In
mathematics, the spectral theory of ordinary differential equations is the part of
spectral theory In mathematics, spectral theory is an inclusive term for theories extending the eigenvector and eigenvalue theory of a single square matrix to a much broader theory of the structure of operators in a variety of mathematical spaces. It is a result ...
concerned with the determination of the
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 color ...
and
eigenfunction expansion
In mathematics, an eigenfunction of a linear operator ''D'' defined on some function space is any non-zero function f in that space that, when acted upon by ''D'', is only multiplied by some scaling factor called an eigenvalue. As an equation, thi ...
associated with a linear
ordinary differential equation
In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contras ...
. In his dissertation
Hermann Weyl
Hermann Klaus Hugo Weyl, (; 9 November 1885 – 8 December 1955) was a German mathematician, theoretical physicist and philosopher. Although much of his working life was spent in Zürich, Switzerland, and then Princeton, New Jersey, he is ass ...
generalized the classical
Sturm–Liouville theory In mathematics and its applications, classical Sturm–Liouville theory is the theory of ''real'' second-order ''linear'' ordinary differential equations of the form:
for given coefficient functions , , and , an unknown function ''y = y''(''x'') ...
on a finite
closed interval
In mathematics, a (real) interval is a set of real numbers that contains all real numbers lying between any two numbers of the set. For example, the set of numbers satisfying is an interval which contains , , and all numbers in between. Othe ...
to second order
differential operators with singularities at the endpoints of the interval, possibly semi-infinite or infinite. Unlike the classical case, the spectrum may no longer consist of just a countable set of eigenvalues, but may also contain a continuous part. In this case the eigenfunction expansion involves an integral over the continuous part with respect to a
spectral measure
In mathematics, the spectral theory of ordinary differential equations is the part of spectral theory concerned with the determination of the spectrum and eigenfunction expansion associated with a linear ordinary differential equation. In his diss ...
, given by the
Titchmarsh–
Kodaira formula. The theory was put in its final simplified form for singular differential equations of even degree by Kodaira and others, using
von Neumann's
spectral theorem
In mathematics, particularly linear algebra and functional analysis, a spectral theorem is a result about when a linear operator or matrix can be diagonalized (that is, represented as a diagonal matrix in some basis). This is extremely useful b ...
. It has had important applications in
quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, q ...
,
operator theory
In mathematics, operator theory is the study of linear operators on function spaces, beginning with differential operators and integral operators. The operators may be presented abstractly by their characteristics, such as bounded linear oper ...
and
harmonic analysis
Harmonic analysis is a branch of mathematics concerned with the representation of functions or signals as the superposition of basic waves, and the study of and generalization of the notions of Fourier series and Fourier transforms (i.e. an e ...
on
semisimple Lie group
In mathematics, a Lie algebra is semisimple if it is a direct sum of simple Lie algebras. (A simple Lie algebra is a non-abelian Lie algebra without any non-zero proper ideals).
Throughout the article, unless otherwise stated, a Lie algebra is ...
s.
Introduction
Spectral theory In mathematics, spectral theory is an inclusive term for theories extending the eigenvector and eigenvalue theory of a single square matrix to a much broader theory of the structure of operators in a variety of mathematical spaces. It is a result ...
for second order ordinary differential equations on a compact interval was developed by
Jacques Charles François Sturm Jacques Charles François Sturm (29 September 1803 – 15 December 1855) was a French mathematician.
Life and work
Sturm was born in Geneva (then part of France) in 1803. The family of his father, Jean-Henri Sturm, had emigrated from Strasbourg ...
and
Joseph Liouville
Joseph Liouville (; ; 24 March 1809 – 8 September 1882) was a French mathematician and engineer.
Life and work
He was born in Saint-Omer in France on 24 March 1809. His parents were Claude-Joseph Liouville (an army officer) and Thérèse ...
in the nineteenth century and is now known as
Sturm–Liouville theory In mathematics and its applications, classical Sturm–Liouville theory is the theory of ''real'' second-order ''linear'' ordinary differential equations of the form:
for given coefficient functions , , and , an unknown function ''y = y''(''x'') ...
