Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 40, 2015, 329-341

THE BOUND STATES OF 3D Y-JUNCTION WAVEGUIDES

Pauliina Uusitalo

University of Oulu, Department of Electrical Engineering
P.O. Box 4500, 90014 Oulu, Finland; pauliina.uusitalo 'at' oulu.fi

Abstract. The spectrum of the Dirichlet problem for Laplace operator is studied in three terminal tubes. The cross-section of the tubes is either a circle or a square. We show that these Y-shaped waveguides always have at least one eigenvalue in the discrete spectrum. In the regular case, that is, the angle between the cylinders is 2π/3, there exists exactly one eigenvalue in the discrete spectrum. While the angle 2α between the arms is varying, we show that the number of the bound states remains to be one for α ∈ (arctan(3/4),π/2] when the cross-section of the tubes is square. However, when the angle becomes sharp enough, the number of eigenvalues in the discrete spectrum increases. Moreover, it is shown that the eigenvalues are monotonously increasing when the angle 2α is in the interval (0,2π/3) and are monotonously decreasing when 2α ∈ (2π/3,π].

2010 Mathematics Subject Classification: Primary 35Q40, 35J05, 58C40, 81Q10.

Key words: Quantum waveguide, spectrum, bound state.

Reference to this article: P. Uusitalo: The bound states of 3D Y-junction waveguides. Ann. Acad. Sci. Fenn. Math. 40 (2015), 329-341.

Full document as PDF file

doi:10.5186/aasfm.2015.4023