bf H}^{*}}\varphi}+(1-\|e^{-it{\bf H}^{*}}\varphi\|^{2})J$ where $\rho_{e^{-it{\bf H}^{*}}\varphi}({\bf A})=(e^{-it{\bf H}^{*}}\varphi,{\bf A}e^{-it{\bf H}^{*}}\varphi)$ for any ${\bf A}\in\cal A$.

Let for any $\epsilon\in(0,1)$ the regularized group $T_{\epsilon}^{t},\,t\in\mathbb{R},$ of transformation of the set $\Sigma_{p}({\cal H})$ is generated by the regularized Hamiltonian ${\bf L}_{\epsilon}$ by the equality $T_{\epsilon}^{t}\rho_{\varphi}=\rho_{\varphi,\epsilon}^{t}$ where

[TABLE]

and $U^{t}_{\epsilon}=e^{-it{\bf H}_{\epsilon}},\,t\in R$ (see ((30))).

Since the one-parameter families of maps are defined on the set of the states $\{\rho_{\varphi},\,\varphi\in{\cal H}$ and this set of states is dense in the set $\Sigma({\cal A}_{comp})$ in the topology of point-wice convergence (see [22, 8]) then it can be uniquely extended onto the whole set $\Sigma({\cal A}_{comp}).$

Theorem 3. The sequence of regularized groups $\{T_{\epsilon}^{t}\}$ converges to the one-parameter family of maps $T_{comp}^{t}$ in the following sense:

for any ${\bf A}\in{\cal A}_{comp}$, (of the type ${\bf A}=\lambda\cdot 1+K$) the equality

[TABLE]

*holds for any $L>0,\varphi\in H$. *

The statement of the theorem 3 is the consequence of the lemma 5 and theorem 1.

Proof. Let ${\bf K}$ be a compact self-adjoint operator in the space $H$ and let $\sigma$ is a positive number. Then there is the finite number of unique vectors $\phi_{1},...,\phi_{m}$ and the finite number of real numbers $c_{1},...,c_{m}$ such that $\|{\bf K}-\sum\limits_{k=1}^{m}c_{k}{\bf P}_{\phi_{k}}\|_{B(H)}<{1\over 2}\sigma$ where ${\bf P}_{\phi_{k}}$ is the operator of orthogonal projection on the one-dimensional subspace ${\rm lin}(\phi_{k})$ of the space $\cal H$.

Therefore $|\rho^{t}_{\varphi,\epsilon}({\bf K})-\rho^{t}_{\varphi,\epsilon}(\sum\limits_{k=1}^{m}c_{k}{\bf P}_{\phi_{k}})|\leq{1\over 2}\sigma$. Since $\rho^{t}_{\varphi,\epsilon}(\sum\limits_{k=1}^{m}c_{k}{\bf P}_{\phi_{k}})=\sum\limits_{k=1}^{m}c_{k}\rho^{t}_{\varphi,\epsilon}({\bf P}_{\phi_{k}})=\sum\limits_{k=1}^{m}c_{k}|(u_{\epsilon}(t),\phi_{k})|^{2}$ then according to the theorem 1 and Rieman oscilation theorem $\lim\limits_{\epsilon\to 0}\rho^{t}_{\varphi,\epsilon}(\sum\limits_{k=1}^{m}c_{k}{\bf P}_{\phi_{k}})=\sum\limits_{k=1}^{m}c_{k}|\lim\limits_{\epsilon\to 0}[(u_{\epsilon}(t),\phi_{k})]|^{2}=\sum\limits_{k=1}^{m}c_{k}|\int\limits_{0}^{+\infty}(\varphi(x+bt)\phi_{k}(x))dx|^{2}$.

If $u^{*}(t,x)=\phi(x+bt),\,(t,x)\in\mathbb{R}_{+}\times\mathbb{R}_{+}$, then $\lim\limits_{\epsilon\to 0}\rho^{t}_{\varphi,\epsilon}(\sum\limits_{k=1}^{m}c_{k}{\bf P}_{\phi_{k}})=(u^{*}(t,\cdot),(\sum\limits_{k=1}^{m}c_{k}{\bf P}_{\phi_{k}})u^{*}(t,\cdot))$.

