ABSTRACT

We consider the Schrödinger operators on graphs with a finite or countable number of edges and Schrödinger operators on branched manifolds of variable dimension. In particular, a description of self-adjoint extensions of symmetric Schrödinger operator, initially defined on a smooth function, whose support does not contain the branch points of the graph and branch points of the manifold. These results are obtained for graphs with a single vertex, graphs with multiple vertices and graphs with a single vertex and countable set of rays.

Keywords:The Schrödinger Equation; Schrödinger Operators on Graphs and Branched Manifolds; Self-Adjoint Extensions

1. Introduction

Differential operators on graphs and other branched manifolds have applications to the description of a number of processes in quantum mechanics and biology. Fundamentals of the theory of differential equations on graphs presented in the monograph [1], in which a number of examples of physical problems leading to the study of differential operators on graphs. In the articles [2-4] spectral properties of such operators are investigated by the dynamic properties of the evolution determined by the Schrödinger equation on the graph. In the articles [5-8] we study the set of self-adjoint extensions Schrödinger operator defined initially in the space of compactly supported smooth functions, whose support do not contain the branch points of the graph ([6-8]) or points of changing the operator type [5]. Feynman approximation formulas for the unitary semigroups defined by some of the self-adjoint extensions are founded in the article [7]. This article contains the consideration of the Laplace operators on graphs with a finite or countable number of edges. This article is a continuation of studies [7] in which we studied the graph with a finite set of edges are considered.

The relevant problem under consideration consists of recently considerable interest in the description of particle dynamics on graphs, branched dendrites and other manifolds from mathematical physics and quantum mechanics. Mathematically, the operation of differentiation function is uniquely defined for functions on region or on a smooth manifold, which needs to be clarified for the functions defined on manifolds, containing the branch point. The purpose of this study is to determine the action of the Schrödinger operator on functions defined on a manifold with a finite set of branch points. For this purpose, we define the Schrödinger operator in the space finite and infinitely differentiable functions whose support does not contain the branching points. Schrödinger operator 𝐋 on a graph is called a self-adjoint extension of the operator. In this article we describe the set of all operators of Schrödinger operators on a graph in terms of conditions on the set of limit values at the branch point functions in the domain of 𝐋 and its derivative. In this article we obtained the results with a single vertex (they represent a union n of semidirect with a common vertex), graphs with multiple vertices and graphs with a single vertex and a countable set of rays and the set of all operators of Schrödinger on branched manifold in terms of conditions on the set of limit values on a manifold of branching functions in the domain of the operator 𝐋.

In this article we found general description of a set of self-adjoint extensions, of the operator, as on graphs and branched manifolds.

2. Formulation of Problem and Notation

We study the Schrödinger operator on the graph Γ, defining the processes of diffusion and quantum dynamics on a graph both on branched manifold. Following [1] terminology graph Γ is called finite or countable collection of smooth one-dimensional manifolds (called edges of the graph), each of which is diffeomorphic to the ray or interval [0,1]. The boundary points of the edges will be called vertices of the graph. Each vertex of a graph is a boundary point of a non-empty set of edges of a graph.

Assumed that on Γ given Borel measure, we determine the requirement that its restriction to each edge coincides with the standard Lebesgue measure, then.

Let -vector space of infinitely differentiable complex-valued functions on Γ with compact support not containing the vertices, and operator is linear operator defined on a linear space  by the equation

(2.1)

in which the functions, , -real-valued, bounded and continuous everywhere except at the vertex function on Γ, function takes on each edge a constant value and for all .

We say that Γ is branched manifolds, if Γ defined as the union of instances of regions

we assume that for each 𝛼 region is -dimensional bounded domain in the space

with dimensional smooth boundary. The boundary of the manifold Γ is defined as the union of instances of boundaries regions where

Point is called a branch point of the manifold Γ, if it is a boundary point of at least two different regions where

Assumed that on Γ given Borel measure, we determine the requirement that its restriction to each regions coincides with the standard Lebesgue measure space Then the space of square-integrable in the Lebesgue measure on the set of complex-valued functions Γ admits the representation

Let -vector space of infinitely differentiable complex-valued functions on Γ with compact support not containing branch points of the manifold, and -linear operator defined on by relation where

(2.2)

in which the functions, , -real-valued, bounded and continuous everywhere except at the branching points of Γ, function takes on each region a constant value and for all . Here -restricting a function on the region

Definition: The linear self-adjoint operator 𝐋 in the space is called Hamiltonian of quantum system with mass in the electromagnetic filed if 𝐋 is self-adjoint extension of the operator

We investigate the properties of the Cauchy problem for the Schrödinger equation

(2.3)

with the initial condition

(2.4)

Here 𝐋-symmetric operator in a Hilbert space is an extension operator of, given on linear manifold by the equation (2.1) or (2.2). The purpose of the article is to describe the set of all self-adjoint extensions of the operator, that may act as generators of the unitary group of the Cauchy problem (2.3), (2.4) for the Schrödinger equation.

