In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, two
linear operator
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pre ...
s are called isospectral or cospectral if they have the same
spectrum
A spectrum (plural ''spectra'' or ''spectrums'') is a condition that is not limited to a specific set of values but can vary, without gaps, across a continuum. The word was first used scientifically in optics to describe the rainbow of colors i ...
. Roughly speaking, they are supposed to have the same
sets of
eigenvalue
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...
s, when those are counted with
multiplicity
Multiplicity may refer to: In science and the humanities
* Multiplicity (mathematics), the number of times an element is repeated in a multiset
* Multiplicity (philosophy), a philosophical concept
* Multiplicity (psychology), having or using multi ...
.
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 most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** ''The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...
.
In infinite dimensions, the spectrum need not consist solely of isolated eigenvalues. However, the case of a
compact operator
In functional analysis, a branch of mathematics, a compact operator is a linear operator T: X \to Y, where X,Y are normed vector spaces, with the property that T maps bounded subsets of X to relatively compact subsets of Y (subsets with compact ...
on a
Hilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
(or
Banach space
In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...
) 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 is that of the
Laplace operator
In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \nabla is the ...
on a domain in R
2. 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
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 overtones, via the use of mathematical theory.
"Can One Hear the Shape of a Drum?" is the title of a 1966 article ...
.
Finite dimensional spaces
In the case of operators on finite-dimensional vector spaces, for
complex
Complex commonly refers to:
* Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe
** Complex system, a system composed of many components which may interact with each ...
square matrices, the relation of being isospectral for two
diagonalizable matrices
In linear algebra, a square matrix A is called diagonalizable or non-defective if it is similar to a diagonal matrix, i.e., if there exists an invertible matrix P and a diagonal matrix D such that or equivalently (Such D are not unique.) ...
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'')
−1''AM''(''t'') depending on a
parameter
A parameter (), generally, is any characteristic that can help in defining or classifying a particular system (meaning an event, project, object, situation, etc.). That is, a parameter is an element of a system that is useful, or critical, when ...
''t'' in a complicated way. This is an evolution of a matrix that happens inside one similarity class.
A fundamental insight in
soliton
In mathematics and physics, a soliton or solitary wave is a self-reinforcing wave packet that maintains its shape while it propagates at a constant velocity. Solitons are caused by a cancellation of nonlinear and dispersive effects in the medium ...
theory was that the
infinitesimal
In mathematics, an infinitesimal number is a quantity that is closer to zero than any standard real number, but that is not zero. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally referr ...
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 In mathematics, in the theory of integrable systems, a Lax pair is a pair of time-dependent matrices or operators that satisfy a corresponding differential equation, called the ''Lax equation''. Lax pairs were introduced by Peter Lax to discuss sol ...
s (P,L) giving rise to analogous equations, by
Peter Lax
Peter David Lax (born Lax Péter Dávid; 1 May 1926) is a Hungarian-born American mathematician and Abel Prize laureate working in the areas of pure and applied mathematics.
Lax has made important contributions to integrable systems, fluid dyn ...
, 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
Laplace–Beltrami operator
In differential geometry, the Laplace–Beltrami operator is a generalization of the Laplace operator to functions defined on submanifolds in Euclidean space and, even more generally, on Riemannian and pseudo-Riemannian manifolds. It is named af ...
(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 Willard Milnor (born February 20, 1931) is an American mathematician known for his work in differential topology, algebraic K-theory and low-dimensional holomorphic dynamical systems. Milnor is a distinguished professor at Stony Brook Uni ...
. He constructed a pair of flat tori of 16 dimension, using arithmetic lattices first studied by
Ernst Witt
Ernst Witt (26 June 1911 – 3 July 1991) was a German mathematician, one of the leading algebraists of his time.
Biography
Witt was born on the island of Alsen, then a part of the German Empire. Shortly after his birth, his parents moved the ...
. 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
In mathematics, the Selberg trace formula, introduced by , is an expression for the character of the unitary representation of a Lie group on the space of square-integrable functions, where is a cofinite discrete group. The character is given b ...
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 mathematics, class field theory (CFT) is the fundamental branch of algebraic number theory whose goal is to describe all the abelian Galois extensions of local and global fields using objects associated to the ground field.
Hilbert is credit ...
. 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
is a Japanese mathematician and author of many books and essays on mathematics and mathematical sciences. He is professor emeritus of both Meiji University and Tohoku University. He is also distinguished professor of emeritus at Meiji in recogni ...
found a general method of construction based on a
covering space A covering of a topological space X is a continuous map \pi : E \rightarrow X with special properties.
Definition
Let X be a topological space. A covering of X is a continuous map
: \pi : E \rightarrow X
such that there exists a discrete spa ...
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 on the trace formula, via the
Selberg zeta function
The Selberg zeta-function was introduced by . It is analogous to the famous Riemann zeta function
: \zeta(s) = \prod_ \frac
where \mathbb is the set of prime numbers. The Selberg zeta-function uses the lengths of simple closed geodesics inste ...
. Sunada noticed that the method of constructing number fields with the same
Dedekind zeta function
In mathematics, the Dedekind zeta function of an algebraic number field ''K'', generally denoted ζ''K''(''s''), is a generalization of the Riemann zeta function (which is obtained in the case where ''K'' is the field of rational numbers Q). It ca ...
could be adapted to compact manifolds. His method relies on the fact that if ''M'' is a finite covering of a compact Riemannian manifold
''M''
0 with ''G'' the
finite group
Finite is the opposite of infinite. It may refer to:
* Finite number (disambiguation)
* Finite set, a set whose cardinality (number of elements) is some natural number
* Finite verb, a verb form that has a subject, usually being inflected or marked ...
of
deck transformation A covering of a topological space X is a continuous map \pi : E \rightarrow X with special properties.
Definition
Let X be a topological space. A covering of X is a continuous map
: \pi : E \rightarrow X
such that there exists a discrete spa ...
s and ''H''
1, ''H''
2 are subgroups of ''G'' meeting each conjugacy class of ''G'' in the same number of elements, then the manifolds ''H''
1 \ ''M'' and ''H''
2 \ ''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 ( ; Polish: ''Marek Kac''; August 3, 1914 – October 26, 1984) was a Polish American mathematician. His main interest was probability theory. His question, " Can one hear the shape of a drum?" set off research into spectral theory, the i ...
'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 . On the other hand,
Alan Reid proved that certain isospectral arithmetic hyperbolic manifolds in are commensurable.
See also
*
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 overtones, via the use of mathematical theory.
"Can One Hear the Shape of a Drum?" is the title of a 1966 article ...
*
Spectral geometry Spectral geometry is a field in mathematics which concerns relationships between geometric structures of manifolds and spectra of canonically defined differential operators. The case of the Laplace–Beltrami operator on a closed Riemannian m ...
Notes
References
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{{Functional analysis
Spectral theory