Radially isospectral annular membranes

^** Email: h.gottlieb{at}griffith.edu.au By utilizing transformations of both the radial coordinate andthe radial wave function, densities of annular membranes whichare radially isospectral to any given radial density are produced.In particular, new families of annular membrane densities arefound which are radially isospectral to a uniform membrane.Some new generalizations for completely isospectral annularmembranes (including angle-dependent densities) are also given.     

Symmetries for the Ablowitz–Ladik Hierarchy: Part I. Four‐Potential Case

In the paper, we first investigate symmetries of isospectral and non‐isospectral four‐potential Ablowitz–Ladik hierarchies. We express these hierarchies in the form of u_n,t= L^mH⁽⁰⁾ , where m is an arbitrary integer (instead of a nature number) and L is the recursion operator. Then by means of the zero‐curvature representations of the isospectral and non‐isospectral flows, we construct symmetries for the isospectral equation hierarchy as well as non‐isospectral equation hierarchy, respectively. The symmetries, respectively, form two centerless Kac‐Moody‐Virasoro algebras. The recursion operator L is proved to be hereditary and a strong symmetry for this isospectral equation hierarchy. Besides, we make clear for the relation between four‐potential and two‐potential Ablowitz–Ladik hierarchies. The even order members in the four‐potential Ablowitz–Ladik hierarchies together with their symmetries and algebraic structures can be reduced to two‐potential case. The reduction keeps invariant for the algebraic structures and the recursion operator for two potential case becomes L² .     

Spectral Properties of Four-Dimensional Compact Flat Manifolds

We study the spectral properties of a large class of compact flat Riemannian manifolds of dimension 4, namely, those whose corresponding Bieberbach groups have the canonical lattice as translation lattice. By using the explicit expression of the heat trace of the Laplacian acting on p-forms, we determine all p-isospectral and L-isospectral pairs and we show that in this class of manifolds, isospectrality on functions and isospectrality on p-forms for all values of p are equivalent to each other. The list shows for any p, 1 ≤ p ≤ 3, many p-isospectral pairs that are not isospectral on functions and have different lengths of closed geodesics. We also determine all length isospectral pairs (i.e. with the same length multiplicities), showing that there are two weak length isospectral pairs that are not length isospectral, and many pairs, p-isospectral for all p and not length isospectral. Mathematics Subject Classifications (2000): 58J53, 58C22, 20H15.     

Equivariant isospectrality and Sunada’s method

We construct pairs and continuous families of isospectral yet locally non-isometric orbifolds via an equivariant version of Sunada’s method. We also observe that if a good orbifold ${\mathcal{O}}We construct pairs and continuous families of isospectral yet locally non-isometric orbifolds via an equivariant version of Sunada’s method. We also observe that if a good orbifold O{\mathcal{O}} and a smooth manifold M are isospectral, then they cannot admit non-trivial finite Riemannian covers M₁ ?O{M_1 \to\mathcal{O}} and M ₂ → M where M ₁ and M ₂ are isospectral manifolds.     

Length spectra and<Emphasis Type="Italic">p</Emphasis>-spectra of compact flat manifolds

We compare and contrast various length vs Laplace spectra of compact flat Riemannian manifolds. As a major consequence we produce the first examples of pairs of closed manifolds that are isospectral on p-forms for some p ≠ 0, but have different weak length spectrum. For instance, we give a pair of 4-dimensional manifolds that are isospectral on p-forms for p = 1, 3and we exhibit a length of a closed geodesic that occurs in one manifold but cannot occur in the other. We also exhibit examples of this kind having different injectivity radius and different first eigenvalue of the Laplace spectrum on functions. These results follow from a method that uses integral roots of the Krawtchouk polynomials. We prove a Poisson summation formula relating the p-eigenvalue spectrum with the lengths of closed geodesics. As a consequence we show that the Laplace spectrum on functions determines the lengths of closed geodesics and, by an example, that it does not determine the complex lengths. Furthermore we show that orientability is an audible property for closed flat manifolds. We give a variety of examples, for instance, a pair of manifolds isospectral on functions (resp. Sunada isospectral) with different multiplicities of length of closed geodesies and a pair with the same multiplicities of complex lengths of closed geodesies and not isospectral on p-forms for any p, or else isospectral on p-forms for only one value of p ≠ 0.     

