Traces for Sturm–Liouville Operators with Frozen Argument on Star Graphs

Hu, Youjun; Huang, Zhen-You; Yang, Chuan‐Fu

doi:10.1007/s00025-020-1165-x

Cited by 16 publications

(14 citation statements)

References 15 publications

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“…Put Y n (z) := q n (2z), n = 0, k − 1. By virtue of (25), these polynomials Y n (z) satisfy the recurrent relations (20). Substituting q n (z) = Y n (z/2) into (37), we get…”

Section: Corollary 1 the Determinant Of The Matrixmentioning

confidence: 98%

“…Put Y n (z) := i −n q n (2iz), n = 0, k − 1. Using (25), one can easily check that the polynomials Y n (z) satisfy the recurrent relations (20). Substituting q n (z…”

Section: Corollary 1 the Determinant Of The Matrixmentioning

confidence: 99%

“…for any possible γ ∈ C \ {0} was studied, and a complete solution was obtained for the inverse problem of recovering the potential q(x) from the corresponding spectrum (spectra). Further aspects of recovering the operator as well as its spectral properties were studied in [17][18][19][20].…”

Section: Introductionmentioning

confidence: 99%

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Sturm-Liouville-type operators with frozen argument and Chebyshev polynomials

Tsai¹,

Liu²,

Buterin³

et al. 2021

Preprint

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The paper deals with Sturm-Liouville-type operators with frozen argument of the form y := −y (x) + q(x)y(a), y (α) (0) = y (β) (1) = 0, where α, β ∈ {0, 1} and a ∈ [0, 1] is an arbitrary fixed rational number. Such nonlocal operators belong to the so-called loaded differential operators, which often appear in mathematical physics. We focus on the inverse problem of recovering the potential q(x) from the spectrum of the operator . Our goal is two-fold. Firstly, we establish a deep connection between the so-called main equation of this inverse problem and Chebyshev polynomials of the first and the second kinds. This connection gives a new perspective method for solving the inverse problem. In particular, it allows one to completely describe all non-degenerate and degenerate cases, i.e. when the solution of the inverse problem is unique or not, respectively. Secondly, we give a complete and convenient description of iso-spectral potentials in the space of complex-valued integrable functions.

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“…Put Y n (z) := q n (2z), n = 0, k − 1. By virtue of (25), these polynomials Y n (z) satisfy the recurrent relations (20). Substituting q n (z) = Y n (z/2) into (37), we get…”

Section: Corollary 1 the Determinant Of The Matrixmentioning

confidence: 98%

“…Put Y n (z) := i −n q n (2iz), n = 0, k − 1. Using (25), one can easily check that the polynomials Y n (z) satisfy the recurrent relations (20). Substituting q n (z…”

Section: Corollary 1 the Determinant Of The Matrixmentioning

confidence: 99%

See 1 more Smart Citation

Sturm-Liouville-type operators with frozen argument and Chebyshev polynomials

Tsai¹,

Liu²,

Buterin³

et al. 2021

Preprint

View full text Add to dashboard Cite

show abstract

“…Indeed, for 𝜈 = 1, formula ( 26) is obvious. Further, let it hold for any 𝜈 ≤ n. Then expanding the determinant in (26) for 𝜈 = n + 1 with respect to the elements of the last row, we obtain the last equality in (25). Finally, expanding the determinant…”

Section: Conflict Of Interestmentioning

confidence: 99%

“…Other aspects of recovering the operator 𝓁 as well as its spectral properties were studied in previous research. [23][24][25][26] Returning to Inverse Problem 1, we remind that in earlier work, [15][16][17] it was reduced to some linear functional equation with respect to the potential q(x) (see Equation (10) in the next section), which was referred to as main equation of the inverse problem. For rational values of a, the main equation can be represented as linear system (13) with a special k × k-matrix A (𝛼,𝛽) 𝑗,k , whose rank appeared to be ranging between k − 1 and k. These two possibilities, in turn, correspond to the degenerate and non-degenerate cases, respectively.…”

Section: Introductionmentioning

confidence: 99%

Sturm–Liouville‐type operators with frozen argument and Chebyshev polynomials

Tsai

Liu

Buterin

et al. 2022

Math Methods in App Sciences

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In recent years, there appeared a growing interest in the inverse spectral theory for functional-differential operators with frozen argument. Such operators are nonlocal and belong to the so-called loaded differential operators, which frequently appear in mathematics as well as natural sciences and engineering. However, to various classes of nonlocal operators, classical methods of the inverse spectral theory are not applicable. For this reason, it is relevant to develop new methods and approaches for solving inverse problems for operators of this type. We establish a deep connection between the inverse problem for operators with frozen argument and Chebyshev polynomials of the first and the second kinds. Appearing to be of an interest from the point of view of the Chebyshev polynomials theory itself, this connection gives a new perspective method for studying inverse problems for operators with frozen argument. In particular, it allows one to completely describe all non-degenerate and degenerate cases, that is, when the corresponding inverse problem is uniquely solvable or not, respectively. Moreover, it gives a convenient description of all isospectral potentials in the degenerate case, which is demonstrated by some illustrative examples.

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On damping a control system with global aftereffect on quantum graphs: Stochastic interpretation

Buterin

2024

Math Methods in App Sciences

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Quantum graphs model processes in complex systems represented as spatial networks in various fields of natural science and technology. An example is the oscillations of elastic string networks, the nodes of which, besides the continuity conditions, also obey the Kirchhoff conditions, expressing the balance of tensions. In this paper, we propose a new look at quantum graphs as temporal networks, which means that the variable parametrizing the edges of a graph is interpreted as time, while each internal vertex is a branching point giving several different scenarios for the further trajectory of a process. Then Kirchhoff‐type conditions may also arise. Namely, they will be satisfied by such a trajectory of the process that is optimal with account of all the scenarios simultaneously. By employing the recent concept of global delay, we extend the problem of damping a first‐order control system with aftereffect, considered earlier only on an interval, to an arbitrary tree graph. The first means that the delay, imposed starting from the initial moment of time, associated with the root of the tree, propagates through all internal vertices. Bringing the system into the equilibrium and minimizing the energy functional with account of the anticipated probability of each scenario, we come to a variational problem. Then, we establish its equivalence to a self‐adjoint boundary value problem on the tree for some second‐order equations involving both the global delay and the global advance. The unique solvability of both problems is proved. We also illustrate that the interval case when the coefficients of the equation are discrete stochastic processes in discrete time can be viewed as the extension to a tree.

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Traces for Sturm–Liouville Operators with Frozen Argument on Star Graphs

Cited by 16 publications

References 15 publications

Sturm-Liouville-type operators with frozen argument and Chebyshev polynomials

Sturm-Liouville-type operators with frozen argument and Chebyshev polynomials

Sturm–Liouville‐type operators with frozen argument and Chebyshev polynomials

On damping a control system with global aftereffect on quantum graphs: Stochastic interpretation

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