. In modern language it is an application of the
spectral theorem
In mathematics, particularly linear algebra and functional analysis, a spectral theorem is a result about when a linear operator or matrix can be diagonalized (that is, represented as a diagonal matrix in some basis). This is extremely useful b ...
for
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 ...
s due to
David Hilbert. In his dissertation, published in 1910,
Hermann Weyl
Hermann Klaus Hugo Weyl, (; 9 November 1885 – 8 December 1955) was a German mathematician, theoretical physicist and philosopher. Although much of his working life was spent in Zürich, Switzerland, and then Princeton, New Jersey, he is ass ...
extended this theory to second order ordinary differential equations with
singularities at the endpoints of the interval, now allowed to be infinite or semi-infinite. He simultaneously developed a spectral theory adapted to these special operators and introduced
boundary condition
In mathematics, in the field of differential equations, a boundary value problem is a differential equation together with a set of additional constraints, called the boundary conditions. A solution to a boundary value problem is a solution to ...
s in terms of his celebrated dichotomy between ''limit points'' and ''limit circles''.
In the 1920s
John von Neumann
John von Neumann (; hu, Neumann János Lajos, ; December 28, 1903 – February 8, 1957) was a Hungarian-American mathematician, physicist, computer scientist, engineer and polymath. He was regarded as having perhaps the widest cove ...
established a general spectral theorem for
unbounded self-adjoint operator
In mathematics, a self-adjoint operator on an infinite-dimensional complex vector space ''V'' with inner product \langle\cdot,\cdot\rangle (equivalently, a Hermitian operator in the finite-dimensional case) is a linear map ''A'' (from ''V'' to ...
s, which
Kunihiko Kodaira
was a Japanese mathematician known for distinguished work in algebraic geometry and the theory of complex manifolds, and as the founder of the Japanese school of algebraic geometers. He was awarded a Fields Medal in 1954, being the first Japane ...
used to streamline Weyl's method. Kodaira also generalised Weyl's method to singular ordinary differential equations of even order and obtained a simple formula for the
spectral measure
In mathematics, the spectral theory of ordinary differential equations is the part of spectral theory concerned with the determination of the spectrum and eigenfunction expansion associated with a linear ordinary differential equation. In his diss ...
. The same formula had also been obtained independently by
E. C. Titchmarsh in 1946 (scientific communication between
Japan and the
United Kingdom
The United Kingdom of Great Britain and Northern Ireland, commonly known as the United Kingdom (UK) or Britain, is a country in Europe, off the north-western coast of the European mainland, continental mainland. It comprises England, Scotlan ...
had been interrupted by
World War II
World War II or the Second World War, often abbreviated as WWII or WW2, was a world war that lasted from 1939 to 1945. It involved the World War II by country, vast majority of the world's countries—including all of the great power ...
). Titchmarsh had followed the method of the German mathematician
Emil Hilb
Emil Hilb (born 26 April 1882 in Stuttgart; died 6 August 1929 in Würzburg) was a German-Jewish mathematician who worked in the fields of special functions, differential equations, and difference equations. He was one of the authors of the ' ...
, who derived the eigenfunction expansions using
complex function theory
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebra ...
instead of
operator theory
In mathematics, operator theory is the study of linear operators on function spaces, beginning with differential operators and integral operators. The operators may be presented abstractly by their characteristics, such as bounded linear oper ...
. Other methods avoiding the spectral theorem were later developed independently by Levitan, Levinson and Yoshida, who used the fact that the
resolvent of the singular differential operator could be approximated by
compact
Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to:
* Interstate compact
* Blood compact, an ancient ritual of the Philippines
* Compact government, a type of colonial rule utilized in British ...
resolvents corresponding to
Sturm–Liouville problems for proper subintervals. Another method was found by
Mark Grigoryevich Krein
Mark Grigorievich Krein ( uk, Марко́ Григо́рович Крейн, russian: Марк Григо́рьевич Крейн; 3 April 1907 – 17 October 1989) was a Soviet mathematician, one of the major figures of the Soviet school of fu ...
