Các Vấn Đề Đảo Ngược cho Phép Toán Dirac trên Đồ Thị Sao

Springer Science and Business Media LLC - Tập 39 - Trang 161-175 - 2022
Dai Quan Liu1, Chuan Fu Yang1
1School of Mathematics and Statistics, Nanjing University of Science and Technology, Nanjing, P. R. China

Tóm tắt

Theo công trình trước đó, chúng tôi sẽ nghiên cứu một số bài toán đảo ngược cho phép toán Dirac trên một đồ thị sao đều. Đã chứng minh rằng hàm Weyl gọi là xác định duy nhất các tiềm năng. Hơn nữa, chúng tôi chú ý đến bài toán đảo ngược của việc khôi phục các tiềm năng từ dữ liệu phổ, bao gồm các giá trị riêng và ma trận trọng số, và trình bày một thuật toán xây dựng. Công cụ cơ bản trong bài báo này là phương pháp ánh xạ phổ được phát triển bởi Yurko.

Từ khóa

#phép toán Dirac #đồ thị sao #hàm Weyl #bài toán đảo ngược #dữ liệu phổ

Tài liệu tham khảo

Adamyan, V., Langer, H., Tretter, C., et al.: Dirac-Krein systems on star graphs. Integr. Equ. Oper. Theory, 86(1), 121–150 (2016) Berkolaiko, G., Kuchment, P.: Introduction to Quantum Graphs, American Mathematical Society, Providence, RI, 2013 Blinova, I. V., Popov, I. Y.: Quantum graph with the Dirac operator and resonance states completeness. Operator Theory: Advances and Applications, 268, 111–124 (2018) Bolte, J., Harrison, J.: Spectral statistics for the Dirac operator on graphs. J. Phys. A: Math. Gen., 36, 2747–2769 (2003) Bondarenko, N. P.: Spectral analysis for the matrix Sturm-Liouville operator on a finite interval. Tamkang J. Math., 42(3), 305–327 (2011) Bondarenko, N. P.: An inverse problem for the non-self-adjoint matrix Sturm-Liouville operator. Tamkang J. Math., 50(1), 71–102 (2019) Bondarenko, N. P.: Spectral analysis of the matrix Sturm-Liouville operator. Bound. Value Probl., 2019, Paper No. 178, 17 pp. (2019) Bondarenko, N. P.: Constructive solution of the inverse spectral problem for the matrix Sturm-Liouville operator. Inv. Probl. Sci. Eng., 28(9), 1307–1330 (2020) Bondarenko, N. P.: Spectral analysis of the Sturm-Liouville operator on the star-shaped graph. Math. Meth. Appl. Sci., 43, 471–485 (2020) Bondarenko, N. P.: Spectral data characterization for the Sturm-Liouville operator on the star-shaped graph. Anal. Math. Phys., 10(4), Paper No. 83, 28 pp. (2020) Bulla, W., Trenkler, T.: The free Dirac operator on compact and noncompact graphs. J. Math. Phys., 31, 1157–1163 (1990) Freiling, G., Yurko, V. A.: Inverse Sturm-Liouville Problems and Their Applications, NOVA Science Publishers, New York, 2001 Hu, Y. T., Bondarenko, N. P., Shieh, C. T., Yang, C. F.: Traces and inverse nodal problems for Dirac-type integro-differential operators on a graph. Appl. Math. Comput., 363, Paper No. 124606, 10 pp. (2019) Kiss, M.: An n-dimensional Ambarzumian type theorem for Dirac operators. Inverse Problems, 20(5), 1593–1597 (2004) Kottos, T., Smilansky, U.: Quantum chaos on graphs. Phys. Rev. Lett., 79(24), 4794–4797 (1997) Kuchment, P., Kunyansky, L.: Differential operators on graphs and photonic crystals. Adv. Comput. Math., 16(24), 263–290 (2002) Langese, J., Leugering, G., Schmidt, J.: Modelling, Analysis and Control of Dynamic Elastic Multi-Link Structures, Birkhäuser, Boston, 1994 Li, S. Y.: The eigenvalue problem and its inverse spectrum problem for a class of differential operators. (Chinese) Acta Math. Sci., 16(4), 391–403 (1996) Liu, D. Q., Yang, C. F.: Partial inverse problems for Dirac operators on star graphs. Mediterr. J. Math., 17(6), Paper No. 180, 14 pp. (2020) Liu, D. Q., Yang, C. F.: Inverse spectral problems for Dirac operators on a star graph with mixed boundary conditions. Math. Methods Appl. Sci., 44(13), 10663–10672 (2021) Liu, D. Q., Yang, C. F.: Weight matrices of the Dirac operator on a star graph. Math. Meth. Appl. Sci., 44(18), 14114–14125 (2021) Mikhaylov, A., Mikhaylov, V. S., Murzabekova, G.: Inverse dynamic and spectral problems for the one-dimensional Dirac system on a finite tree. J. Inverse Ill-Posed Probl., 26(5), 673–680 (2018) Pauling, L.: The diamagnetic anisotropy of aromatic molecules. J. Chem. Phys., 4(10), 673–677 (1936) Pokornyi, Y. V., Borovskikh, A. V.: Differential equations on networks (geometric graphs). J. Math. Sci., 119(6), 691–718 (2004) Puyda, D. V.: On inverse spectral problems for self-adjoint Dirac operators with general boundary conditions. Methods Funct. Anal. Topology, 19(4), 346–363 (2013) Yang, C. F., Huang, Z. Y.: Inverse spectral problems for 2m-dimensional canonical Dirac operators. Inverse Problems, 23(6), 2565–2574 (2007) Yang, C. F., Yang, X. P.: Some Ambarzumyan-type theorems for Dirac operators. Inverse Problems, 25(9), Paper No. 095012, 13 pp. (2009) Yang, C. F., Pivovarchik, V. N., Huang, Z. Y.: Ambarzumyan-type theorems on star graphs. Oper. Matrices, 5(1), 119–131 (2011) Yang, C. F., Huang, Z. Y.: Spectral asymptotics and regularized traces for Dirac operators on a star-shaped graph. Appl. Anal., 91(9), 1717–1730 (2012) Yang, C. F., Wang, F., Huang, Z. Y.: Ambarzumyan theorems for Dirac operators. Acta Math. Appl. Sin. Engl. Ser., 37(2), 287–298 (2021) Yurko, V. A.: Inverse problems for the matrix Sturm-Liouville equation on a finite interval. Inverse Problems, 22(4), 1139–1149 (2006) Yurko, V. A.: Inverse spectral problems for differential operators on spatial networks. Russian Math. Surveys, 71(3), 539–584 (2016)