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We consider the Schrodinger operators on graphs with a finite or countable number of edges and Schr?dinger operators on branched manifolds of variable dimension. In particular, a description of self-adjoint extensions of symmetric Schr?dinger 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.

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 [

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 this article we found general description of a set of self-adjoint extensions, of the operator

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 [

Assumed that on Γ given Borel measure, we determine the requirement that its restriction to each edge

Let

in which the functions

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

with

Point

Assumed that on Γ given Borel measure, we determine the requirement that its restriction to each regions

Let

in which the functions

Definition: The linear self-adjoint operator 𝐋 in the space

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

with the initial condition

Here 𝐋-symmetric operator in a Hilbert space

Graph Γ with one vertex, is defined as the union of

Here

Assumed that for all

Operator

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

Theorem 1. Let

self-adjoint if and only if the matrix

Proof. If

Hence

Traces

Corollary 1. If

with domain

Proof. If

Hence

Traces

Theorem 1 gives a description of a wide class of self-adjoint extensions of the operator

Theorem 2. The operator 𝐋 is self-adjoint if and only if its domain of definition

Proof. Let

where

If

Element

Let

(3.3), and therefore

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

The operator 𝐋 is self-adjoint if and only if

Theorem 3. The operator 𝐋 is self-adjoint if and only if its domain of definition

Proof. Let

where

If

Element

Let

(3.7), and therefore

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

The operator 𝐋 is self-adjoint if and only if

which proves Theorem 3.

In the present article, a graph with multiple vertices is understood by one-dimensional cellular of complex [

We introduce the operators

Theorem 4. Let

adjoint if and only if the matrix

Proof. If

Hence

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Corollary 2. If

𝐋 with domain

Proof. If

Hence

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description of this graph is defined by the following structures [

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

Theorem 5. Let

Proof. If

Hence

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Corollary 3. If

Proof. If

Hence

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The assumption

The operator

Let the components

The boundary values of the normal derivative is denoted by

We introduce the Hilbert space

We define space of boundary values

Boundary value

We introduce in the space

And operator

Theorem 6. Let performed assumption

Then the self-adjoint operator

Proof. Since

Hence

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that

Corollary 4. Let performed assumption

Then the self-adjoint operator

Proof. Since performed assumption

Hence

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that

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 [