Isospectral
Encyclopedia
In mathematics
, two linear operators are called isospectral or cospectral if they have the same spectrum. Roughly speaking, they are supposed to have the same sets of eigenvalues, when those are counted with multiplicity.
The theory of isospectral operators is markedly different depending on whether the space is finite or infinite dimensional. In finite-dimensions, one essentially deals with square matrices
. Two diagonalizable matrices
are isospectral just when they are similar.
In infinite dimensions, the spectrum need not consist solely of isolated eigenvalues. However, the case of a compact operator
on a Hilbert space
(or Banach space
) is still tractable, since the eigenvalues are at most countable with at most a single limit point λ = 0. The most studied isospectral problem in infinite dimensions that of the Laplace operator
on a domain in R2. Two such domains are called isospectral if their Laplacians are isospectral. The problem of inferring the geometrical properties of a domain from the spectrum of its Laplacian is often known as hearing the shape of a drum
.
square matrices, the relation of being isospectral for two diagonalizable matrices
is just similarity. This doesn't however reduce completely the interest of the concept, since we can have an isospectral family of matrices of shape A(t) = M(t)−1AM(t) depending on a parameter
t in a complicated way. This is an evolution of a matrix that happens inside one similarity class.
A fundamental insight in soliton
theory was that the infinitesimal
analogue of that equation, namely
was behind the conservation laws that were responsible for keeping solitons from dissipating. That is, the preservation of spectrum was an interpretation of the conservation mechanism. The identification of so-called Lax pair
s (P,L) giving rise to analogous equations, by Peter Lax
, showed how linear machinery could explain the non-linear behaviour.
There are many examples of isospectral manifolds which are not isometric. The first example was given in 1964 by John Milnor
. He constructed a pair of flat tori of 16 dimension, using arithmetic lattices first studied by Ernst Witt
. After this example, many isospectral pairs in dimension two and higher were constructed (for instance, by M. F. Vignéras, A. Ikeda, H. Urakawa, C. Gordon). In particular , based on the Selberg trace formula
for PSL(2,R) and PSL(2,C), constructed examples of isospectral, non-isometric closed hyperbolic 2-manifolds and 3-manifolds as quotients of hyperbolic 2-space and 3-space by arithmetic subgroups, constructed using quaternion algebras associated with quadratic extensions of the rationals by class field theory
. In this case Selberg's trace formula shows that the spectrum of the Laplacian fully determines the length spectrum, the set of lengths of closed geodesics in each free homotopy class, along with the twist along the geodesic in the 3-dimensional case.
In 1985 Toshikazu Sunada
found a general method of construction based on a covering space technique, which, either in its original or certain generalized versions, came to be known as the Sunada method or Sunada construction. Like the previous methods it is based the trace formula, via
through the Selberg zeta function. Sunada noticed that the method of constructing number fields with the same Dedekind zeta function
could be adapted to compact manifolds. His method relies on the fact that if M is a finite covering of a compact Riemannian manifold
M0 with G the finite group
of deck transformations and H1, H2 are subgroups of G meeting each conjugacy class of G in the same number of elements, then the manifolds H1 \ M and H2 \ M are isospectral
but not necessarily isometric. Although this does not recapture the arithmetic examples of Milnor and Vignéras, Sunada's method yields many known examples of isospectral manifolds. It led C. Gordon, D. Webb and S. Wolpert to the discovery in 1991 of a counter example to Mark Kac
's problem "Can one hear the shape of a drum?
" An elementary treatment, based on Sunada's method, was later given in .
Sunada's idea also stimulated the attempt to find isospectral examples which could not be obtained by his technique. Among many examples, the most striking one is a simply connected example of .
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, two linear operators are called isospectral or cospectral if they have the same spectrum. Roughly speaking, they are supposed to have the same sets of eigenvalues, when those are counted with multiplicity.
The theory of isospectral operators is markedly different depending on whether the space is finite or infinite dimensional. In finite-dimensions, one essentially deals with square matrices
Matrix (mathematics)
In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...
