Isospectral Pairs Research Articles

The isospectral problem for flat tori from three perspectives

Flat tori are among the only types of Riemannian manifolds for which the Laplace eigenvalues can be explicitly computed. In 1964, Milnor used a construction of Witt to find an example of isospectral nonisometric Riemannian manifolds, a striking and concise result that occupied one page in the Proceedings of the National Academy of Science of the USA. Milnor’s example is a pair of 16-dimensional flat tori, whose set of Laplace eigenvalues are identical, in spite of the fact that these tori are not isometric. A natural question is, What is the lowest dimension in which such isospectral nonisometric pairs exist? This isospectral question for flat tori can be equivalently formulated in analytic, geometric, and number theoretic language. We explore this question from all three perspectives and describe its resolution by Schiemann in the 1990s. Moreover, we share a number of open problems.

Recursion Operators for Multidimensional Integrable PDEs

We present a novel construction of recursion operators for integrable second-order multidimensional PDEs admitting isospectral scalar Lax pairs with Lax operators being first-order scalar differential operators linear in the spectral parameter. Our approach, illustrated by several examples and applicable to many other PDEs of the kind in question, employs an ansatz for the sought-for recursion operator of the equation under study based on the Lax pair for the latter.

Hückel theory and distinguishing between isospectral molecules: 1,4-divinylbenzene and 2-phenylbutadiene, and tetramethylenemethane and cyclobutadiene + carbon

Although Huckel molecular orbital theory (HMO) has been generally outdated as a computational method for the energetics of organic molecules, there is still much we can learn from it. This paper explores two compounds, 1,4-divinylbenzene and 2-phenylbutadiene, which, according to Huckel theory, are isospectral, i.e., they have the same eigenvalues and orbital energy levels. However, by using a small, well-defined perturbation, we show it is mathematically possible to distinguish between these two compounds. We suggest that this is a general procedure for distinguishing isospectral pairs of molecules as opposed to the two species being identical.

Some symmetries and similarity solutions of the long-water wave hierarchy

We introduce isospectral and non-isospectral Lax pairs, then apply the Tu scheme to generate the isospectral integrable hierarchy and the non-isospectral hierarchy, whose Hamiltonian structure, hereditary operator, symmetries are followed to obtain. In addition, a kind of Bäcklund transformation of the long-water wave hierarchy for the isospectral hierarchy is constructed. Through reductions of the isospectral hierarchy, we again get the long-water wave system whose similarity solutions, nonlinear self-adjointness and the non-invariant solutions are investigated, respectively, by the use of symmetry analysis. Finally, we make use of the weight method of the variables appearing in the long-water wave system to analyze the conservation laws of the system.

Torsional rigidity, isospectrality and quantum graphs

We study torsional rigidity for graph and quantum graph analogs of well-known pairs of isospectral non-isometric planar domains. We prove that such isospectral pairs are distinguished by torsional rigidity.

Isospectral Kähler graphs

We give some basic ways to construct Kahler graphs which are compound graphs having principal and auxiliary graphs. By use of these methods we give some examples of isospectral pairs of Kahler graphs.

Hearing shapes of few electrons quantum drums: A configuration–interaction study

The – highly remarkable – existence of non-congruent yet vibrationally isospectral shapes has been first proved theoretically and then also tested experimentally – by using electromagnetic waves in cavities, vibrating smectic films or electrons in nanostructures. In this context, we address the question whether isospectrality holds if two or more electrons interact electrostatically, using the accurate configuration–interaction method, in a discrete representation of the Bilby and Hawk shapes. Isospectral pairs offer an unique possibility to test how identical sets of single-particle energies may combine differently in the few-electrons eigenmodes, due to different wave functions spatial distributions. Our results point towards the break down of isospectrality in the presence of interactions. Thus one should be able to ”hear” the shapes of few electrons quantum drums. Interestingly however, for the analyzed two and three electrons cases, there exists an interaction strength (which can be tuned by changing the size of the shapes), for which the ground states energies of Bilby and Hawk coincide, but not the excited states as well. Wigner localization is studied and shown to occur at about the same size for both Bilby and Hawk shapes. Next, an exercise is proposed to use the two-electrons charge density of the Bilby and Hawk ground states in the phase extraction scheme as proposed by Moon et al. (2008). Results show that out-of-phase regions appear if the linear size of the shapes exceeds the Bohr radius as occupation of higher Slater determinants becomes significant.

$\Gamma $-extensions of the spectrum of an orbifold

We introduce the \Gamma-extension of the spectrum of the Laplacian of a Riemannian orbifold, where \Gamma is a finitely generated discrete group. This extension, called the \Gamma-spectrum, is the union of the Laplace spectra of the \Gamma-sectors of the orbifold, and hence constitutes a Riemannian invariant that is directly related to the singular set of the orbifold. We compare the \Gamma-spectra of known examples of isospectral pairs and families of orbifolds and demonstrate that it many cases, isospectral orbifolds need not be \Gamma-isospectral. We additionally prove a version of Sunada's theorem that allows us to construct pairs of orbifolds that are \Gamma-isospectral for any choice of \Gamma.

