Spectral asymptotics and regularized traces for Dirac operators on a star-shaped graph

References

• Kottos , T and Smilansky , U . 1997 . Quantum chaos on graphs . Phys. Rev. Lett. , 79 : 4794 – 4797 .
   Web of Science ®Google Scholar
• Kottos , T and Smilansky , U . 1999 . Periodic orbit theory and spectral statistics for quantum graphs . Ann. Phys. , 274 : 76 – 124 .
   Web of Science ®Google Scholar
• F. Gesztesy, R. Ratnaseelan, and G. Teschl, The KdV hierarchy and associated trace formulas, in Proceedings of the International Conference on Applications of Operator Theory, I. Gohberg, P. Lancaster, and P.N. Shivakumar, eds., Operator Theory: Advances and Applications, Vol. 87, Birkháuser, Basel, 1996, pp. 125–163
   Google Scholar
• Kaup , DJ and Newell , AC . 1978 . An exact solution for a derivative nonlinear Schrödinger equation . J. Math. Phys. , 19 : 798 – 801 .
   Web of Science ®Google Scholar
• Trubowitz , E . 1977 . The inverse problem for periodic potentials . Commun. Pure Appl. Math. , 30 : 321 – 337 .
   Web of Science ®Google Scholar
• Roth , JP . 1984 . “ Le spectre du Laplacien sur un graphe ” . In Théorie du potentiel, Proceedings of the Colloque Jacques Deny, Orsay 1983, Lecture Notes in Mathematics , Edited by: Moko bodzki , G and Pinchon , D . Vol. 1096 , 521 – 539 . Berlin : Springer .
   Google Scholar
• Roth , JP . 1983 . Spectre du Laplacien sur un graphe . C.R. Acad. Sci. Paris , 296 : 793 – 795 .
   Google Scholar
• Gutzwiller , MC . 1971 . Periodic orbits and classical quantization conditions . J. Math. Phys. , 12 : 343 – 358 .
   Web of Science ®Google Scholar
• Gutzwiller , MC . 1990 . Chaos in Classical and Quantum Mechanics , New York : Springer .
   Google Scholar
• Bolte , J and Harrison , J . 2003 . Spectral statistics for the Dirac operator on graphs . J. Phys. A: Math. Gen. , 36 : 2747 – 2769 .
   Google Scholar
• J. Bolte and S. Endres, Trace formulae for quantum graphs, in Analysis on Graphs and its Applications, P. Exner, J.P. Keating, P. Kuchment, T. Sunada, and Teplyaer, eds., Proc. Symp. Pure Math., Vol. 77, American Mathematical Society, Providence, RI, 2008, pp. 247–259
   Google Scholar
• Currie , S and Watson , BA . 2008 . Green's functions and regularized traces of Sturm-Liouville operators on graphs . Proc. Edinb. Math. Soc. (Series 2) , 51 : 315 – 335 .
   Web of Science ®Google Scholar
• V. Kostrykin, J. Potthoff, and R. Schrader, Heat kernels on metric graphs and a trace formula, preprint (2007). Available at arXiv: math-ph/0701009
   Google Scholar
• Winn , B . 2006 . “ On the trace formula for quantum star graphs ” . In Quantum Graphs and their Applications , Contemporary Mathematics, Vol. 41 Edited by: Berkolaiko , G , Carlson , R , Fulling , SA and Kuchment , P . 293 – 307 . Providence , RI : American Mathematical Society .
   Google Scholar
• Yang , CF and Yang , JX . 2007 . Large eigenvalues and traces of the Sturm-Liouville equations on star-shaped graphs . Methods Appl. Anal. , 14 : 179 – 196 .
   Google Scholar
• Bulla , W and Trenkler , T . 1990 . The free Dirac operator on compact and noncompact graphs . J. Math. Phys. , 31 : 1157 – 1163 .
   Web of Science ®Google Scholar
• Pivovarchik , VN . 2007 . Inverse problem for the Sturm-Liouville equation on a star-shaped graph . Math. Nachrichten , 280 : 1595 – 1619 .
   Web of Science ®Google Scholar
• Pivovarchik , VN and Woracek , H . 2009 . Sums of Nevanlinna functions and differential equations on star-shaped graphs . Oper. Matrices , 3 : 451 – 501 .
   Web of Science ®Google Scholar
• Yurko , VA . 2005 . Inverse spectral problems for Sturm-Liouville operators on graphs . Inverse Probl. , 21 : 1075 – 1086 .
   Web of Science ®Google Scholar
• Levitan , BM and Sargsjan , IS . 1991 . Sturm-Liouville and Dirac Operators , Dordrecht : Kluwer .
   Google Scholar
• Shi , QC . 1993 . Some trace formulae for one dimensional Dirac systems . Acta Math. Sci. , 18 : 316 – 321 .
   Google Scholar
• Naimark , M . 1968 . Linear Differential Operators: II , New York : Ungar . (transl. from the second Russian edition)
   Google Scholar
• Ahlfors , L . 1966 . Complex Analysis , New York : McGraw-Hill .
   Google Scholar