3. Graph with One Vertex

Graph Γ with one vertex, is defined as the union of instances of semidirect with a common origin, called the vertex of the graph. Assumed that on Γ given Borel measure defined by the requirement that its restriction to each semidirect coincides with the standard Lebesgue measure, then. Let -vector space of infinitely differentiable complex-valued functions on Γ with compact support not containing the point and -linear operator, defined on by the relation

Here -restriction of a function on semidirect.

Assumed that for all number and the function and we denote in the point

Operator with domain of definition  is densely defined and symmetric. The domain adjoint operator is a linear subspace The restriction of any function on semidirect possess boundary values at the vertex, which we denote by where the symbol means This is also true for the first derivatives of these restrictions, which use similar notation.

Von Neumann theorem ([9,10]) provides a description of a set of self-adjoint extensions of symmetric operators. We obtain an explicit description of a set of self-adjoint extensions of the operator in terms of conditions on a linear subspace in the space of boundary values

Theorem 1. Let, and The operator 𝐋 with domain

self-adjoint if and only if the matrix satisfies the equality

Proof. If and then we have the equality

Hence

Traces take arbitrary values, therefore the equality is necessary and sufficient for inclusion which proves Theorem 1.

Corollary 1. If and -diagonal matrices and the matrix elements are defined by the formula

respectively, and where then the operator 𝐋

with domain self-adjoint if and only if the matrices and satisfy the equality

Proof. If and then we have the equality

Hence

Traces take arbitrary values, therefore the equality is necessary and sufficient for inclusion which proves the corollary 1.

Theorem 1 gives a description of a wide class of self-adjoint extensions of the operator but does not describe the totality of self-adjoint extensions. This makes the next theorem.

Theorem 2. The operator 𝐋 is self-adjoint if and only if its domain of definition consists of the functions in the space boundary values satisfy the equality where rank of the matrix equals and the matrix is is self-adjoint:

Proof. Let, and We denote by set of solutions of linear equations

(3.1)

where is the fundamental matrix and is a matrix of independent constants. Substituting each of the solutions of the fundamental equation specifying the domain, we obtain by following the relation of the fundamental matrix, with the matrix of the system of equation (3.1)

(3.2)

If and domain of the operator 𝐋 defined by a system of equation (3.1), then for any rightly the equality

Element satisfies condition

(3.3)

Let -basis in the linear then each column of matrix satisfies

(3.3), and therefore

(3.4)

Of (3.2) and (3.4), it follows that the matrix can be selected

The operator 𝐋 is self-adjoint if and only if so if -matrix of the columns of the basis vectors in the subspace then if and only if is also the matrix of the columns of the basis vectors in the subspace that is, any of its column satisfies the system of equaation (3.1). And this is equivalent to the system of equations which proves Theorem 2.

Theorem 3. The operator 𝐋 is self-adjoint if and only if its domain of definition consists of the functions in the space boundary values satisfy the equality where rank of the matrix equals and the matrix is is self-adjoint:

Proof. Let, and We denote by set of solutions of linear equations

(3.5)

where is the fundamental matrix and is a matrix of independent constants. Substituting each of the solutions of the fundamental equation specifying the domain, we obtain by following the relation of the fundamental matrix, with the matrix of the system of equation (3.5)

(3.6)

If and domain of the operator 𝐋 defined by a system of equation (3.5), then for any rightly the equality

Element satisfies condition

(3.7)

Let -basis in the linear then each column of matrix satisfies

(3.7), and therefore

(3.8)

Of (3.6) and (3.8), it follows that the matrix can be selected

The operator 𝐋 is self-adjoint if and only if so if -matrix of the columns of the basis vectors in the subspace then if and only if is also the matrix of the columns of the basis vectors in the subspace that is, any of its column satisfies the system of equation (3.5). And this is equivalent to the system of equations

which proves Theorem 3.

4. Graph with Multiple Vertices

In the present article, a graph with multiple vertices is understood by one-dimensional cellular of complex [3]. Let graph Γ, be a collection of vertices from each of which proceeds edges representing the infinity semidirect or line segments that connect vertex with other vertices. We fix on each edges parametrization of the natural parameters. In this case, the edges of semidirect parameter increases from the boundary points and the edges of intervals, the orientation is chosen arbitrarily. Let -initial point of the edges semidirect, -initial point of edges interval, -end point of edges interval. Let -the collection of all boundary points of the edges We define the function on the set so that if -beginning of edges and if -end of the edges, denoted by diagonal matrix with numbers on the diagonal.