Fermion in a multi-kink-antikink soliton background, and exotic supersymmetry

We construct a fermion system in a multi-kink-antikink soliton background, and present in an explicit form all its trapped configurations (bound state solutions) as well as scattering states. This is achieved by exploiting an exotic N=4 centrally extended nonlinear supersymmetry of completely isospectral pairs of reflectionless Schrodinger systems with potentials to be n-soliton solutions for the Korteweg-de Vries equation. The obtained reflectionless Dirac system with a position-dependent mass is shown to possess its own exotic nonlinear supersymmetry associated with the matrix Lax-Novikov operator being a Darboux-dressed momentum. In the process, we get an algebraic recursive representation for the multi-kink-antikink backgrounds, and establish their relation to the the modified Korteweg-de Vries equation. We also indicate how the results can be related to the physics of self-consistent condensates based on the Bogoliubov-de Gennes equations.

Phase extraction in disordered isospectral shapes

The phase of the electronic wave function is not directly measurable but, quite remarkably, it becomes accessible in pairs of isospectral shapes, as recently proposed in the experiment by Moon et al. [Science 319, 782 (2008)]. The method is based on a special property, called transplantation, which relates the eigenfunctions of the isospectral pairs, and allows us to extract the phase distributions, if the amplitude distributions are known. We numerically simulate such a phase extraction procedure in the presence of disorder, which is introduced both as Anderson disorder and as roughness at edges. With disorder, the transplantation can no longer lead to a perfect fit of the wave functions, however we show that a phase can still be extracted-defined as the phase that minimizes the misfit. Interestingly, this extracted phase coincides with (or differs negligibly from) the phase of the disorder-free system, up to a certain disorder amplitude, and a misfit of the wave functions as high as ~5%, proving a robustness of the phase extraction method against disorder. However, if the disorder is increased further, the extracted phase shows a puzzle structure, no longer correlated with the phase of the disorder-free system. A discrete model is used, which is the natural approach for disorder analysis. We provide a proof that discretization preserves isospectrality and the transplantation can be adapted to the discrete systems.

Filters connecting isospectral quadratic systems

This paper concerns n×n matrix polynomials of the form L(λ)=Mλ2+Dλ+K. We study isospectral pairs L(λ), L˜(λ) and show how to construct linear matrix functions F(λ),F˜(λ) (called filters) for which F˜(λ)L(λ)=L˜(λ)F(λ). The ultimate objective is to design filters in such a way that L˜(λ) is diagonal – where this is possible. The main result of this paper is a constructive theorem demonstrating that two systems are isospectral if and only if there exists an associated family of coprime filters. Well-established techniques are used in the construction of filters using “standard pairs” for matrix polynomials and “structure preserving transformations” for their linearizations. In this paper, attention is confined to systems with nonsingular leading coefficient, M.

Hearing shapes of drums: Mathematical and physical aspects of isospectrality

In a celebrated paper '"Can one hear the shape of a drum?"' M. Kac [Amer. Math. Monthly 73, 1 (1966)] asked his famous question about the existence of nonisometric billiards having the same spectrum of the Laplacian. This question was eventually answered positively in 1992 by the construction of noncongruent planar isospectral pairs. This review highlights mathematical and physical aspects of isospectrality.

Formula omitted]-cohomology and spectral properties of flat manifolds of diagonal type

We study the cohomology groups with Z 2 -coefficients for compact flat Riemannian manifolds of diagonal type M Γ = Γ ∖ R n by explicit computation of the differentials in the Lyndon–Hochschild–Serre spectral sequence. We obtain expressions for H j ( M Γ , Z 2 ) , j = 1 , 2 and give an effective criterion for the non-vanishing of the second Stiefel–Whitney class w 2 ( M Γ ) . We apply the results to exhibit isospectral pairs with special cohomological properties; for instance, we give isospectral 5-manifolds with different H 2 ( M Γ , Z 2 ) , and isospectral 4-manifolds M , M ′ having the same Z 2 -cohomology where w 2 ( M ) = 0 and w 2 ( M ′ ) ≠ 0 . We compute the Z 2 -cohomology of all generalized Hantzsche–Wendt n -manifolds for n = 3 , 4 , 5 and we study H 2 and w 2 for a large n -dimensional family, K n , with explicit computation for a subfamily of examples due to Lee and Szczarba.