. Two diagonalizable matrices
Diagonalizable matrix
In linear algebra, a square matrix A is called diagonalizable if it is similar to a diagonal matrix, i.e., if there exists an invertible matrix P such that P −1AP is a diagonal matrix...
are isospectral just when they are similar.
In infinite dimensions, the spectrum need not consist solely of isolated eigenvalues. However, the case of a compact operator
Compact operator
In functional analysis, a branch of mathematics, a compact operator is a linear operator L from a Banach space X to another Banach space Y, such that the image under L of any bounded subset of X is a relatively compact subset of Y...
on a Hilbert space
Hilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...
(or Banach space
Banach space
In mathematics, Banach spaces is the name for complete normed vector spaces, one of the central objects of study in functional analysis. A complete normed vector space is a vector space V with a norm ||·|| such that every Cauchy sequence in V has a limit in V In mathematics, Banach spaces is the...
) is still tractable, since the eigenvalues are at most countable with at most a single limit point λ = 0. The most studied isospectral problem in infinite dimensions that of the Laplace operator
Laplace operator
In mathematics the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a function on Euclidean space. It is usually denoted by the symbols ∇·∇, ∇2 or Δ...
on a domain in R2. Two such domains are called isospectral if their Laplacians are isospectral. The problem of inferring the geometrical properties of a domain from the spectrum of its Laplacian is often known as hearing the shape of a drum
Hearing the shape of a drum
To hear the shape of a drum is to infer information about the shape of the drumhead from the sound it makes, i.e., from the list of basic harmonics, via the use of mathematical theory...
.
Finite dimensional spaces
In the case of operators on finite-dimensional vector spaces, for complexComplex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
square matrices, the relation of being isospectral for two diagonalizable matrices
Diagonalizable matrix
In linear algebra, a square matrix A is called diagonalizable if it is similar to a diagonal matrix, i.e., if there exists an invertible matrix P such that P −1AP is a diagonal matrix...
is just similarity. This doesn't however reduce completely the interest of the concept, since we can have an isospectral family of matrices of shape A(t) = M(t)−1AM(t) depending on a parameter
Parameter
Parameter from Ancient Greek παρά also “para” meaning “beside, subsidiary” and μέτρον also “metron” meaning “measure”, can be interpreted in mathematics, logic, linguistics, environmental science and other disciplines....
t in a complicated way. This is an evolution of a matrix that happens inside one similarity class.
A fundamental insight in soliton
Soliton
In mathematics and physics, a soliton is a self-reinforcing solitary wave that maintains its shape while it travels at constant speed. Solitons are caused by a cancellation of nonlinear and dispersive effects in the medium...
theory was that the infinitesimal
Infinitesimal
Infinitesimals have been used to express the idea of objects so small that there is no way to see them or to measure them. The word infinitesimal comes from a 17th century Modern Latin coinage infinitesimus, which originally referred to the "infinite-th" item in a series.In common speech, an...
analogue of that equation, namely
- A ′ = [A, M] = AM − MA
was behind the conservation laws that were responsible for keeping solitons from dissipating. That is, the preservation of spectrum was an interpretation of the conservation mechanism. The identification of so-called Lax pair
Lax pair
In mathematics, in the theory of integrable systems, a Lax pair is a pair of time-dependent matrices or operators that describe the corresponding differential equations. They were introduced by Peter Lax to discuss solitons in continuous media...
s (P,L) giving rise to analogous equations, by Peter Lax
Peter Lax
Peter David Lax is a mathematician working in the areas of pure and applied mathematics. He has made important contributions to integrable systems, fluid dynamics and shock waves, solitonic physics, hyperbolic conservation laws, and mathematical and scientific computing, among other fields...
, showed how linear machinery could explain the non-linear behaviour.