Since $|(u^{*}(t,\cdot),{\bf K}u^{*}(t,\cdot))-(u^{*}(t,\cdot),(\sum\limits_{k=1}^{m}c_{k}{\bf P}_{\phi_{k}})u^{*}(t,\cdot))|<{1\over 2}\sigma$ then

${\overline{\lim\limits_{\epsilon\to 0}}}|(u^{*}(t,\cdot),{\bf K}u^{*}(t,\cdot))-\rho^{t}_{\varphi,\epsilon}({\bf K})|\leq\sigma$.

Since $\sigma>0$ is arbitrary number then $\lim\limits_{\epsilon\to 0}\rho^{t}_{\varphi,\epsilon}({\bf K})=(u^{*}(t,\cdot),{\bf K}u^{*}(t,\cdot))$ for arbitrary $t\geq 0$ and arbitrary compact operator $\bf K$. The theorem 3 is proved.

Theorem 4. *The family of maps $T_{comp}^{t},\,t\in R$, is not a group. But the restrictions $T_{comp}^{t}(b),\,t\in R_{\pm}$ are both semigroups of maps of the set $\Sigma({\cal A}_{comp})$ into itself: $T^{t}_{comp}T^{\tau}_{comp}=T^{t+\tau}_{comp},\,t,\tau\in\mathbb{R}_{+}.$ *

Proof. The function $T_{comp}^{t},\,t\in\mathbb{R}$, is not a group since, in particular, the equalities $T_{comp}^{-t}T_{comp}^{t}\rho_{\varphi}=\rho_{\varphi}=T_{comp}^{t}T_{comp}^{-t}\rho_{\varphi}$ can’t hold both for any $t>0$. The semigroup properties of the limit one-parameter family of maps $T_{comp}^{t},\,t\in\mathbb{R}_{\pm},$ is the consequence of the semigroup properties of operator-functions $e^{-it{\bf H}^{*}},\,t\geq 0$ with $b<0$, and $e^{-it{\bf H}},\,t\geq 0$ with $b>0$.

Remark 3. The state $\sigma_{\varphi}^{t}=T_{comp}^{t}\rho_{\varphi}$ on the $C^{*}$-algebra ${\cal K}$ can be extended onto the hole $C^{*}$-algebra $B(H)$.

Remark 4. Only one of two semigroup $T_{comp}^{t},\,t\in\mathbb{R}_{\pm},$ is the semigroup of inverse maps.

The convergence of the sequence of regularized state dynamics on the algebra ${\cal A}_{mult}$ and the failure of the semigroup properties

Theorem 5. Let $b>0$ and let ${\cal A}_{mult}$ be the abelian $C^{*}$-algebra of operators ${\bf A}_{f}$ of multiplication on the arbitrary measurable essentially bounded function $f(x)\in L_{\infty}(\mathbb{R}_{+})$ in the space $L_{2}(\mathbb{R}_{+})$. Then for any $\varphi\in L_{2}(\mathbb{R}_{+})$ and for any operator ${\bf A}_{f}\in{\cal A}_{mult}$ the following equality

[TABLE]

holds for any $t\geq 0$.

Theorem 6. The one-parameter family of maps $T^{t}_{mult},\,t\geq 0,$ acting in the set $\Sigma({\cal A}_{mult})$ according to (34), does not satisfy the semigroup property $T^{t}_{mult}T^{\tau}_{mult}=T^{t+\tau}_{mult},\,t,\tau\in\mathbb{R}_{+}.$

The dynamical maps $T^{t}_{mult},\,t\geq 0,$ can be presented by two one-parameter family of maps in the space $H=L_{2}(\mathbb{R}_{+})$:

1) the first one-parameter family of maps ${\bf V}_{t},\,t\geq 0,$ acting in the space $H=L_{2}(\mathbb{R}_{+})$ by the formula

${\bf V}_{t}u(x)=u(x+bt),\,x\geq 0$;

2) the second one-parameter family of maps ${\bf W}_{t},\,t\geq 0,$ acting in the space $H=L_{2}(\mathbb{R}_{+})$ by the formular

${\bf W}_{t}\varphi(x)=\varphi(bt-x),\,x\in[0,bt],$**

${\bf W}_{t}\varphi(x)=0,\,x>bt.$* *

Proof. Let us note that ${\bf V}_{t}^{*}{\bf V}_{t}={\bf P}_{[bt,+\infty)}$ anf ${\bf W}_{t}^{*}={\bf W}_{t},\,forall\,t\geq 0,$ ${\bf W}_{0}=\bf 0$ and ${\bf W}_{t}^{*}{\bf W}_{t}={\bf P}_{[0,bt]}$. Then, in particular

