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Chelyabinsk Physical and Mathematical Journal. 2024■ Vol. 9, iss. 4. P. 682-688.
INFLUENCE OF QUANTUM THE ASYMPTOTICS OF THE
A. G. Belolipetskaia", I. Y. Popovb
ITMO University, Saint Petersburg, Russia "[email protected], [email protected]
DOI: 10.47475/2500-0101-2024-9-4-682-688
GRAPH PARAMETERS ON NUMBER OF RESONANCES
We study quantum graphs with the Dirac operator on edges and the Kirchhoff coupling condition at the vertices. The number of resonances is determined numerically. It is revealed how the parameters of the quantum graph (particularly, the topological structure and the volume of the quantum graph as well as the parameters of the Dirac operator) affect the number of resonances.
Keywords: quantum graph, Dirac operator, asymptotics, resonance.
Introduction
Quantum graph models began to be used as early as in 1930s, but it is only in the last few decades that quantum graphs have gained increased interest. Currently, there is a huge amount of literature on quantum graphs; a detailed description of the history of the development of quantum graph theory and an extensive bibliography can be found in works [1; 2].
The study of resonances is not only an interesting theoretical problem, but also a theory that is actively used in various branches of physics. There are several approaches to studying resonances. The main ones are the Lax-Phillips scattering theory [3], complex scaling [4-7], functional model [8] and asymptotic methods [9-13]. Depending on the physical problem being solved, the Schrodinger operator or the Dirac operator can be considered. For example, E.B. Davies and A. Pushnitski in the article [14] obtained results for a quantum graph on the edges of which the Schrodinger operator acts, and Kirchhoff's connection conditions are used as coupling conditions at the vertices. A similar problem with coupling conditions at vertices of a general form was also studied; the results can be seen, for example, in works [15; 16]. In addition to the above, results were also obtained when a magnetic field [17] was added to the system. If the Dirac operator acts on the edges, then there are few results [18]. The completeness problem for the resonance states for quantum graphs was studied in [19-23]
In this article, we study the high-energy asymptotics of the resonances of the Dirac operator on a quantum graph. Using examples of specific models, we demonstrated how the type of asymptotics is influenced by the parameters of the operator and the parameters of the quantum graph, such as the topological structure and volume of the quantum graph.
When studying resonances on a quantum graph, we consider graphs of the following topological structure: a compact internal part, and a finite number of edges of infinite
The work was supported by Russian Science Foundation (grant number 23-21-00096 (https://rscf.ru/en/project/23-21-00096/)).
length attached. On one edge the Dirac operator D acts as follows:
ai
Dfk -I ™ dx
— ^ ai
ox
where i is the imaginary unit, a is the operator parameter which can be different for each edge of the quantum graph.
In this article, we consider the Kirchhoff coupling conditions. In this regard, the operator D on the curve in dimensionless units has the following domain:
domD - jf - ^ , g ^(v) - 0, p(v) - 0, v E dG
where ^ E W1(G \ V(G)) n ACloc(G), W}(G \ V(G)) is the Sobolev space, ACioc(G) is the space of absolutely continuous functions, V(G) is the the set of all vertices of the graph G, E G \ V(G) is an edge of G.
For further research, let us write down a system of differential equations characterizing the posed problem Dfk - Afk in more details:
^k(x) + ai • ^k(x) - A^k(x), (D
-^k(x) + ai • ^k(x) - A^k(x). ()
Let's consider the situation when A - a. Let's express the function (x) from the second equation of the system (1) and substitute it into the first equation of system (1). After simplification, we obtain the following equation: — <^k(x) — (A — ai)2 • (x) - 0.
The coupling conditions at the vertex after such transformations have the following form:
E
1 Vk(x) - 0.
A ai
Let us introduce the following notation for the function describing the number of resonances whose modulus does not exceed R > 0: N(R) - # {k : k E Ares, |k| ^ R} , where Ares is the set of all resonances.
The main result, concerning to the asymptotics of the number of resonances being the roots of equation F(k) - 0, is presented for the function N(R, F), which calculates the number of roots of the function F(k) taking into account their multiplicity, not exceeding the parameter R: N(R, F) - #{k : F(k) - 0, |k| < R}.