Isospectral manifolds
Two closed Riemannian manifolds are said to be isospectral if the eigenvalues of their Laplacians, counted multiplicities, coincide. One of fundamental problems in spectral geometry is to ask to what extent the eigenvalues determine the geometry of a given manifold.There are many examples of isospectral manifolds which are not isometric. The first example was given in 1964 by John Milnor
John Milnor
John Willard Milnor is an American mathematician known for his work in differential topology, K-theory and dynamical systems. He won the Fields Medal in 1962, the Wolf Prize in 1989, and the Abel Prize in 2011. Milnor is a distinguished professor at Stony Brook University...
. He constructed a pair of flat tori of 16 dimension, using arithmetic lattices first studied by Ernst Witt
Ernst Witt
Ernst Witt was a German mathematician born on the island of Als . Shortly after his birth, he and his parents moved to China, and he did not return to Europe until he was nine....
. After this example, many isospectral pairs in dimension two and higher were constructed (for instance, by M. F. Vignéras, A. Ikeda, H. Urakawa, C. Gordon). In particular , based on the Selberg trace formula
Selberg trace formula
In mathematics, the Selberg trace formula, introduced by , is an expression for the character of the unitary representation of G on the space L2 of square-integrable functions, where G is a Lie group and Γ a cofinite discrete group...
for PSL(2,R) and PSL(2,C), constructed examples of isospectral, non-isometric closed hyperbolic 2-manifolds and 3-manifolds as quotients of hyperbolic 2-space and 3-space by arithmetic subgroups, constructed using quaternion algebras associated with quadratic extensions of the rationals by class field theory
Class field theory
In mathematics, class field theory is a major branch of algebraic number theory that studies abelian extensions of number fields.Most of the central results in this area were proved in the period between 1900 and 1950...
. In this case Selberg's trace formula shows that the spectrum of the Laplacian fully determines the length spectrum, the set of lengths of closed geodesics in each free homotopy class, along with the twist along the geodesic in the 3-dimensional case.
In 1985 Toshikazu Sunada
Toshikazu Sunada
is a Japanese mathematician and author of many books and essays on mathematics and mathematical sciences. He is professor of mathematics at Meiji University, Tokyo, and is also professor emeritus of Tohoku University, Tohoku, Japan...
found a general method of construction based on a covering space technique, which, either in its original or certain generalized versions, came to be known as the Sunada method or Sunada construction. Like the previous methods it is based the trace formula, via
through the Selberg zeta function. Sunada noticed that the method of constructing number fields with the same Dedekind zeta function
Dedekind zeta function
In mathematics, the Dedekind zeta function of an algebraic number field K, generally denoted ζK, is a generalization of the Riemann zeta function—which is obtained by specializing to the case where K is the rational numbers Q...
could be adapted to compact manifolds. His method relies on the fact that if M is a finite covering of a compact Riemannian manifold
M0 with G the finite group
Finite group
In mathematics and abstract algebra, a finite group is a group whose underlying set G has finitely many elements. During the twentieth century, mathematicians investigated certain aspects of the theory of finite groups in great depth, especially the local theory of finite groups, and the theory of...
of deck transformations and H1, H2 are subgroups of G meeting each conjugacy class of G in the same number of elements, then the manifolds H1 \ M and H2 \ M are isospectral
but not necessarily isometric. Although this does not recapture the arithmetic examples of Milnor and Vignéras, Sunada's method yields many known examples of isospectral manifolds. It led C. Gordon, D. Webb and S. Wolpert to the discovery in 1991 of a counter example to Mark Kac
Mark Kac
Mark Kac was a Polish mathematician. His main interest was probability theory. His question, "Can one hear the shape of a drum?" set off research into spectral theory, with the idea of understanding the extent to which the spectrum allows one to read back the geometry. Kac completed his Ph.D...
's problem "Can one hear the shape of a drum?
Hearing the shape of a drum
To hear the shape of a drum is to infer information about the shape of the drumhead from the sound it makes, i.e., from the list of basic harmonics, via the use of mathematical theory...
" An elementary treatment, based on Sunada's method, was later given in .
Sunada's idea also stimulated the attempt to find isospectral examples which could not be obtained by his technique. Among many examples, the most striking one is a simply connected example of .