; his use of ''direction functionals'' was subsequently generalised by
Izrail Glazman
Azrael (; , 'God has helped'; ) is the angel of death in some Abrahamic religions, namely Islam, Christian popular culture and some traditions of Judaism. He is also referenced in Sikhism.
Relative to similar concepts of such beings, Azrael h ...
to arbitrary ordinary differential equations of even order.
Weyl applied his theory to
Carl Friedrich Gauss
Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refe ...
's
hypergeometric differential equation, thus obtaining a far-reaching generalisation of the transform formula of
Gustav Ferdinand Mehler (1881) for the
Legendre differential equation, rediscovered by the Russian physicist
Vladimir Fock
Vladimir Aleksandrovich Fock (or Fok; russian: Влади́мир Алекса́ндрович Фок) (December 22, 1898 – December 27, 1974) was a Soviet physicist, who did foundational work on quantum mechanics and quantum electrodynamic ...
in 1943, and usually called the
Mehler–Fock transform. The corresponding ordinary differential operator is the radial part of the
Laplacian operator on 2-dimensional
hyperbolic space
In mathematics, hyperbolic space of dimension n is the unique simply connected, n-dimensional Riemannian manifold of constant sectional curvature equal to -1. It is homogeneous, and satisfies the stronger property of being a symmetric space. ...
. More generally, the
Plancherel theorem for
SL(2,R)
In mathematics, the special linear group SL(2, R) or SL2(R) is the group of 2 × 2 real matrices with determinant one:
: \mbox(2,\mathbf) = \left\.
It is a connected non-compact simple real Lie group of dimension 3 ...
of
Harish Chandra
Harish-Chandra FRS (11 October 1923 – 16 October 1983) was an Indian American mathematician and physicist who did fundamental work in representation theory, especially harmonic analysis on semisimple Lie groups.
Early life
Harish-Chandra ...
and
Gelfand–
Naimark can be deduced from Weyl's theory for the hypergeometric equation, as can the theory of
spherical function Spherical function can refer to
* Spherical harmonics
*Zonal spherical function In mathematics, a zonal spherical function or often just spherical function is a function on a locally compact group ''G'' with compact subgroup ''K'' (often a maximal ...
s for the
isometry groups of higher dimensional hyperbolic spaces. Harish Chandra's later development of the Plancherel theorem for general real
semisimple Lie group
In mathematics, a Lie algebra is semisimple if it is a direct sum of simple Lie algebras. (A simple Lie algebra is a non-abelian Lie algebra without any non-zero proper ideals).
Throughout the article, unless otherwise stated, a Lie algebra is ...
s was strongly influenced by the methods Weyl developed for eigenfunction expansions associated with singular ordinary differential equations. Equally importantly the theory also laid the mathematical foundations for the analysis of the
Schrödinger equation
The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of th ...
and
scattering matrix in
quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, q ...
.
Solutions of ordinary differential equations
Reduction to standard form
Let ''D'' be the second order differential operator on ''(a,b)'' given by
where ''p'' is a strictly positive continuously differentiable function and ''q'' and ''r'' are continuous
real-valued functions.
For ''x''
0 in (''a'', ''b''), define the
Liouville transformation ψ by
If
is the
unitary operator
In functional analysis, a unitary operator is a surjective bounded operator on a Hilbert space that preserves the inner product. Unitary operators are usually taken as operating ''on'' a Hilbert space, but the same notion serves to define the c ...
defined by
then
and
Hence,
where
and
The term in ''g' '' can be removed using an
Euler
Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...
integrating factor
In mathematics, an integrating factor is a function that is chosen to facilitate the solving of a given equation involving differentials. It is commonly used to solve ordinary differential equations, but is also used within multivariable calc ...
