×

Spectral asymptotics and regularized traces for Dirac operators on a star-shaped graph. (English) Zbl 1260.34056

The authors consider the following boundary value problem for Dirac systems on a star-shaped graph consisting of \(d\) segments with equal length \(d\geq 2\). \[ \left[B\frac{d}{dx}+Q_j(x)\right]y_j(x)=\lambda y_j(x),\quad j=1,\dotsc,d,\;x\in[0,\pi], \] where \[ y_{j}(\lambda,x)=(y_{j,1}(\lambda,x),y_{j,2}(\lambda,x))^T \] with the boundary condition \[ y_{j,1}(\lambda,0)=0,\quad j=1,\dotsc,d \] or \[ y_{j,2}(\lambda,0)=0\quad j=1,\dotsc,d \] and the standard conditions \[ y_{j,1}(\lambda,\pi)=y_{i,1}(\lambda,\pi),\quad i,j=1,\dotsc,d \] and \[ \sum_{j=1}^dy_{j,2}(\lambda,\pi)=0, \] where \(B=\left(\begin{matrix} 0&1\\ -1&1 \end{matrix}\right)\) and \(Q_j\) is a \(2\times 2\) real matrix with \(C^1([0,\pi])-\)entries.
The authors discuss the asymptotic expression of eigenvalues of the above operators. They also investigate the trace formula for operators in terms of the coefficients of the operators.

MSC:

34B45 Boundary value problems on graphs and networks for ordinary differential equations
34L15 Eigenvalues, estimation of eigenvalues, upper and lower bounds of ordinary differential operators
34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
34L20 Asymptotic distribution of eigenvalues, asymptotic theory of eigenfunctions for ordinary differential operators
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] DOI: 10.1103/PhysRevLett.79.4794 · doi:10.1103/PhysRevLett.79.4794
[2] DOI: 10.1006/aphy.1999.5904 · Zbl 1036.81014 · doi:10.1006/aphy.1999.5904
[3] F. Gesztesy, R. Ratnaseelan, and G. Teschl,The KdV hierarchy and associated trace formulas, inProceedings 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 · Zbl 0865.35116
[4] DOI: 10.1063/1.523737 · Zbl 0383.35015 · doi:10.1063/1.523737
[5] DOI: 10.1002/cpa.3160300305 · Zbl 0403.34022 · doi:10.1002/cpa.3160300305
[6] Roth JP, Théorie du potentiel, Proceedings of the Colloque Jacques Deny, Orsay 1983, Lecture Notes in Mathematics 1096 pp 521– (1984)
[7] Roth JP, C.R. Acad. Sci. Paris 296 pp 793– (1983)
[8] DOI: 10.1063/1.1665596 · doi:10.1063/1.1665596
[9] Gutzwiller MC, Chaos in Classical and Quantum Mechanics (1990) · doi:10.1007/978-1-4612-0983-6
[10] DOI: 10.1088/0305-4470/36/11/307 · Zbl 1038.05057 · doi:10.1088/0305-4470/36/11/307
[11] J. Bolte and S. Endres,Trace formulae for quantum graphs, inAnalysis 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 · Zbl 1153.81493
[12] Currie S, Proc. Edinb. Math. Soc. (Series 2) 51 pp 315– (2008)
[13] V. Kostrykin, J. Potthoff, and R. Schrader, Heat kernels on metric graphs and a trace formula, preprint (2007). Available at arXiv: math-ph/0701009 · Zbl 1155.34017
[14] DOI: 10.1090/conm/415/07876 · doi:10.1090/conm/415/07876
[15] Yang CF, Methods Appl. Anal. 14 pp 179– (2007)
[16] DOI: 10.1063/1.529025 · Zbl 0699.47032 · doi:10.1063/1.529025
[17] DOI: 10.1002/mana.200410567 · Zbl 1135.34304 · doi:10.1002/mana.200410567
[18] DOI: 10.7153/oam-03-26 · Zbl 1190.30022 · doi:10.7153/oam-03-26
[19] DOI: 10.1088/0266-5611/21/3/017 · Zbl 1089.34009 · doi:10.1088/0266-5611/21/3/017
[20] DOI: 10.1007/978-94-011-3748-5 · doi:10.1007/978-94-011-3748-5
[21] Shi QC, Acta Math. Sci. 18 pp 316– (1993)
[22] Naimark M, Linear Differential Operators: II (1968)
[23] Ahlfors L, Complex Analysis (1966)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.