[TABLE]

Theorem 7. Let $b>0$ and let ${\cal A}_{mult}$ be the abelian $C^{*}$-algebra of operators ${\bf A}_{f}$ of multiplication on the measurable essentially bounded function $f(x)\in L_{\infty}(R_{+})$ in the space $L_{2}(R_{+})$. Then the equality

[TABLE]

*holds. Here ${\bf V}_{t}^{*}{\bf V}_{t}+{\bf W}_{t}^{*}{\bf W}_{t}={\bf I}$. *

The equality (34) is the consequence of Lemma 5 and the expression of the functions $\psi_{\epsilon}$. In fact, according to (31) and Remarks 1 and 2 we have

[TABLE]

[TABLE]

where $r_{\epsilon}=\phi_{\epsilon}-v$.

Since $\|r_{e}\|_{H}\to 0$ and $\lim\limits_{\epsilon\to 0}(\psi_{\epsilon},g)=0$ for any vector $g\in H$ then the equality (34) holds. Then for any $\varphi\in L_{2}(R_{+})$ and for any operator ${\bf A}_{f}\in{\cal A}_{mult}$ the following equality

[TABLE]

holds. Hence the theorem 7 and the equality (36) is proved.

The absence of semigroup property $T^{t}T^{\tau}\neq T^{t+\tau},\,t,\tau\in\mathbb{R}_{+}$ is the consequence of the following equalities: if $T^{t}\phi(x)=\phi(bt-x)$ for any $\phi\in H$ with ${\rm supp}\phi\in[0,bt]$ then $T^{\tau}\phi(bt-x)=\phi(b\tau-bt+x)\neq\phi(b(\tau+t)-x)$. The equality (35) for the function ${\bf W}^{t},\,t\geq 0,$ can be easy checked.

Remark 5. The dynamics (37) of the set of states $\Sigma({\cal A}_{mult})$ on the obsevable algebra ${\cal A}_{mult}$ is the realization of Kraus representation of one-parametric family of completely positive mapping of the set $\Sigma({\cal A}_{mult})$, i.e. the representation of the type

[TABLE]

where $\{{\bf V}_{j}(t)\}$ are bounded linear operators satisfying the condition

[TABLE]

Since the operators ${\bf V}_{t}$ and ${\bf W}_{t}$ in the formula (37) satisfy the equalities ${\bf V}^{*}_{t}{\bf V}_{t}={\bf P}_{[bt,+\infty)}={\bf P}_{H_{0}}(t)$ and ${\bf W}_{t}^{*}{\bf W}_{t}={\bf P}_{[0,bt)}={\bf P}_{H_{1}}(t)$, then according to Wold (see [23]) decomposition $H=H_{0}(t)\oplus H_{1}(t)$ for isometric operators $\exp(it{\bf L})$ the pair of operators ${\bf V}_{b}(t),\,{\bf W}(t)$ satisfies the condition (39). Hence the presentation (37) of the limit dynamics on $\Sigma({\cal A}_{mult})$ is the realization of Kraus decomposition (38) with two terms. It should be noted that the first family of operators ${\bf V}_{b}(t),\,t\geq 0,$ is one-parametric $C_{0}$-semigroup but the second family of operators ${\bf W}(t),\,t\geq 0,$ is not satisfy the semigroup property according to the relations (39), (40). Thus we obtain that the limit dynamic $T^{t},\,t\geq 0,$ of the set of states on the algebra ${\cal A}_{mult}$ is the Kraus decomposition of one-parameter family of completely positive mappings of the set $\Sigma({\cal A}_{mult})$ with two terms.

The pure and mixed states on the algebras ${\cal A}_{comp}$

Any element $\bf A$ of the ring ${\cal K}_{comp}$ of compact self-adjoint operators is defined by the ortho normal basis $\{e_{k}\}$ of eigenvectors of compact self-adjoint operator $\bf A$ and the sequence of its eigenvalues $\{a_{k}\}\in c_{0}$, where $c_{0}$ is Banach space of infinitezimal sequences endowing with supremum-norm. Since $c_{0}^{*}=l_{1}$ then the space of linear functionals on the ring ${\cal K}_{comp}$ is the space of trace-class operators $T_{1}({\cal H})$. Any element $\rho\in T_{1}({\cal H})$ is defined by the ortho normal basis of eigenvectors of thace-class self-adjoint operator $\rho$ and the sequence of its eigenvalues $\{p_{k}\}\in l_{1}$.