. If ''S' ''/''S'' = −''R''/2, then ''h'' = ''Sg'' satisfies
where the
potential
Potential generally refers to a currently unrealized ability. The term is used in a wide variety of fields, from physics to the social sciences to indicate things that are in a state where they are able to change in ways ranging from the simple r ...
''V'' is given by
The differential operator can thus always be reduced to one of the form
Existence theorem
The following is a version of the classical
Picard existence theorem for second order differential equations with values in a
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 ve ...
E.
Let α, β be arbitrary elements of E, ''A'' a
bounded operator
In functional analysis and operator theory, a bounded linear operator is a linear transformation L : X \to Y between topological vector spaces (TVSs) X and Y that maps bounded subsets of X to bounded subsets of Y.
If X and Y are normed vecto ...
on ''E'' and ''q'' a continuous function on
'a'', ''b''
Then, for ''c'' = ''a'' or ''b'', the differential equation
:''Df'' = ''Af''
has a unique solution ''f'' in ''C''
2(
'a'',''b''E) satisfying the initial conditions
:''f''(''c'') = β, ''f'' '(''c'') = α.
In fact a solution of the differential equation with these initial conditions is equivalent to a solution
of the
integral equation
In mathematics, integral equations are equations in which an unknown function appears under an integral sign. In mathematical notation, integral equations may thus be expressed as being of the form: f(x_1,x_2,x_3,...,x_n ; u(x_1,x_2,x_3,...,x_n ...
:''f'' = ''h'' + ''T'' ''f''
with ''T'' the bounded linear map on ''C''(
'a'',''b'' E) defined by
where ''K'' is the
Volterra kernel
:''K''(''x'',''t'')= (''x'' − ''t'')(''q''(''t'') − ''A'')
and
:''h''(''x'') = α(''x'' − ''c'') + β.
Since , , ''T''
k, , tends to 0, this integral equation has a unique solution given by the
Neumann series
:''f'' = (''I'' − ''T'')
−1 ''h'' = ''h'' + ''T'' ''h'' + ''T''
2 ''h'' + ''T''
3 ''h'' + ⋯
This iterative scheme is often called ''Picard iteration'' after the French mathematician
Charles Émile Picard.
Fundamental eigenfunctions
If ''f'' is twice continuously differentiable (i.e. ''C''
2) on (''a'', ''b'') satisfying ''Df'' = λ''f'', then ''f'' is called an
eigenfunction
In mathematics, an eigenfunction of a linear operator ''D'' defined on some function space is any non-zero function f in that space that, when acted upon by ''D'', is only multiplied by some scaling factor called an eigenvalue. As an equation, ...
of ''L'' with
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 denot ...
λ.
* In the case of a compact interval
'a'', ''b''and ''q'' continuous on
'a'', ''b'' the existence theorem implies that for ''c'' = ''a'' or ''b'' and every complex number λ there a unique ''C''
2 eigenfunction ''f''
λ on
'a'', ''b''with ''f''
λ(c) and ''f'' '
λ(c) prescribed. Moreover, for each ''x'' in
'a'', ''b'' ''f''
λ(x) and ''f'' '
λ(x) are
holomorphic function
In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex de ...
s of λ.
* For an arbitrary interval (''a'',''b'') and ''q'' continuous on (''a'', ''b''), the existence theorem implies that for ''c'' in (''a'', ''b'') and every complex number λ there a unique ''C''
2 eigenfunction ''f''
λ on (''a'', ''b'') with ''f''
λ(c) and ''f'' '
λ(c) prescribed. Moreover, for each ''x'' in (''a'', ''b''), ''f''
λ(x) and ''f'' '
λ(x) are
holomorphic function
In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex de ...
s of λ.
Green's formula
If ''f'' and ''g'' are ''C''
2 functions on (''a'', ''b''), the
Wronskian ''W''(''f'', ''g'') is defined by
:''W''(''f'', ''g'') (x) = ''f''(''x'') ''g'' '(''x'') − ''f'' '(''x'') ''g''(''x'').