We introduce the operators and the space of boundary values of functions from and their derivatives, linearly isomorphic to space Through we denote the collection limit function values on edges of the boundary, which is the point and by denote by -dimensional vector of for the vector limit values of the derivative use similar notation, and let denoted in

Theorem 4. Let, and The operator 𝐋 with domain

adjoint if and only if the matrix satisfies the equality

Proof. If and then we have the equality

Hence

Traces take arbitrary values, therefore the equality is necessary and sufficient for inclusion which proves Theorem 4.

Corollary 2. If and -diagonal matrices and the matrix elements are defined by the formula

respectively, and where then the operator

𝐋 with domain self-adjoint if and only if the matrices and satisfy the equality

Proof. If and then we have the equality

Hence

Traces take arbitrary values, therefore the equality is necessary and sufficient for inclusion which proves the corollary 2.

5. Graph with One Vertex and with a Countable Set of Rays

description of this graph is defined by the following structures [11]. In this case we denote by -locally finite non-negative countably additive measure on N such that denoted by—Hilbert space of boundary values with the norm

The restriction of any function on semidirect possesses the boundary values at the vertex:

This is also true for the first derivatives of these restrictions We denote by and diagonal matrices and their matrix elements are given by the formula

and respectively.

Theorem 5. Let , and The operator with domain

self-adjoint if and only if the operator is self-adjoint in the space

Proof. If and then we have the equality

Hence

Traces take arbitrary values in the space therefore the equality

is necessary and sufficient for inclusion which proves Theorem 5.

Corollary 3. If and and are operators in the space given by diagonal matrices with elements on the diagonal, respectively, where

Then the operator with domain self-adjoint if and only if the operators and acting in the space satisfy the equality

(5.1)

Proof. If and then we have the equality

Hence

Traces take arbitrary values in the space therefore the equality

is necessary and sufficient for inclusion as if and only if then for the self-adjoint operator is necessary and sufficient to satisfy the equality (5.1).

6. Schrödinger Operators on Branched Manifolds

The assumption. Let the function m takes the constant values on each region for all

, and satisfy condition Through we denote the limiting values of the vector function on the boundary

The operator with domain densely defined and symmetric. The domain

adjoint operator is a linear subspace

Let the components manifold constitute a semidirect, finite intervals and regions. In the case of one-dimensional region boundary value is a set of complex numbers on the boundary represents one or two points. In the case boundary value is an element of the space According to the trace theorem ([12]). Through denote the collection of limit values function on the boundary Similarly, the limit value of the derivative constriction in the direction of the outer normal to boundary in the case semidirect represents a an element of space, in case of a limited interval-element of space and in the case of dimension -element of space

The boundary values of the normal derivative is denoted by where is vector of the external relative to normal to the

We introduce the Hilbert space

We define space of boundary values where and similarly,

Boundary value function is an element of the space and the boundary value its normal derivative-an element of the space

We introduce in the space operators and Operator acts on each element as an operator of multiplication by a function

And operator acts on each element as an operator of multiplication by a function

Theorem 6. Let performed assumption about functions and for any Let -linear operator in space with a dense domain the values of which belongs in linear manifold Let -linear manifold of functions boundary values are related to the boundary values of the derivatives in the direction of the outward normal by relation

Then the self-adjoint operator is necessary and sufficient to satisfy the equality

Proof. Since for any then from conditions and then we have the equality

Hence

Traces take arbitrary values, therefore the equality is necessary and sufficient for inclusion Since the domain of definition operator is determined by the equation

that  then implies that

Corollary 4. Let performed assumption about functions Let -linear operator in space with a dense domain the values of which belongs in linear manifold Let -linear manifold of functions boundary values are related to the boundary values of the derivatives in the direction of the outward normal by relation

Then the self-adjoint operator is necessary and sufficient to satisfy the equality

Proof. Since performed assumption then from conditions and then we have the equality

Hence

Traces take arbitrary values, therefore the equality is necessary and sufficient for inclusion Since the domain of definition operator is determined by the equation

that then implies that

7. Conclusion

In this paper we describe the set of all Schrödinger operators on graph and branched manifold, defined as a self-adjoint extension of the operator, originally defined on smooth functions with supports, not contained in the branch points manifold. Thus, given a description of the various options, we determine the Laplace operator on the space of the functions defined on a branched manifold. Description of the definition of each of the self-adjoint extensions is given in terms of linear relations satisfied by the limit at the branch points and the boundary points of the graph function value in the domain of operator and the its derivative. Each of the Laplace operators corresponds to the Markov process, whose behavior in a neighborhood of branch points, we determined by the choice of the domain of the Laplace operator, obtained in this paper results, which is an extension of the study work [8] describes the self-adjoint extensions of a graph with a single vertex and two edges, to the case of a graph with an arbitrary number of edges. In addition, this paper summarizes the results of [6] in the case of Laplace operators, for which the linear relation in the space of boundary values that define the domain of the operator, do not admit the possibility of expressing the limit function values at the boundary points and branch points of the graph of the limiting values of its derivative.

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