Kyungpook Mathematical Journal 2021; 61(2): 249-255
Published online June 30, 2021
Copyright © Kyungpook Mathematical Journal.
On the Spectrum Discreteness for the Magnetic Schrödinger Operator on Quantum Graphs
Igor Y. Popov* and Anna G. Belolipetskaia
Received: December 24, 2019; Revised: July 30, 2020; Accepted: November 23, 2020
The aim of this work is to study the discreteness of the spectrum of the Schröodinger operator on infinite quantum graphs in a magnetic field. The problem was solved on a set of quantum graphs of a special kind.
Keywords: quantum graph, spectrum, magnetic field
For the operator describing a physical system, it is an ongoing problem to describe which properties the system characterise when the spectrum of the operator is discrete. This problem has been solved in various special cases. For example, Molchanov proposed in  a criterion for a potential to provide the discreteness of the Hamiltonian spectrum in the 1-dimensional case. Necessary and sufficient conditions for a self-adjoint operator on a line related to a general second-order expression to have discrete spectrum are presented in the article . The discreteness of the spectrum of the non-magnetic Schrödinger operator has been studied, for example, in [1, 2, 11, 16]. In the case of a magnetic field, one works in the space of complex functions, which complicates the task. Studies of the magnetic Schr\"odinger operator were carried out in [3, 6, 7, 10, 14], but no rigorous criteria have been proved for the discreteness of the spectrum of the Schrödinger operator on quantum graphs in a magnetic field. The mathematical modeling of the physical system in this article is based on the theory of quantum graphs. A rigorous proof of the correctness of their use was offered in , and the mathematical theory of quantum graphs was treated in [4, 9].
In this article, we confine our attention to some specific quantum graphs only. The class of these graphs is described below.
A quantum graph belongs to the class
(1) any two vertices are connected by no more than a finite set of edges,
(2) the length of the edges of the graph is bounded below by a positive constant,
(3) for any fixed vertex
vand for any marked edges (the sum of the lengths of all marked edges is equal to infinity) there is a path psatisfying the following properties:
pstarts at vertex v;
pis isomorphic to the half-line;
pcontains marked edges (not necessarily all), the sum of their lengths is equal to
Unfortunately, this definition is not illustrative. Two examples of quantum graphs belonging to the class
The domain of the Schrödinger operator on a curve in
The operator acts on each edge of the quantum graph as follows (in dimensionless units):
Note that the variable
We deal with the spectral problem
The main result of this article is the following theorem.
Consider a quantum graph that belongs to the class
For any fixed λ there exist values for the boundary conditions
Note that from the assumption of the theorem it is known that function
where the function
We make the following conversion:
where the path between the roots
We carry out the following transformation:
After some transformations, the following inequalities can be obtained:
Thus, the following inequality holds:
The right-hand side of equation (3.6) is real, which means that after integration it will also be real. Therefore, the left side of equation (3.6) before and after integration is also real. Therefore, the expression (3.12) is a real function. Thus, the inequality (3.13) in terms of the source variables will look like this:
Note that due to
To summarize, we considered quantum graphs of a certain topological structure (see Definition) with the Schrödinger operator corresponding to the scalar potential of a special form (3.2). A theorem was formulated and proved for the quantum graphs which states that for any fixed eigenvalue, there is a set of constants characterizing the boundary conditions such that the eigenfunction has finitely many zeros. This theorem is not yet the criterion for the discreteness of the spectrum of the Schrödinger operator on a quantum graph in a magnetic field, but allows it to be studied. The obtained result can be useful in physical applications related to the transport properties of nanosystems.
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