Green's formula - which in this one-dimensional case is a simple integration by parts - states that for ''x'', ''y'' in (''a'', ''b'')
When ''q'' is continuous and ''f'', ''g'' ''C''
2 on the compact interval
'a'', ''b'' this formula also holds for ''x'' = ''a'' or ''y'' = ''b''.
When ''f'' and ''g'' are eigenfunctions for the same eigenvalue, then
so that ''W''(''f'', ''g'') is independent of ''x''.
Classical Sturm–Liouville theory
Let
'a'', ''b''be a finite closed interval, ''q'' a real-valued continuous function on
'a'', ''b''and let ''H''
0 be the
space of C
2 functions ''f'' on
'a'', ''b''satisfying the
Robin boundary conditions
with
inner product
In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
In practise usually one of the two standard boundary conditions:
*
Dirichlet boundary condition
In the mathematical study of differential equations, the Dirichlet (or first-type) boundary condition is a type of boundary condition, named after Peter Gustav Lejeune Dirichlet (1805–1859). When imposed on an ordinary or a partial differenti ...
''f''(''c'') = 0
*
Neumann boundary condition
In mathematics, the Neumann (or second-type) boundary condition is a type of boundary condition, named after Carl Neumann.
When imposed on an ordinary or a partial differential equation, the condition specifies the values of the derivative appli ...
''f'' '(''c'') = 0
is imposed at each endpoint ''c'' = ''a'', ''b''.
The differential operator ''D'' given by
acts on ''H''
0. A function ''f'' in ''H''
0 is called an
eigenfunction
In mathematics, an eigenfunction of a linear operator ''D'' defined on some function space is any non-zero function f in that space that, when acted upon by ''D'', is only multiplied by some scaling factor called an eigenvalue. As an equation, ...
of ''D'' (for the above choice of boundary values) if ''Df'' = λ ''f'' for some complex number λ, the corresponding
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 denot ...
.
By Green's formula, ''D'' is formally
self-adjoint
In mathematics, and more specifically in abstract algebra, an element ''x'' of a *-algebra is self-adjoint if x^*=x. A self-adjoint element is also Hermitian, though the reverse doesn't necessarily hold.
A collection ''C'' of elements of a sta ...
on ''H''
0, since the Wronskian ''W(f,g)'' vanishes if both ''f,g'' satisfy the boundary conditions:
:(''Df'', ''g'') = (''f'', ''Dg'') for ''f'', ''g'' in ''H''
0.
As a consequence, exactly as for a
self-adjoint matrix in finite dimensions,
*the eigenvalues of ''D'' are real;
*the
eigenspace
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 ...
s for distinct eigenvalues are
orthogonal
In mathematics, orthogonality is the generalization of the geometric notion of '' perpendicularity''.
By extension, orthogonality is also used to refer to the separation of specific features of a system. The term also has specialized meanings in ...
.
It turns out that the eigenvalues can be described by the
maximum-minimum principle of
Rayleigh Rayleigh may refer to:
Science
*Rayleigh scattering
*Rayleigh–Jeans law
*Rayleigh waves
*Rayleigh (unit), a unit of photon flux named after the 4th Baron Rayleigh
*Rayl, rayl or Rayleigh, two units of specific acoustic impedance and characte ...
–
Ritz
Ritz or The Ritz may refer to:
Facilities and structures Hotels
* The Ritz Hotel, London, a hotel in London, England
* Hôtel Ritz Paris, a hotel in Paris, France
* Hotel Ritz (Madrid), a hotel in Madrid, Spain
* Hotel Ritz (Lisbon), a hotel in ...
(see below). In fact it is easy to see ''a priori'' that the eigenvalues are bounded below because the operator ''D'' is itself ''bounded below'' on ''H''
0:
:*
for some finite (possibly negative) constant
.
In fact integrating by parts
For Dirichlet or Neumann boundary conditions, the first term vanishes and the inequality holds with ''M'' = inf ''q''.