The functional $\rho$ on the space of compact operators $\sigma(H)$ is called continuous if for any sequence $\{{\bf A}_{n}\}$ of compact operators the relation $\rho({\bf A}_{n})\to 0$ is the consequence of the condition $\|{\bf A}_{n}\|_{B(H)}\to 0$. The functional $\rho$ on the space of compact operators $\sigma(H)$ is called ultraweak continuous if the condition $\rho({\bf A}_{n})\to\rho({\bf A})$ satisfies for any sequence $\{{\bf A}_{n}\}$ of compact operators which is monotone increasing to the operator ${\bf A}\in B(H)$ such that ${\bf A}_{n}x\to{\bf A}x\,\forall\,x\in{\cal H}$.

Let $\Sigma({\cal K}_{comp})$ be the set of nonnegative continuuois functionals $\rho$ on the ring ${\cal K}_{comp}$ such that $\rho({\bf P}_{n})\to 1$ for arbitrary sequence of projectors which monotone nondecreasing to unite operator.

Lemma 6. The set $\Sigma({\cal K}_{comp})$ is the intersection of the unique sphere with the positive cone of the space of trace class operators $T_{1}(H).$

In fact any nonnegative element $\rho\in T_{1}(H)$ defines the nonnegative linear continuous functional on the ring ${\cal K}_{comp}$. Conversely any nonnegative linear continuous functional $\rho$ on the ring ${\cal K}_{comp}$ defines nonnegative bounded quadratic form on the space $\cal H$ such that associated with nonnegative self-adjoint trace class operator ${\bf A}_{\rho}\in T_{1}(H)$. Therefore the set $S({\cal K}_{comp})$ of nonnegative linear continuous functionals on the ring ${\cal K}_{comp}$ coincides with the cone of nonnegative elements in the space $T_{1}(H)$. The normalization condition for the functional $\rho\in S({\cal K}_{comp})$ is equivalent to the belonging of nonnegative trace class operator to the unite sphere of the space $T_{1}({\cal H})$.

Note that any continuous nonnegative functional $\rho\in S({\cal K}_{comp})$ is ultraweak continuous.

Since the set $\Sigma({\cal K}_{comp})$ of the states on the ring $\cal K$ is the convex compact in Banach space $T_{1}({\cal H})$ then the set of extreme points of the set $\Sigma({\cal K}_{comp})$ coincides with the set of one dimentional orthogonal projectors.

The algebra ${\cal A}_{comp}$ is the result of the adjointing of the unite operator $\bf I$ to the ring ${\cal K}_{comp}$. The set $\Sigma({\cal K}_{comp})$ of the states on the ring ${\cal K}_{comp}$ is the part of the set of the states on the algebra ${\cal A}_{comp}$ and is the set of all ulraweak continuous states on the algebra ${\cal A}_{comp}$. The arbitrary state on the algebra ${\cal A}_{comp}$ wich has no ulraweak continuity component is the state $J\in\Sigma({\cal A}_{comp})$ such that $J({\bf A})=0\ \forall\ {\bf A}\in{\cal K}(H)$ and $J({\bf I})=1$.

Remark 6. The state $J$ on the algebra of operators ${\cal A}_{comp}$ is not normal (i.e. is not ultraweak continuous) since for any sequence of finite dimentional projectors $\{{\bf P}_{n}\}$ which monotone increase to the unique operator $\bf I$ the condition ${\bf J}({\bf P}_{n})\to{\bf J}({\bf I})$ for $n\to\infty$ is not satisfied: ${\bf J}({\bf P}_{n})=0\ \forall\ n\in\bf N$ but ${\bf J}({\bf I})=1$ by the definition of the state $J$.

Lemma 7. For any state $\rho\in\Sigma({\cal A}_{comp})$ there is the number $\alpha\in[0,1]$ and the state $r\in\Sigma({\cal K}_{comp})$ such that $\rho=\alpha r+(1-\alpha)J$.

The proof of such statement can be finded in the paper [21].

The destruction of pure states on algebras ${\cal A}_{comp}$ by the limiting dynamics

Theorem 8. Let $b>0$ and $T^{t}_{comp},\,t\geq 0,$, be the limit one-parametric semigroup of the maps of the set of states $\Sigma({\cal A}_{comp})$ which is given by the equality (33) (see theorem 3). Then for any unique vector $\varphi\in H$ there is the value

[TABLE]

such that for any $t>T_{*}$

[TABLE]

where $\alpha(t)=\|e^{-it{\bf H}^{*}}\varphi\|_{H}^{2}>0,$ and $\Phi(t,\varphi)=(\alpha(t))^{-{1\over 2}}e^{-it{\bf H}^{*}}\varphi$.

