Isospectral

Abstract. We present a method to construct pairs of isospectral quantum graphs which are not isometric. These graphs are the analogues of the family of isospectral domains in R2 which were first introduced by Gordon, Webb and Wolpert (C. Gordon et.al., 1992), recently enlarged by P. Buser el. al. (P. Buser et.al., 1994), and discussed further by Okada et. al. (Y. Okada et.al., 2001). 

In response to M. Kac s classical paper Can one hear the shape of a drum (M. Kac, 1966), much research effort was invested in two complementary problems - to identify classes of systems for which Kac s question is answered in the affirmative, or to find examples which are isospectral but not isometric. In the present paper we shall focus our attention to quantum graphs and in the following lines will review the subject of isospectrality from this intentionally narrowed point of view. The interested reader is referred to (T. Sunada, 1985 C. Gordon et.al., 1992 S. Chapman, 1995 P. Buser et.al., 1994 Y. Okada et.al., 2001 S. Zelditch, 2004) for a broader view of the field where spectral inversion and its uniqueness are discussed. 

The paper is organized in the following way. For the sake of completeness we shall give a short review of some elementary definitions and facts on quantum graphs. We shall then show how pairs of isospectral domains in R2 can be converted to isospectral pairs of quantum graphs, and discuss their spectra and eigenfunctions. 

The secular condition (11) has to be obeyed separately for the leading orders in e. The diverging term implies that (p,0 = (4> 1, (pi, (ps, 0. The term proportional to e requires (po = —Ao4>i. The remaining condition can be expressed as (—AiAq 1 + Ao)(po = 0. This condition, in turn, can be reformulated for vectors of dimension 5 obtained from (po above by disregarding the second and fifth components. The resulting 5x5 matrices for the two isospectral graphs are... 

Direct computation shows that det Ai(a, b k) = det Au(a, b k) for all k, which proves isospectrality. The graphs which result from this contraction are rather different from the original ones (i.e., they have vertices with v = 4). They remain metrically distinct even when a = b. The above discussion shows that their isospectrality is based on the same algebraic roots,however, in some disguise. Note, that taking the limit a — 0 gives two identical graphs. 

The main disadvantage of (15) is that the number of directed bonds 2B may be quite large, (in the example discussed presently, 2B = 30 while the matrix A has dimension 7. However, the tree structure of most of the isospectral graphs under consideration here can be used to obtain a much simplified, and almost explicit form of the secular function. This simplification is the object of the present appendix. 

However, the secular functions for the pairs of graphs coincide in both version - thus providing explicit proofs of isospectrality. 

This is identical to Eq. (6.45) with A = D and X - 1 = N/2. One also notes that the spectrum of the Poschl-Teller potential in one dimension is identical to that of the Morse potential in one dimension. These two potentials are therefore called isospectral. This identity arises from the fact that, as mentioned in Chapter 3, the two algebras 0(2) and U(l) are isomorphic. The situation is different in three dimensions, where this is no longer the case. 

After the completion of the text of this article a remarkable discovery has been made in the spectral theory of benzenoid molecules. In the 1980s serveral authors tried to find isospectral benzenoid systems (i.e. benzenoids having equal spectra, cf Sect. 3). These efforts were, however, not successful. Finally, Cioslowski... 

It has been recently shown [140] that the above example is the smallest possible and that it is unique. All isospectral benzenoids constructed by Babic s method have odd numbers of vertices and, consequently, have no Kekule structures. In the present moment (December 1991) isospectral Kekulean benzenoids are not known. 

The characteristic polynomial is a molecular descriptor able to discriminate well among several graphs however, some nonisomorphic graphs have the same characteristic polynomial (same eigenvalues), and for this reason they are called isospectral graphs [Herndon, 1974a]. Moreover, the sum of the coefficients of the characteristic polynomial can be used as a scalar molecular descriptor. 

Balasubramanian, K. and Basak, S.C. (1998). Characterization of Isospectral Graphs Using Graph Invariants and Derived Orthogonal Parameters. J.Chem.Inf.Comput.Sci., 38,367-373. 

Herndon, W.C. (1974a). Isospectral Molecules. Tetrahedron Lett., 8,671-674. 

Hosoya, H. (1994). Topological Twin Graphs. Smallest Pair of Isospectral Polyhedral Graphs with Eight Vertices. J.Chem.lnf.Comput.Sci.,34,428-431. 

Rucker, C. and Rticker, G. (1992). Understanding the Properties of Isospectral Points and Pairs in Graphs The Concept of Qrthogonal Relation. J.Math.Chem., 9,207-238. 

Rtlcker, G. and Rucker, C. (1991a). Isocodal and Isospectral Points, Edges, and Pairs in Graphs and How to Cope with Them in Computerized Symmetry Recognition. J.Chem.Inf.Com-putScL,31,422-421. 

A standard set of solvable potentials with critical behavior can be found in many text books on quantum mechanics [49,50], like the usual square-well potentials and other piecewise constant potentials. Also there are many potentials that are solvable only at d - 1 or for three-dimensional, v waves like the Hulthen potential, the Eckart potential, and the Posch-Teller potential. These potentials belong to a class of potentials, called shape-invariant potentials, that are exactly solvable using supersymmetric quantum mechanics [51,52], There are also many approaches to make isospectral deformation of these potentials [51,53] therefore it is possible to construct nonsymmetrical potentials with the same critical behavior as that of the original symmetric problem. 

Characterization of isospectral graphs using graph invariants and derived orthogonal parameters. 

Hosoya, H. (1994) Topological twin graphs. Smallest pair of isospectral polyhedral graphs with eight vertices. /. Chem. Inf. Comput. Sci., 34, 428-431. 

Rucker, C. and Rucker, G. (1992) Understanding the properties of isospectral points and pairs in graphs the concept of orthogonal relation. J. Math. Chem., 9, 207-238. 

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