Schrödinger Operators on Graphs and Branched Manifolds

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.


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][3][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][6][7][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][7][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 0 L in the space 0,0 C ∞ finite and infinitely differentiable functions whose support does not contain the branching points.Schrödinger operator  on a graph is called a self-adjoint extension of the operator 0 L .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  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 .
In this article we found general description of a set of self-adjoint extensions, of the operator 0 L , as on graphs and branched manifolds.

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 Γ i (called edges of the graph), each of which is diffeomorphic to the ray . 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 j Γ coincides with the standard Lebesgue measure, then ( ) ( ) -vector space of infinitely differentiable complex-valued functions on Γ with compact support not containing the vertices, and operator 0 0 j = ⊕ L L is linear operator defined on a linear space ( ) ( ) in which the functions m , B , C -real-valued, bounded and continuous everywhere except at the vertex func- tion on Γ, function m takes on each edge Γ j a constant value m , j and 0 m m 0 j ≥ > for all 1, , ; . We say that Γ is branched manifolds, if Γ defined as the union of n instances of regions Γ , with ( ) dimensional smooth boundary α η .The boundary of the manifold Γ is defined as the union of n instances of boundaries regions ( ) ≠ Assumed that on Γ given Borel measure, we determine the requirement that its restriction to each regions Γ α coincides with the standard Lebesgue measure space .
d R α Then the space of square-integrable in the Lebesgue measure on the set of complex-valued functions Γ admits the representation ( ) ( ) -vector space of infinitely differentiable complex-valued functions on Γ with compact support not containing branch points of the manifold, and 0 0 α = ⊕ L L -linear operator defined on in which the functions m , B  , C -real-valued, bounded and continuous everywhere except at the branching points of Γ, function m takes on each region Γ α a constant value m , α and 0 m m 0, α ≥ > for all 1, , n;

Graph with One Vertex
Graph Γ with one vertex, is defined as the union of n instances of semidirect with a common origin Q , called the vertex of the graph.Assumed that on Γ given Borel measure defined by the re- quirement that its restriction to each semidirect Γ j coincides with the standard Lebesgue measure, then ( ) ( ) -vector space of infinitely differentiable complex-valued functions on Γ with compact support not containing the point , Q and 0 Assumed that for all j number m 0 j > and the function is densely defined and symmetric.The domain ( ) The restriction of any function ( ) possess boundary values at the vertex, which we denote by ( ) j 0 , u where the symbol ( )  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 0 L in terms of conditions on a linear subspace in the space of boundary values = and ( )

self-adjoint if and only if the matrix A satisfies the equality
take arbitrary values, therefore the equality ( ) ( ) is necessary and sufficient for inclusion which proves Theorem 1. Corollary 1.If M and  -diagonal matrices and the matrix elements are defined by the formula 1 , m respectively, and self-adjoint if and only if the matrices , A M and  sa- tisfy the equality Traces ( ) 0 u take arbitrary values, therefore the equality ( ) ( ) ( ) where 2n n × Φ is the fundamental matrix and 1 n h × 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) and domain of the operator  defined by a system of Equation (3.1), then for any then each column of matrix V satisfies (3.3), and therefore Of (3.2) and (3.4), it follows that the matrix V can be selected The operator  is self-adjoint if and only if ( ) ( ) A A is self-adjoint: ( ) ≠ and ( ) then each column of matrix V satisfies (3.7), and therefore Of (3.6) and (3.8), it follows that the matrix V can be selected The operator  is self-adjoint if and only if ( ) ( ) L 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.

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 n vertices 1 , , , representing the infinity semidirect or line segments that connect vertex j Q with other vertices.We fix on each edges Γ j 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 d j -initial point of the edges semidirect, a j -initial point of edges Γ j interval, b j -end point of edges Γ j interval.Let c , 1, , N k k =  -the collection of all boundary points of the edges 1 Γ , ,Γ ., respectively, and then the operator self-adjoint if and only if the matrices , A M and  satisfy the equality then we have the equality take arbitrary values, therefore the equality ( ) ( ) ( ) which proves the corollary 2.

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 µ -locally finite non-negative countably additive measure on N such that ( ) 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 ( )

. u′
We denote by Λ, M and  diagonal matrices and their matrix elements are given by the formula ( ) and ( ) = and ( ) take arbitrary values in the space is necessary and sufficient for inclusion we denote the limiting values of the vector function B α  on the boundary .
According to the trace theorem ( ) ( ) 12]).Through u η de- note the collection of ( ) η represents a an element of space  , in case of a limited interval-element of space 2 ,  and in the case of dimension 2 d α ≥ -element of space ( ) The boundary values of the normal derivative is denoted by  where α n is vector of the external relative to Γ α normal to the , We introduce the Hilbert space ( ) ( ) ( ) .
We define space of boundary values ( ) and similarly, ( ) is an element of the space 3 2 , h and the boundary value u n η ∂ ∂ its normal derivative-an element of the space 1 2 .h We introduce in the space h operators M and .F Operator M acts on each element v h ∈ as an op- erator of multiplication by a function

And operator F acts on each element v h
∈ as an operator of multiplication by a function  ( ) Traces u η take arbitrary values, therefore the equality

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

2 .
necessary and sufficient for inclusion ( ) * D , v ∈ L which proves the corollary 1. Theorem 1 gives a description of a wide class of self-adjoint extensions of the operator 0 , L but does not de-scribe the totality of self-adjoint extensions.This makes the next theorem.Theorem The operator  is self-adjoint if and only if its domain of definition ( ) D L consists of the functions in the space

LTheorem 3 .
the columns of the basis vec- tors in the subspace ( ) ( ) if and only if V is also the matrix of the columns of the basis vectors in the subspace ( ) ( ) any of its column satisfies the system of Equaation(3.1).And this is equivalent to the system of equations ( ) The operator  is self-adjoint if and only if its domain of definition

. 5 )
where 2n n × Φ is the fundamental matrix and 1 n h × is a matrix of independent constants.Substituting each of the solutions of the fundamental equation the domain, we obtain by following the relation of the fundamental matrix, with the matrix of the system of Equation (3and domain of the operator  defined by a system of Equation (3.5), then for any

L
the columns of the basis vec- tors in the subspace ( ) ( ) if and only if V is also the matrix of the columns of the basis vectors in the subspace ( ) ( )

=L 4 .
if c k -beginning of edges and ( ) if c k -end of the edges, denoted by S diagonal matrix with numbers ( ) and the space G of boundary values of functions from ( ) denote the collection limit function values on edges of the boundary, which is the point , Let m 1 = , ( )B x 0= and ( )C x 0. =The operator  with domain adjoint if and only if the matrix A satisfies the equality * .then we have the equality

Corollary 2 . 2 1
If M and  -diagonal matrices and the matrix elements are defined by the formula 2 if and only if the operators ,A M and  acting in the space 2 , l satisfy the equality Let the components Γ α manifold Γ constitute a m semidirect, k finite intervals and ( )N m k − + regions.In the case of one-dimensional region Γ α boundary value u u α in the direction of the outer normal n α to boundary α η in the case semidirect α

6 .
Let  performed assumption H about functions m, B,  Let A -linear operator in space h with a dense domain3 2 , related to the boundary values of the derivatives in the direction of the outward normal by relation

4 .
Let  performed assumption H about functions m, B,  C. Let A -linear operator in space h with a dense domain3 2 ,hthe values of which belongs in linear manifold1 2 .related to the boundary values of the derivatives in the direction of the outward normal by relation Since the domain of definition operator A L is determined by the equation A , restricting a function u on the region Γ .