For general Robin boundary conditions the first term can be estimated using an elementary ''Peter-Paul'' version of
Sobolev's inequality:
:: "''Given ε > 0, there is constant R >0 such that , f(x), ''
2 ≤ ε ''(f', f') + R (f, f) for all f in C
1, b
The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline o ...
''"
In fact, since
::, ''f''(''b'') − ''f''(''x''), ≤ (''b'' − ''a'')
1/2·, , ''f'' ', ,
2,
only an estimate for ''f''(''b'') is needed and this follows by replacing ''f''(''x'') in the above inequality by (''x'' − ''a'')
''n''·(''b'' − ''a'')
−''n''·''f''(''x'') for ''n'' sufficiently large.
Green's function (regular case)
From the theory of ordinary differential equations, there are unique fundamental eigenfunctions φ
λ(x), χ
λ(x) such that
* ''D'' φ
λ = λ φ
λ, φ
λ(''a'') = sin α, φ
λ'(''a'') = cos α
* ''D'' χ
λ = λ χ
λ, χ
λ(''b'') = sin β, χ
λ'(''b'') = cos β
which at each point, together with their first derivatives, depend holomorphically on λ. Let
:ω(λ) = W(φ
λ, χ
λ),
be an
entire holomorphic function.
This function ω(λ) plays the rôle of the
characteristic polynomial
In linear algebra, the characteristic polynomial of a square matrix is a polynomial which is invariant under matrix similarity and has the eigenvalues as roots. It has the determinant and the trace of the matrix among its coefficients. The ...
of ''D''. Indeed, the uniqueness of the fundamental eigenfunctions implies that its zeros are precisely the eigenvalues of ''D'' and that each non-zero eigenspace is one-dimensional. In particular there are at most countably many eigenvalues of ''D'' and, if there are infinitely many, they must tend to infinity. It turns out that the zeros of ω(λ) also have mutilplicity one (see below).
If λ is not an eigenvalue of ''D'' on ''H''
0, define the
Green's function
In mathematics, a Green's function is the impulse response of an inhomogeneous linear differential operator defined on a domain with specified initial conditions or boundary conditions.
This means that if \operatorname is the linear differenti ...
by
:''G''
λ(''x'',''y'') = φ
λ (''x'') χ
λ(''y'') / ω(λ) for ''x'' ≥ ''y'' and χ
λ(''x'') φ
λ (''y'') / ω(λ) for ''y'' ≥ ''x''.
This kernel defines an operator on the inner product space C
'a'',''b''via
Since ''G''
λ(''x'',''y'') is continuous on
'a'', ''b''x
'a'', ''b'' it defines a
Hilbert–Schmidt operator
In mathematics, a Hilbert–Schmidt operator, named after David Hilbert and Erhard Schmidt, is a bounded operator A \colon H \to H that acts on a Hilbert space H and has finite Hilbert–Schmidt norm
\, A\, ^2_ \ \stackrel\ \sum_ \, Ae_i\, ^2_ ...
on the Hilbert space completion
''H'' of C
'a'', ''b''= ''H''
1 (or equivalently of the dense subspace ''H''
0), taking values in ''H''
1. This operator carries ''H''
1 into ''H''
0. When λ is real, ''G''
λ(''x'',''y'') = ''G''
λ(''y'',''x'') is also real, so defines a self-adjoint operator on ''H''. Moreover,
* ''G''
λ (''D'' − λ) =I on ''H''
0
* ''G''
λ carries ''H''
1 into ''H''
0, and (''D'' − λ) ''G''
λ = I on ''H''
1.
Thus the operator ''G''
λ can be identified with the
resolvent (''D'' − λ)
−1.
Spectral theorem
Theorem. ''The eigenvalues of D are real of multiplicity one and form an increasing sequence λ
1 < λ
2 < ··· tending to infinity.''
''The corresponding normalised eigenfunctions form an orthonormal basis of'' ''H''
0.