Isospectral Euler–Bernoulli beams via factorization and the Lie method

We obtain isospectral Euler–Bernoulli beams by using factorization and Lie symmetry techniques. The canonical Euler–Bernoulli beam operator is factorized as the product of a second-order linear differential operator and its adjoint. The factors are then reversed to obtain isospectral beams. The factorization is possible provided the coefficients of the factors satisfy a system of non-linear ordinary differential equations. The uncoupling of this system yields a single non-linear third-order ordinary differential equation. This ordinary differential equation, called the principal equation, is analyzed, reduced or solved using Lie group methods. We show that the principal equation may admit a one-dimensional or three-dimensional symmetry Lie algebra. When the principal system admits a unique symmetry, the best we can do is to depress its order by one. We obtain a one-parameter family of invariant solutions in this case. The maximally symmetric case is shown to be isomorphic to a Chazy equation which is solved in closed form to derive the general solution of the principal equation. The transformations connecting isospectral pairs are obtained by numerically solving systems of ordinary differential equations using the fourth-order Runge–Kutta method.

Equiconducting molecular conductors

Non-isomorphic single-molecule conductors can have transmission functions that are identical at all electron energies, within the Hückel tight-binding approximation. These equiconducting cases follow from the graph-theoretical nature of the transmission function, and the existence of isospectral chemical graphs. Systematic constructions of equiconducting molecular graphs are presented, relying on the notions of isospectral pairs and isospectral vertices. These allow identification of non-isomorphic equiconducting connection patterns for a given molecule, and infinite families of equiconducting non-isomorphic molecules.

Can one hear the shape of a Lie-type geometry?

In a celebrated paper ‘Can one hear the shape of a drum?’ (Kac 1966 Am. Math. Mon. 73 1–23) Kac asked his famous question on the consequences of isospectrality, which was eventually answered negatively in Gordon et al (1992 Invent. Math. 110 1–22) by the construction of non-congruent planar isospectral pairs. In this paper, we pose a similar question for Lie-type geometries, give partial answers and show that the question can be reduced to an important geometrical problem.

Coupling commutator pairs and integrable systems

According to classification of the matrix Lie algebras, a type of explicit Lie algebras are constructed which can be decomposed into a few Lie subalgebras. These subalgebras constitute several coupling commutator pairs from which some continuous multi-integrable couplings could be generated if the proper isospectral Lax pairs could be set up. Then the above Lie algebras are again decomposed into a kind of Lie algebras which are also closed under the matrix multiplication. From such the Lie algebras, some discrete multi-integrable couplings could be worked out. Finally, a few examples are given. However, the Hamiltonian structures of the (continuous and discrete) integrable couplings obtained by the above Lie algebras cannot be computed by using the trace identity or the quadratic-form identity, which is a strange and interesting problem. The phenomenon indicates that the importance of the Lie-algebra classification. The problem also needs us to try to find an efficient scheme to deal with.

Isospectral drums in , involution graphs and Euclidean TI-domains

The widely investigated question ‘Can one hear the shape of a drum?’ which Kac posed in his published lecture (Kac 1966 Am. Math. Mon. 73 1–23) was eventually answered negatively in Gordon et al (1992 Invent. Math. 110 1–22) by construction of isospectral pairs in . Up to present, all known planar counter examples are constructed by a certain tiling method, and in this communication, we call such examples isospectral Euclidean TI-domains. From counter examples of this type, one can construct a pair of (finite) involution graphs. In this communication, we address the question as to how the isospectrality of the domains for the Laplacian influences the cospectrality of the involution graphs.

PSLn(q) as operator group of isospectral drums

In a paper by M Kac (1966 Am. Math. Mon. 73 1–23), Kac asked his famous question 'Can one hear the shape of a drum?', which was answered negatively in Gordon et al (1992 Invent. Math. 110 1–22) by construction of planar isospectral pairs. In Buser et al (1994 Int. Math. Res. Not. 9), it is observed that all operator groups associated with the known counter examples are isomorphic to one of PSL3(2), PSL3(3), PSL4(2) and PSL3(4). We show that if (D1, D2) is a pair of non-congruent planar isospectral domains constructed from unfolding a polygonal base-tile and with associated operator group PSLn(q), then (n, q) belongs to this very restricted list.

Kac's question, planar isospectral pairs and involutions in projective space: II. Classification of generalized projective isospectral data

In Am. Math. Monthly (73 1–23 (1966)), Kac asked his famous question 'Can one hear the shape of a drum?', which was eventually answered negatively in Gordon et al (1992 Invent. Math. 110 1–22) by construction of planar isospectral pairs. Giraud (2005 J. Phys. A: Math. Gen. 38 L477–83) observed that most of the known examples can be generated from solutions of a certain equation which involves a set of involutions of an n-dimensional projective space over some finite field. He then generated all possible solutions for n = 2, when the involutions fix the same number of points. In Thas (2006 J. Phys. A: Math. Gen. 39 L385–8) we showed that no other examples arise for any other dimension, still assuming that the involutions fix the same number of points. In this paper we study the problem for involutions not necessarily fixing the same number of points, and solve the problem completely.