Partial Inverse Problems for Dirac Operators on Star Graphs
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Partial Inverse Problems for Dirac Operators on Star Graphs Dai-Quan Liu
and Chuan-Fu Yang
Abstract. Partial inverse problems for Dirac operators on star graphs are studied. We consider Dirac operators on the graphs, and prove that the potential on one edge is uniquely determined by part of its spectra and part of the potential provided that the potentials on the remaining edges are given a priori. This extends the results of Horv´ ath to Dirac operators on the graphs. Mathematics Subject Classification. 34A55, 34B45, 34L40, 47E05. Keywords. Partial inverse problem, Star graph, Dirac operator, Horv´ ath-type theorem.
1. Introduction In this paper, we investigate inverse spectral theory for canonical Dirac systems on star-shaped graphs. Let G be a star graph consisting of d edges with equal length meeting at a common vertex, where the common vertex is called the central vertex and the other vertices are called pendant vertices. We parameterize each edge by x ∈ [0, π] such that x = 0 corresponding to the pendant vertex and x = π corresponding to the central vertex. Consider Dirac systems yj,2 + (pj (x) + qj (x))yj,1 = λyj,1 , (1.1) x ∈ (0, π), j = 1, d, −yj,1 + (qj (x) − pj (x))yj,2 = λyj,2 , on the edges of G with the boundary conditions yj,1 (0, λ) = 0,
j = 1, d,
(1.2)
yj,2 (0, λ) = 0,
j = 1, d,
(1.3)
or
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180
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D.-Q. Liu and C.-F. Yang
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at the pendant vertices, and the standard matching conditions y1,1 (π, λ) = yj,1 (π, λ), d
j = 2, d, (1.4)
yj,2 (π, λ) = 0,
j=1
at the central vertex, or δS -type couplings y1,2 (π, λ) = yj,2 (π, λ), d
yj,1 (π, λ) = 0,
j = 2, d, (1.5)
j=1
at the central vertex, where λ is a spectral parameter, pj (x), qj (x), j = 1, d, are real continuous functions. Denote by T (1) , T (2) , T (3) and T (4) the oper2d ators acting in Hilbert space 1 L2 [0, π] for the problem (1.1),(1.2), (1.4), or (1.1), (1.3), (1.4), or (1.1), (1.2), (1.5), or (1.1), (1.3), (1.5), respectively. We are interested in partial inverse problems for these operators. Differential operators on graphs, also called quantum graphs, arise in many fields of science and engineering, such as quantum mechanics, organic chemistry, nanotechnology, microelectronics and so on, see [1,14,15,20–22] and the references therein. Especially, [21] gives various physical models for quantum graphs and [1] provides a good overview of this subject. Until now great success has been achieved for Sturm-Liouville operators on graphs. We refer the readers to see the survey paper by Yurko [27]. In recent years, partial inverse problems for Sturm-Liouville operators on graphs have attracted many scholar’s attention, which consist in recovering the potentials on part of the graph when the potentials on the remaining part are given. These problems go back to the researches by Yang [24], in which the author gave some uniqueness results for partial inverse problems on graphs. After then, many scholars contributed to these problems. Especially, Bondarenko et al. derived the reconstruction
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