Let us introduce several supporting definitions for a quantum graph.
Definition 1. An edge is called internal if it has a finite length. Otherwise, we call the edge external.
Definition 2. We call a vertex internal if all the edges coming from it are internal. Otherwise, we call the vertex external.
Definition 3. We call an external vertex balanced if it is true that the number of internal edges is equal to the number of external edges containing it. Otherwise, we call the external vertex unbalanced.
1. Presence of balanced vertices
Consider a quantum graph that consists of two vertices and two edges, one of finite length and the second of infinite length. The topological structure of this graph is shown in Fig. 1.
Fig. 1. An example of a quantum graph that has a balanced vertex
For this quantum graph, it is simple to find a solution of equations (1) in the form of linear combination of basic solutions. The coupling conditions gives one a linear system of homogeneous equations for determination of the coefficients of the linear combination. Non-trivial solution exists if the determinant of the system is zero. Let us choose the following order of the equations in the system which determines the lines of the determinant. The first line corresponds to the continuity equation for the internal vertex, the second and the third lines correspond to the continuity equations for the external vertex. The fourth line corresponds to the Kirchhoff equation for the internal vertex, the fifth line corresponds to the Kirchhoff equation for the external vertex. The first and the second columns correspond to the coefficients of the exponents of the solution on the internal edge, the third column corresponds to the coefficients of the exponent on the external edge. The fourth and the fifth columns correspond to the value at the internal and external vertices, respectively. Thus, the determinant has the following form:
gi(A-mi)l g-i(A-mi)l q _1 Q
1 1 Q Q -1
Q Q 1 Q -1
(A - iai)ei(A-iai)l -(A - iai)e-i(A-iai)l Q Q Q
A - ia1 -(A - ia1) A - ia2 Q Q
= (A - ia1) ■ (-(A - ia1)(ei(A-iai)l - e-i(A-iai)l) + (A - ia2)(ei(A-iai)l + e-i(A-iai)l)).
Multiplying the resulting characteristic equation by ei(A-mi)l, we obtain the following equation:
-(A - ia1)(e2i(A-iai)l - 1) + (A - za2)(e2i(A-mi)l + 1) = Q. (2)
Let's rewrite equation (2), introducing notation for the real and imaginary parts A = x + iy:
-e2(y-ai)l sin(2x/)(a1 - a2) + 2x = Q, (3)
e2(y-ai)l cos(2x/)(a1 - a2) + 2y - a1 - a2 = Q. ()
By simplifying this system of equations, one can obtain the following equation:
2x ■ cos(2x/) + (2y - a1 - a2) ■ sin(2x1) = Q
This transition is correct, since it is easy to check that the points x at which the expressions cos(2x/) and sin(2x1) turn to 0 are not roots of the system (3). Whence we get that the imaginary part of A is equal to the following:
a1 + a2 , , , ,
y = —2--x ' cts(2x1). (4)
We substitute the resulting value for y in (4) into the first equation in the system (3), simplifying, and obtain the following equation:
(a2 - a1 )e-2(--ctg(2xl))l sin(2x1) + 2x = Q. (5)
Note that the equation (5) does not depend on the parameters a1, a2, but depends only on their difference, so we introduce a new notation a = a2 - a1. We rewrite the equation (5) in terms of a parameters:
ae-«l+2xl-ctg(2xi) sin(2x1) + 2x = Q.
(6)
Let us carry out a numerical study of the number of roots of the equation (6) for a - 0. The results are presented in Fig. 2 and Fig. 3. The parameter a - 0 does not affect the number of roots of equation (6), but only affects the location of the roots on the x axis. As a grows, the roots move further away from + ^r and tend to y + ^, where k E Z. Numerical confirmation can be seen in Fig. 2, Fig. 3.