Flat manifolds isospectral on p-forms

We study isospectrality on p-forms of compact flat manifolds by using the equivariant spectrum of the Hodge-Laplacian on the torus. We give an explicit formula for the multiplicity of eigenvalues and a criterion for isospectrality. We construct a variety of new isospectral pairs, some of which are the first such examples in the context of compact Riemannian manifolds. For instance, we give pairs of flat manifolds of dimension n=2p, p≥2, not homeomorphic to each other, which are isospectral on p-forms but not on q-forms for q∈p, 0≤q≤n. Also, we give manifolds isospectral on p-forms if and only if p is odd, one of them orientable and the other not, and a pair of 0-isospectral flat manifolds, one of them Kähler, and the other not admitting any Kähler structure. We also construct pairs, M, M′ of dimension n≥6, which are isospectral on functions and such that β_p(M)<β_p(M’), for 04 and ? ₂ ² , respectively.     

Isospectral potentials and conformally equivalent isospectral metrics on spheres,balls and Lie groups

We construct pairs of conformally equivalent isospectral Riemannian metrics ?1g and ?2g on spheres Sⁿ and balls Bⁿ⁺¹ for certain dimensions n, the smallest of which is n=7, and on certain compact simple Lie groups. In the case of Lie groups, the metric g is left-invariant. In the case of spheres and balls, the metric g not the standard metric but may be chosen arbitrarily close to the standard one. For the same manifolds (M, g) we also show that the functions ?₁ and ?₂ are isospectral potentials for the Schrödinger operator ?₂\gD + \gf. To our knowledge, these are the first examples of isospectral potentials and of isospectral conformally equivalent metrics on simply connected closed manifolds.     

The Inverse Problem for a Discrete Periodic Schrodinger Operator

We study isospectral sets for a discrete 1D Schrodinger operator on ℤ with an (N + 1)-periodic potential. We show that for small odd potentials, the isospectral set consists of 2^{(N + 1)/2} elements, while for large potentials, the isospectral set consists of (N +1)! elements. Moreover, asymptotics for endpoints of the spectrum of the Schrodinger operator for small (and large) potentials are determined. Bibliography: 5 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 315, 2004, pp. 96–101.     

Lie algebraic structures of some (1+2)-dimensional Lax integrable systems

The paper proposes an approach to constructing the symmetries and their algebraic structures for isospectral and nonisospectral evolution equations of (1+2)-dimensional systems associated with the linear problem of Sato theory. To do that, we introduce the implicit representations of the isospectral flows {K_m} and nonisospectral flows {σ_n} in the high dimensional cases. Three examples, the Kodomstev–Petviashvili system, BKP system and new CKP system, are considered to demonstrate our method.     

Isospectral flows that preserve matrix structure

The matrix A = (a_ij) ∈ S_n is said to lie on a strict undirected graph G if a_ij = 0 (i ≠ j) whenever (i, j) is not in E(G). If S is skew-symmetric, the isospectral flow maintains the spectrum of A. We consider isospectral flows that maintain a matrix A(t) on a given graph G. We review known results for a graph G that is a (generalised) path, and construct isospectral flows for a (generalised) ring, and a star, and show how a flow may be constructed for a general graph. The analysis may be applied to the isospectral problem for a lumped-mass finite element model of an undamped vibrating system. In that context, it is important that the flow maintain other properties such as irreducibility or positivity, and we discuss whether they are maintained.     

On recognition by spectrum of the simple groups B 3(q), C 3(q), and D 4(q)

The spectrum of a finite group is the set of its element orders. Two groups are isospectral whenever they have the same spectra. We consider the classes of finite groups isospectral to the simple symplectic and orthogonal groups B ₃(q), C ₃(q), and D ₄(q). We prove that in the case of even characteristic and q > 2 these groups can be reconstructed from their spectra up to isomorphisms. In the case of odd characteristic we obtain a restriction on the composition structure of groups of this class.     

Isospectral orbifolds with different maximal isotropy orders

We construct pairs of compact Riemannian orbifolds which are isospectral for the Laplace operator on functions such that the maximal isotropy order of singular points in one of the orbifolds is higher than in the other. In one type of examples, isospectrality arises from a version of the famous Sunada theorem which also implies isospectrality on p-forms; here the orbifolds are quotients of certain compact normal homogeneous spaces. In another type of examples, the orbifolds are quotients of Euclidean and are shown to be isospectral on functions using dimension formulas for the eigenspaces developed in [12]. In the latter type of examples the orbifolds are not isospectral on 1-forms. Along the way we also give several additional examples of isospectral orbifolds which do not have maximal isotropy groups of different size but other interesting properties. All three authors were partially supported by DFG Sonderforschungsbereich 647.     

Generalizations of the neumann system—a curve theoretical approach part III: Order n systems

The Neumann system is a well-known algebraically completely integrable Hamiltonian system. Its geometry has roots in hyperelliptic curve theory and the isospectral deformation theory of Hill's operator. In this paper generalizations of the Neumann system are found for n-sheeted Riemann surfaces and the isospectral deformation theory of operators of order n. Trace formulas, Lax pairs, and constants of motion are found. The new systems are shown to be algebraically completely integrable.     