''The kth eigenvalue of D is given by the
minimax principle''
''In particular if q
1 ≤ q
2, then''
In fact let ''T'' = ''G''
λ for λ large and negative. Then ''T'' defines a
compact self-adjoint operator on the Hilbert space ''H''.
By the
spectral theorem
In mathematics, particularly linear algebra and functional analysis, a spectral theorem is a result about when a linear operator or matrix can be diagonalized (that is, represented as a diagonal matrix in some basis). This is extremely useful b ...
for compact self-adjoint operators, ''H'' has an orthonormal basis consisting of eigenvectors ψ
''n'' of ''T'' with
''T''ψ
''n'' = μ
''n'' ψ
''n'', where μ
''n'' tends to zero. The range of ''T'' contains ''H''
0 so is dense. Hence 0 is not an eigenvalue of ''T''. The resolvent properties of ''T'' imply that ψ
''n'' lies in ''H''
0 and that
:''D'' ψ
''n'' = (λ + 1/μ
''n'') ψ
''n''
The minimax principle follows because if
then λ(''G'')= λ
k for the
linear span
In mathematics, the linear span (also called the linear hull or just span) of a set of vectors (from a vector space), denoted , pp. 29-30, §§ 2.5, 2.8 is defined as the set of all linear combinations of the vectors in . It can be characteri ...
of the first ''k'' − 1 eigenfunctions. For any other (''k'' − 1)-dimensional subspace ''G'', some ''f'' in the linear span of the first ''k'' eigenvectors must be orthogonal to ''G''. Hence λ(''G'') ≤ (''Df'',''f'')/(''f'',''f'') ≤ λ
k.
Wronskian as a Fredholm determinant
For simplicity, suppose that ''m'' ≤ ''q''(''x'') ≤ ''M'' on
,πwith Dirichlet boundary conditions.
The minimax principle shows that
It follows that the resolvent (''D'' − λ)
−1 is a
trace-class operator In mathematics, specifically functional analysis, a trace-class operator is a linear operator for which a trace may be defined, such that the trace is a finite number independent of the choice of basis used to compute the trace. This trace of trace- ...
whenever λ is not an eigenvalue of ''D'' and hence that the
Fredholm determinant det I − μ(''D'' − λ)
−1 is defined.
The Dirichlet boundary conditions imply that
:ω(λ)= φ
λ(''b'').
Using Picard iteration, Titchmarsh showed that φ
λ(''b''), and hence ω(λ), is an
entire function of finite order 1/2:
:ω(λ) = O(e
)
At a zero μ of ω(λ), φ
μ(''b'') = 0. Moreover,
satisfies (''D'' − μ)ψ = φ
μ. Thus
:ω(λ) = (λ − μ)ψ(''b'') + O( (λ − μ)
2).
This implies that
* μ is a simple zero of ω(λ).
For otherwise ψ(''b'') = 0, so that ψ would have to lie in ''H''
0.
But then
:(φ
μ, φ
μ) = ((''D'' − μ)ψ, φ
μ) = (ψ, (''D'' − μ)φ
μ) = 0,
a contradiction.
On the other hand, the distribution of the zeros of the entire function
ω(λ) is already known from the minimax principle.
By the
Hadamard factorization theorem In mathematics, and particularly in the field of complex analysis, the Weierstrass factorization theorem asserts that every entire function can be represented as a (possibly infinite) product involving its zeroes. The theorem may be viewed as an ex ...
, it follows
that
*
for some non-zero constant ''C''.
