ON THE ALGEBRAIC ASPECTS OF SAXON-HUTNER THEOREM Ivaïlo M. Mladenov ∗ and Tsetska G. Rashkova † ∗ Institute of Biophysics, Bulgarian Academy of Sciences, Acad. G. Bonchev Str., Bl. 21, 1113 Sofia, Bulgaria, E-mail: mladenov@bio21.bas.bg † Department of Algebra and Geometry, University of Ruse “A. Kanchev”, Studentska Str. 8, 7017 Ruse, Bulgaria, E-mail: tcetcka@ami.ru.acad.bg Abstract. Here we present some necessary and sufficient conditions for the validity of the Saxon- Hutner conjecture concerning the preservation of the energy gaps into an infinite one-dimensional lattice. Keywords: monodromy, transfer matrix, forbidden energy PACS: 02.10.Yn, 71.55.Ak, 71.55.Jv INTRODUCTION Let us consider the one-dimensional Schrödinger equation d 2 ψ +(E ) x 2 − V (x )ψ = 0 (1) d where ψ = ψ (x) is the wave function, the spectral parameter E is the particle energy and V (x) is a known function - the potential. Quantum mechanics deals with the above equation and its generalizations. When V (x)= 0 and E = k 2 we have a free particle motion. Two linearly indepenedent solutions of (1) are e ikx and e −ikx which represent respectively a particle moving to the right (k > 0) and a particle moving to the left (k < 0). A monodromy operator for the equation (1) with a localized potential is a linear operator acting on the space of free particle states in a special way. Proposition 1. The matrix of the monodromy operator in the basis (e ikx , e −ikx ) is an element of the group SU (1, 1), where SU (1, 1)= {M =  ω + iζ ν + iη Mat(2 C) ; det(M)= 1 ν − iη ω − iζ  ∈ , }. Below we will remind some facts from low-dimensional group theory. The group U (1, 1) of pseudo-unitary matrices of signature (1, 1) is the group of the linear trans- formations of the complex plane preserving the pseudo-hermitian form < z, z >= |z 1 | 2 −|z 2 | 2 . The group SL(2, C) of 2 × 2 unimodular matrices is the group leaving in- variant the natural symplectic structure [ , ] in the Euclidean plane ([ζ , η ] is the ori- ented area of a parallelogram spanned by the vectors ζ , η ) and the group GL(2, R) is 169 Downloaded 13 Feb 2008 to 201.141.73.122. Redistribution subject to AIP license or copyright; see http://proceedings.aip.org/proceedings/cpcr.jsp CP956, XXVI Workshop on Geometrical Methods in Physics edited by P. Kielanowski, A. Odzijewicz, M. Schlichenmaier, and T. Voronov © 2007 American Institute of Physics 978-0-7354-0470-0/07/$23.00