Article
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
Department of Mathematics, ITMO University, Kronverkskiy, 49, Saint Petersburg, 197101, Russia
e-mail : popov1955@gmail.com and annabel1502@mail.ru
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
1. Introduction
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 [12] 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 [13]. 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 [15], and the mathematical theory of quantum graphs was treated in [4, 9].
2. Preliminary
In this article, we confine our attention to some specific quantum graphs only. The class of these graphs is described below.
Definition 2.1.
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
v and for any marked edges (the sum of the lengths of all marked edges is equal to infinity) there is a pathp satisfying the following properties:
-
(i)
p starts at vertexv ;
-
(ii)
p is isomorphic to the half-line;
-
(iii)
p contains marked edges (not necessarily all), the sum of their lengths is equal to
infinity.
-
Unfortunately, this definition is not illustrative. Two examples of quantum graphs belonging to the class
Definition 2.2.
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):
where
Note that the variable
We deal with the spectral problem
The main result of this article is the following theorem.
Theorem 3.1.
Consider a quantum graph that belongs to the class
for any
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
where
We make the following conversion:
where
where the path between the roots
We carry out the following transformation:
where
where
After some transformations, the following inequalities can be obtained:
Thus, the following inequality holds:
Let us simplify the expression on the right hand side of the inequality (3.11), returning to the original variables (see (3.3)):
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.
- S.Akduman and A. Pankov, Schrödinger operators with locally integrable potentials on infinite metric graphs, Appl. Anal., 96(12)(2017), 2149-2161.
- D. Barseghyan, et al., Spectral analysis of a class of Schrödinger operators exhibiting a parameter-dependent spectral transition, J. Phys. A, 49(16)(2016), 165302, 19 pp.
- M. Bellassoued, Stable determination of coefficients in the dynamical Schrödinger equation in a magnetic field, Inverse Problems, 33(5)(2017), 055009, 36 pp.
- G. Berkolaiko (ed.), Quantum graphs and their applications, Proceedings of an AMS-IMS-SIAM Joint Summer Research Conference on Quantum Graphs and Their Appli-cations, June 19-23, 2005, Snowbird, UT, Contemporary Mathematics 415, American Mathematical Soc., 2006.
- G. Berkolaiko and P. Kuchment, Introduction to quantum graphs, American Mathe-matical Society, Providence, 2013.
- M. Bonnefont, et al., Magnetic sparseness and Schrödinger operators on graphs, An-nales Henri Poincare, 21(2020), 1489-1516.
- A. Chatterjee, I. Y. Popov and M. O. Smolkina, Persistent current in a chain of two Holstein-Hubbard rings in the presence of Rashba spin-orbit interaction, Nanosystems: Physics, Chemistry, Mathematics, 10(2019), 50-62.
- D. A. Eremin, E. N. Grishanov, D. S. Nikiforov and I. Y. Popov, Wave dynamics on time-depending graph with Aharonov-Bohm ring, Nanosystems: Physics, Chemistry, Mathematics, 9(2018), 457-463.
- P. Exner, et al. (ed.) Analysis on graphs and its applications, Isaac Newton Institute for Mathematical Sciences, Cambridge, UK, January 8-June 29, 2007, Proceedings of Symposia in Pure Mathematics 77, American Mathematical Soc., 2008.
- E. Korotyaev and N. Saburova, Magnetic Schrödinger operators on periodic discrete graphs, J. Funct. Anal., 272(4)(2017), 1625-1660.
- M. O. Kovaleva and I. Y. Popov, On Molchanov’s condition for the spectrum discrete-ness of a quantum graph Hamiltonian with δ-coupling, Rep. Math. Phys., 76(2)(2015), 171-178.
- A. M. Molchanov, On conditions of spectrum discreteness for self-adjoint differential operators of second order, Proc. Moscow Math. Soc., 2(1953), 169-199.
- R. Oinarov and M. Otelbaev, A criterion for a general Sturm-Liouville operator to have a discrete spectrum, and a related imbedding theorem, Differential Equations, 24(4)(1988), 402-408.
- N. Raymond, Bound states of the magnetic Schrödinger operator, EMS Tracts in Mathematics 27, European Mathematical Society, Berlin, 2017.
- K. Ruedenberg and C. W. Scherr, Free electron network model for conjugated systems. I. theory, J. Chem. Phys., 21(9)(1953), 1565-1581.
- J. Zhao, G. Shi and J. Yan, Discreteness of spectrum for Schrödinger operators with δ-type conditions on infinite regular trees, Proc. Roy. Soc. Edinburgh Sect. A, 147(5)(2017), 1091-1117.