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正则树上满足δ'-型条件的Schr？dinger算子的谱与正交多项式
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Abstract:
本文研究了定义在正则度量树Γn上满足δ'-型顶点条件的Schr？dinger算子的谱结构。文章首先给出了正则量子树分解后得到的量子线图上的算子所满足的顶点条件与正交多项式的关系，然后根据L2(Γn)的空间分解定理和正交多项式根的性质得到了Γn上算子的谱结构。
In this paper, we study the spectral structure of Schr？dinger operators with δ'-type vertex condi-tions on regular metric trees. We first give the relationship between the operators with δ'-type vertex conditions on the quantum graph after the decomposition of the regular quantum tree and the orthogonal polynomials; then we get the spectral structure of Schr？dinger operators on regular metric trees by the space decomposition theorem and the roots’ properties of orthogonal polynomi-als.
| [1] | Pauling, L. (1936) The Diamagnetic Anisotropy of Aromatic Molecules. Journal of Chemical Physics, 4, 673-677.
https://doi.org/10.1063/1.1749766 |
| [2] | Carlson, R. (1997) Hill’s Equation for a Homogeneous Tree. Electronic Journal of Differential Equations, 23, 1-30. |
| [3] | Carlson, R. (2000) Nonclassical Sturm-Liouville Problems and Schr？-dinger Operators on Radial Trees. Electronic, 71, 1-24. |
| [4] | Naimark, K. and Solomyak, M. (2000) Eigenvalue Esti-mates for the Weighted Laplacian on Metric Trees. Proceedings of the London Mathematical Society, 80, 690-724. https://doi.org/10.1112/S0024611500012272 |
| [5] | Alexander, V. and Solomyak, M. (2002) Schr？dinger Operators on Homogeneous Metric Trees: Spectrum in Gaps. Reviews in Mathematical Physics, 14, 421-467. https://doi.org/10.1142/S0129055X02001235 |
| [6] | Solomyak, M. (2003) Laplace and Schr？dinger Operators on Regular Metric Trees: The Discrete Spectrum Case. Function Spaces, Differential Operators and Nonlinear Analysis. Binkh？user, Basel, 161-181.
https://doi.org/10.1007/978-3-0348-8035-0_8 |
| [7] | Solomyak, M. (2004) On the Spectrum of the Laplacian on Regular Metric Trees. Waves in Random Media, 14, 155-171.
https://doi.org/10.1088/0959-7174/14/1/017 |
| [8] | Hess, Z.W. and Stephen, S.P. (2021) Spectra of Regular Quan-tum Trees: Rogue Eigenvalues and Dependence on Vertex Condition. Annales Henri Poincaré, 22, 2531-2561. https://doi.org/10.1007/s00023-021-01035-2 |
| [9] | Zhao, J., Shi, G.L. and Yan, J. (2018) The Discrete Spectrum of Schr？dinger Operators with δ-Type Conditions on Regular Metric Trees. Journal of Spectral Theory, 8, 459-491. https://doi.org/10.4171/JST/202 |
| [10] | Berkolaiko, G. and Kuchment, P. (2013) Introduction to Quantum Graphs, Mathematical Surveys and Monographs. Vol. 186, American Mathematical Society, Providence. |
| [11] | Naimark, K. and Solomyak, M. (2001) Geometry of Sobolev Spaces on Regular Trees and Hardy Inequalities. Russian Journal of Math-ematical Physics, 8, 322-335. |
| [12] | Theodore, S.C. (1978) An Introduction to Orthogonal Polynomials. Gordon and Breach, New York. |
