tailieunhanh - Báo cáo toán học: "Combinatorial Identities from the Spectral Theory of Quantum Graph"

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí toán học quốc tế đề tài: Combinatorial Identities from the Spectral Theory of Quantum Graphs. | Combinatorial Identities from the Spectral Theory of Quantum Graphs Holger Schanz and Uzy Smilanskyi IGeorg-August-Universitat and MPI fur Stromungsforschung Gottingen 37073 Gottingen Germany holger@ Department of Physics of Complex Systems The Weizmann Institute of Science Rehovot 76100 Israel Special volume honoring Professor Aviezri Fraenkel Submitted March 2000 Accepted April 26 2000 Abstract The purpose of this paper is to present a newly discovered link between three seemingly unrelated subjects quantum graphs the theory of random matrix ensembles and combinatorics. We discuss the nature of this connection and demonstrate it in a special case pertaining to simple graphs and to the random ensemble of 2 X 2 unitary matrices. The corresponding combinatorial problem results in a few identities which to the best of our knowledge were not proven previously. Mathematical Reviews Subject Numbers 05C38 90B10 1 Introduction In the present paper we show that some questions arising in the study of spectral correlations for quantum graphs and in the theory of random matrix ensembles can be cast as combinatorial problems. This connection will be explained in detail in the next chapter. As a demonstration of this link we solved in detail a particular system and the corresponding combinatorial work resulted in the following identities i Let n q be arbitrary integers with 1 q n and Then F n q n 1 n 1 n 2 vv v v J q 1 q 1Ă n q A n q 1 X I V J U 1 H V 1 H v 1 min q n q S n q X F n q 1 1 1 2 ii Let s t be arbitrary positive integers and N s t min s t X - - U 1 1 1 t V s V -1 S Ế s 1 1 1 Ps t-1 s t s 3 where PN k x are the Kravtchouk polynomials to be dehned in Eq. 47 . Further let x y be complex with x y 1Ạ 2. Then we have the generating functions 1 x 1 1 Xiv2 s t 2X-i . - A - 4 1 1 4x2 2x 1 1 G2 x X1N s t N - lA x 1- 2 5 and g x y X N s t xy 1 v 1Xx v _2 . 6 s t 1 1 y 1 x y 2xy iii Let m be any positive integer. Then 2m-1 ỉ .