Hence
In particular if 0 is not an eigenvalue of ''D''
Tools from abstract spectral theory
Functions of bounded variation
A function ρ(''x'') of
bounded variation
In mathematical analysis, a function of bounded variation, also known as ' function, is a real number, real-valued function (mathematics), function whose total variation is bounded (finite): the graph of a function having this property is well beh ...
on a closed interval
'a'', ''b''is a complex-valued function such that
its
total variation ''V''(ρ), the
supremum
In mathematics, the infimum (abbreviated inf; plural infima) of a subset S of a partially ordered set P is a greatest element in P that is less than or equal to each element of S, if such an element exists. Consequently, the term ''greatest l ...
of the variations
over all
dissection
Dissection (from Latin ' "to cut to pieces"; also called anatomization) is the dismembering of the body of a deceased animal or plant to study its anatomical structure. Autopsy is used in pathology and forensic medicine to determine the cause ...
s
is finite. The real and imaginary parts of ρ are real-valued functions of bounded variation. If ρ is real-valued and normalised so that ρ(a)=0, it has a canonical decomposition as the difference of two bounded non-decreasing functions:
where ρ
+(''x'') and ρ
–(''x'') are the total positive and negative variation of ρ over
'a'', ''x''
If ''f'' is a continuous function on
'a'', ''b''its
Riemann–Stieltjes integral
In mathematics, the Riemann–Stieltjes integral is a generalization of the Riemann integral, named after Bernhard Riemann and Thomas Joannes Stieltjes. The definition of this integral was first published in 1894 by Stieltjes. It serves as an i ...
with respect to ρ
is defined to be the limit of approximating sums
as the
mesh
A mesh is a barrier made of connected strands of metal, fiber, or other flexible or ductile materials. A mesh is similar to a web or a net in that it has many attached or woven strands.
Types
* A plastic mesh may be extruded, oriented, e ...
of the dissection, given by sup , ''x''
''r''+1 - ''x''
''r'', , tends to zero.
This integral satisfies
and thus defines a
bounded linear functional ''d''ρ on ''C''
'a'', ''b''with
norm , , d''ρ, , =''V''(ρ).
Every bounded linear functional μ on ''C''
'a'', ''b''has an
absolute value , μ, defined for non-negative ''f'' by
The form , μ, extends linearly to a bounded linear form on C
'a'', ''b''with norm , , μ, , and satisfies the characterizing inequality
:, μ(''f''), ≤ , μ, (, ''f'', )
for ''f'' in C
'a'', ''b'' If μ is ''real'', i.e. is real-valued on real-valued functions, then
gives a canonical decomposition as a difference of ''positive'' forms, i.e. forms that are non-negative on non-negative functions.
Every positive form μ extends uniquely to the linear span of non-negative bounded lower
semicontinuous functions ''g'' by the formula
where the non-negative continuous functions ''f''
''n'' increase pointwise to ''g''.
The same therefore applies to an arbitrary bounded linear form μ, so that a function ρ of bounded variation may be defined by
where χ
''A'' denotes the
characteristic function In mathematics, the term "characteristic function" can refer to any of several distinct concepts:
* The indicator function of a subset, that is the function
::\mathbf_A\colon X \to \,
:which for a given subset ''A'' of ''X'', has value 1 at point ...
of a subset ''A'' of
'a'', ''b'' Thus μ = ''d''ρ and , , μ, , = , , ''d''ρ, , .
Moreover μ
+ = ''d''ρ
+ and μ
– = ''d''ρ
–.
This correspondence between functions of bounded variation and bounded linear forms is a special case of the
Riesz representation theorem
:''This article describes a theorem concerning the dual of a Hilbert space. For the theorems relating linear functionals to measures, see Riesz–Markov–Kakutani representation theorem.''
The Riesz representation theorem, sometimes called the ...
.
The
support of μ = ''d''ρ is the complement of all points ''x'' in
'a'',''b''where ρ is constant on some neighborhood of ''x''; by definition it is a closed subset ''A'' of
'a'',''b'' Moreover, μ((1-χ
''A'')''f'') =0, so that μ(''f'') = 0 if ''f'' vanishes on ''A''.
Spectral measure
Let ''H'' be a Hilbert space and
a self-adjoint
bounded operator
In functional analysis and operator theory, a bounded linear operator is a linear transformation L : X \to Y between topological vector spaces (TVSs) X and Y that maps bounded subsets of X to bounded subsets of Y.
If X and Y are normed vecto ...
on ''H'' with
, so that the
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 color ...
of
is contained in