The statement of the theorem 8 is the consequence of the theorem 3; it means that the limit dynamics $T^{t}_{comp},\,t\geq 0,$ of the set of the states $\Sigma({\cal A}_{comp})$ transformes the pure state $\rho_{\varphi}$ into the convex combination of two different pure states. In this case the number $T_{*}=T_{*}(\varphi)$ is the moment of destruction of the pure state.

For the example under the consideration in the theorem 1 the all objects of the statment of the theorem D has the explicit describtion, namely, $\alpha(t)=\int\limits_{bt}^{+\infty}|\varphi(x)|^{2}dx,$ $1-\alpha(t)=\int\limits_{0}^{bt}|\varphi(x)|^{2}dx,$ $T_{*}=\sup\{t\geq 0:\ \int\limits_{0}^{bt}|\varphi(x)|^{2}dx=0\}$, $e^{-it{\bf H}^{*}}\varphi(x)=\varphi(x+bt),\,x\in\mathbb{R}_{+},\,t\geq 0.$

4 Conclusions

In the paper we studied the simple model of Schrodinger equation with the degenerate Hamiltonian. As the measure of simplicity of degenerated Hamiltonian we can consider such characteristic of deviation of degenerated Hamiltonian from the set of self-adjoint operators as the deficient indexes.

The deficient indexes of the degenerated Hamiltonian $\bf H$ in the considered example (6) is equal to (1,0) or (0,1) for different sign of the coefficient $b$ in the expression (7).

The examples of degenerated Hamiltonian with arbitrary finite deficient indexes $(n_{-},n_{+})$ can be given by the first order differential operator in the Cauchy problems for transport-type equation which is considered in the work by [12].

For example, if the function $b$ in the expression (6) is given as the ${{\pi}\over 2}$-periodic continuation of the function $\cos(x),\,x\in[0,{{\pi}\over 2})$, and if $H=L_{2}([0,{{k\pi}\over 2})$ for some $k\in\bf N$ then the differential operator (6) with the domain $D({\bf H})=\oplus_{j=1}^{k}\dot{W}^{1}_{2}([{{(j-1)\pi}\over 2},{{j\pi}\over 2}))$ is the symmetric operator with the deficient indexes $(k,0)$. In this case the dynamic generated by the operator $\bf H$ is the direct summa of the $k$ independent dynamics in the spaces $H_{j}=L_{2}([{{(j-1)\pi}\over 2},{{j\pi}\over 2})),\ j=1,...,k$, any of each is unitary equivalent to the dynamic generated by the operator (6).

Similar results can be obtained for the Hamiltonian with the degeneration on the exterior of the ball $B_{R}$ (see [8], example 6.3).

Any isometric semigroup $\bf U$ admits the Wold decomposition (see [23]) $H=H_{0}\oplus H_{1}$ such that the subspaces $H_{0},H_{1}$ are reduce the operators ${\bf U}(t)$ of the semigroup $\bf U$, the restriction ${\bf U}|_{H_{0}}$ is the unitary semigroup and the restriction ${\bf U}|_{H_{1}}$ is one-sided shift. The unitary component of dynamics is well-known object and the shift component is isomorphic to the dynamics which is presented by the isometric semigroup $e^{-it{\bf H}^{*}}$ in the example in the part 2 with the operator $\bf H$ (see (6)) with $b>0$. In this sence the example (6) describes the characteristic properties of dynamics generated by the arbitrary degenered symmetric Hamiltonian with nontrivial indexes of deficience.

Thus if the degenerate Hamiltonian $\bf H$ is maximal symmetric operator then the dynamics generated by Schrodinger equation with this Hamiltonian is isomorphic to one of the two dynamics:

– isometric but not unitary (in the case $n_{+}=0$) as in the example with the condition $b\leq 0$;

– dissipative and destroying the pure states (in the case $n_{+}>0$) as in the example with $b>0$.

5 Acknowledgments

This work is supported by the Russian Science Foundation under grant 14-11-00687 and performed in the Steklov Mathematical Institute of Russian Academy of Sciences, Moscow, Russia.