Fig. 2. Dependence of the location of the roots of Fig. 3. Dependence of the location of the roots of equation (6) on a for x E [25.43, 25.48] equation (6) on a for x E [51.46, 51.49]
In the article [18], it was shown that the asymptotic behavior of the function of the number of resonances has the following form:
2W
N (R, F)-—R + O(1), (7)
n
N
where 0 ^ W ^ V - j is the sum of the lengths of all end edges. The function (7)
j=i
and the function of the number of resonances, constructed numerically, are presented in Fig. 4. Since the accuracy of the root increases with increasing R, then with a fixed step, starting from a certain moment, some roots cannot be found. Therefore, the function diverges from the asymptote, but this is only due to an inaccuracy in the calculations.
400 ■ 350 ■ 300 ■ 250 ■
N
1 T 200 ■
150 ■ 100 ■ 50 ■ 0
0 25 50 75 j? 100 125 150 175 200
Fig. 4. Asymptotic behavior of the number of resonances function, I = n
It is worth noting that the form of the function (7) demonstrates the influence of the volume of the quantum graph. Namely, in the quantum graph under consideration, there should be a directly proportional relationship between the length of the final edge and the asymptote slope. Let's check this dependence numerically. The results are presented in Fig. 5.
2. Lack of balanced vertices
Consider a quantum graph that consists of two vertices connected by an edge of finite length, and one of the vertices has n > 1 edges of infinite length attached to it. Thus, this graph on Fig. 6 does not have balanced vertices.
Fig. 6. An example of a quantum Let us write a determinant for this quantum
graph that has no balanced vertices graph that describes the system of coupling conditions equations in the same way as in the previous section:
An
B1 B2 B3 B4 B5 B6 B7 Bg B9
where B2,B4,Bg are zero matrices, B5 is the identity matrix, B3 is a diagonal matrix whose elements are -1, matrices B1,B7,B8 have the following form:
B1
3i(A-rno )l „-i(A-i«o)l
1
1
B
(A - ia0)ei(A-iao)l - (A - ia0)e-i(A-iao)l 7 (A - ia0) -(A - ia0)
B8
Q ... Q
(A - ia1) ... (A - ian)
Using the method of mathematical induction, it can be shown that the determinant of An (8) is equal to the following:
Ara = (A - a0) ■ ( -(A - a0)(z - z) + ^(A - afc)(z + z)
fc=1
Thus, to study resonances, it is necessary to study the roots of the following equation:
n
-(A - a0)(z - z) + > (A - afc)(z + z) = Q.
fc=1
Write equation (9) in a slightly different form, grouping the coefficients for z and z:
z ■ ( (n - 1)A + a0 - Y^ aA + z ■ (n + 1)A - a0 - afc ) = Q. (10)
fc=1 / V fc=1
The form of equation (10) demonstrates that the parameters of the Dirac operator on
edges of infinite length do not individually affect the resonances, but only their total
value.
References
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Article received 01.10.2023.
Corrections received 05.08.2024.
Челябинский физико-математический журнал. 2024. Т. 9, вып. 4- С. 682-688.
УДК 530.145 Б01: 10.47475/2500-0101-2024-9-4-682-688
ВЛИЯНИЕ ПАРАМЕТРОВ КВАНТОВОГО ГРАФА НА АСИМПТОТИКУ ЧИСЛА РЕЗОНАНСОВ
А. Г. Белолипецкая", И. Ю. Попов6
Университет ИТМО, Санкт-Петербург, Россия
"[email protected], [email protected]
Изучены квантовые графы, на рёбрах которых действует оператор Дирака, а условием связи в вершинах которого является условие Кирхгофа. Для них число резонансов найдено численно. Показано, как параметры квантового графа, такие как топологическая структура, объём квантового графа, а также параметры оператора Дирака влияют на количество резонансов.
Ключевые слова: квантовый граф, оператор Дирака, асимптотика, 'резонанс.
Поступила в 'редакцию 01.10.2023. После переработки 05.08.2024.
Сведения об авторах
Белолипецкая Анна Геннадьевна, инженер, Научно-образовательный центр математики, Университет ИТМО, Санкт-Петербург, Россия; [email protected]. Попов Игорь Юрьевич, доктор физико-математических наук, профессор, Научно-образовательный центр математики, Университет ИТМО, Санкт-Петербург, Россия; [email protected].
Работа поддержана Российским научным фондом (грант 23-21-00096, https://rscf.ru/en/project/23-21-00096/).