Isospectral polygons, planar graphs and heat content

Given a pair of planar isospectral, nonisometric polygons constructed as a quotient of the plane by a finite group, we construct an associated pair of planar isospectral, nonisometric weighted graphs. Using the natural heat operators on the weighted graphs, we associate to each graph a heat content. We prove that the coefficients in the small time asymptotic expansion of the heat content distinguish our isospectral pairs. As a corollary, we prove that the sequence of exit time moments for the natural Markov chains associated to each graph, averaged over starting points in the interior of the graph, provides a collection of invariants that distinguish isospectral pairs in general.

     

Factorization of nonlinear supersymmetry in one-dimensional quantum mechanics. I: General classification of reducibility and analysis of the third-order algebra

We study possible factorizations of supersymmetric (SUSY) transformations in the one-dimensional quantum mechanics into chains of elementary Darboux transformations with nonsingular coefficients. A classification of irreducible (almost) isospectral transformations and of related SUSY algebras is presented. A detailed analysis of SUSY algebras and isospectral operators is performed for the third-order case. Bibliography: 29 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 335, 2006, pp. 22–49.     

On a relation between spectral theory of lens spaces and Ehrhart theory

In this article Ehrhart quasi-polynomials of simplices are employed to determine isospectral lens spaces in terms of a finite set of numbers. Using the natural lattice associated with a lens space the associated toric variety of a lens space is introduced. It is proved that if two lens spaces are isospectral then the dimension of global sections of powers of a natural line bundle on these two toric varieties are equal and they have the same general intersection number. Also, harmonic polynomial representation of the group SO($n$) is used to provide a more elementary proof for a theorem of Lauret, Miatello and Rossetti on isospectrality of lens spaces.     

Isospectral deformations of metrics on spheres

We construct non-trivial continuous isospectral deformations of Riemannian metrics on the ball and on the sphere in R ⁿ for every n≥9. The metrics on the sphere can be chosen arbitrarily close to the round metric; in particular, they can be chosen to be positively curved. The metrics on the ball are both Dirichlet and Neumann isospectral and can be chosen arbitrarily close to the flat metric. Oblatum 19-VI-2000 & 21-II-2001?Published online: 4 May 2001     

You can hear the local orientability of an orbifold

A Riemannian orbifold is a mildly singular generalization of a Riemannian manifold which is locally modeled on the quotient of a connected, open manifold under a finite group of isometries. If all of the isometries used to define the local structures of an entire orbifold are orientation preserving, we call the orbifold locally orientable. We use heat invariants to show that a Riemannian orbifold which is locally orientable cannot be Laplace isospectral to a Riemannian orbifold which is not locally orientable. As a corollary we observe that a Riemannian orbifold that is not locally orientable cannot be Laplace isospectral to a Riemannian manifold.     

Lax-type isospectral deformations on nilmanifolds

A one-parameter familyg(t) (0 t T) of Riemannian metric on a compact manifold is called an isospectral deformation of a metricg(0) if the Laplace-Beltrami operators associated to the metricsg(t) have the same spectra. Examples of non-trivial isospectral deformations were constructed on solvmanifolds for the first time by C.S. Gordon and E. Wilson on the basis of Kirillov theory. This paper considers the isospectral deformations on nilmanifolds from the dynamical point of view. First, we see for certain isospectral deformations that the associated Hamiltonian systems of geodesic flows are decomposed into a collection of reduced systems which are left invariant as Hamiltonian systems under the deformations. This fact is formulated by the classical Lax equations. Next, by using a quantization procedure, we attempt to obtain Lax equations for the reduced Laplacians from the classical Lax equations. As a result, we show that certain isospectral deformations by Gordon-Wilson are represented by the Lax equations.     

Two expanding integrable systems and quasi-Hamiltonian function associated with an equation hierarchy

A Lie algebra sl(2) which is isomorphic to the known Lie algebra A₁ is introduced for which an isospectral Lax pair is presented, whose compatibility condition leads to a soliton-equation hierarchy. By using the trace identity, its Hamiltonian structure is obtained. Especially, as its reduction cases, a Sine equation and a complex modified KdV(cmKdV) equation are obtained,respectively. Then we enlarge the sl(2) into a bigger Lie algebra sl(4) so that a type of expanding integrable model of the hierarchy is worked out. However, the soliton-equation hierarchy is not integrable couplings. In order to generate the integrable couplings, an isospectral Lax pair is introduced. Under the frame of the zero curvature equation, we generate an integrable coupling whose quasi-Hamiltonian function is derived by employing the variational identity. Finally, two types of computing formulas of the constant γ are obtained, respectively.