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.