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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 ...
, 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 colors ...
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, th ...
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 contrast ...
. 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 asso ...
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. Other ...
to second order
differential operator In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a function and retur ...
s 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, given by the Titchmarsh
Kodaira is a city located in the western portion of Tokyo Metropolis, Japan. , the city had an estimated population of 195,207 in 93,654 households, and a population density of 9500 persons per km². The total area of the city was . Geography Kodaira ...
formula In science, a formula is a concise way of expressing information symbolically, as in a mathematical formula or a ''chemical formula''. The informal use of the term ''formula'' in science refers to the general construct of a relationship betwe ...
. The theory was put in its final simplified form for singular differential equations of even degree by Kodaira and others, using
von Neumann Von Neumann may refer to: * John von Neumann (1903–1957), a Hungarian American mathematician * Von Neumann family * Von Neumann (surname), a German surname * Von Neumann (crater), a lunar impact crater See also * Von Neumann algebra * Von Ne ...
'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 be ...
. 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, ...
,
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 operators ...
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 ex ...
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 be ...
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 c ...
s due to
David Hilbert David Hilbert (; ; 23 January 1862 – 14 February 1943) was a German mathematician, one of the most influential mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many ...
. 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 asso ...
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 th ...
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 c ...
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 its ...
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 Japanese ...
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. The same formula had also been obtained independently by E. C. Titchmarsh in 1946 (scientific communication between
Japan Japan ( ja, 日本, or , and formally , ''Nihonkoku'') is an island country in East Asia. It is situated in the northwest Pacific Ocean, and is bordered on the west by the Sea of Japan, while extending from the Sea of Okhotsk in the n ...
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 continental mainland. It comprises England, Scotland, Wales and ...
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 ''E ...
, 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 algebraic ...
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 operators ...
. 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 Britis ...
resolvents corresponding to Sturm–Liouville problems for proper subintervals. Another method was found by Mark Grigoryevich Krein; his use of ''direction functionals'' was subsequently generalised by Izrail Glazman 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 refer ...
'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. The ...
. 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 wit ...
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 wa ...
and GelfandNaimark can be deduced from Weyl's theory for the hypergeometric equation, as can the theory of spherical functions for the
isometry group In mathematics, the isometry group of a metric space is the set of all bijective isometries (i.e. bijective, distance-preserving maps) from the metric space onto itself, with the function composition as group operation. Its identity element is the ...
s 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 physics, the ''S''-matrix or scattering matrix relates the initial state and the final state of a physical system undergoing a scattering process. It is used in quantum mechanics, scattering theory and quantum field theory (QFT). More forma ...
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, ...
.


Solutions of ordinary differential equations


Reduction to standard form

Let ''D'' be the second order differential operator on ''(a,b)'' given by Df(x) = -p(x) f''(x) +r(x) f'(x) + q(x) f(x), 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 \psi(x)= \int_^x p(t)^\, dt If U: L^2(a, b) \mapsto L^2(\psi(a), \psi(b)) 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 co ...
defined by (Uf)(\psi(x)) = f(x) \times \left(\psi'(x)\right)^,\ \ \forall x \in (a, b) then U \frac U^ g = g' \psi' + \frac 12 g \frac and \begin U \frac U^ g & = \left( U \frac U^ \right) \times \left( U \frac U^ \right) g \\ & = \frac \left g' \psi' + \frac 12 g \frac \right\cdot \psi' + \frac 12 \left g' \psi' + \frac 12 g \frac \right\cdot \frac \\ & = g'' \psi'^2 + 2 g' \psi'' + \frac 12 g \cdot \left \frac - \frac 12 \frac \right\end Hence, UDU^ g= -g'' + R g' + Q g, where R = \frac and Q = q - \frac + \frac 4 - \frac 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 ...
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 calcul ...
. If ''S' ''/''S'' = −''R''/2, then ''h'' = ''Sg'' satisfies (S UDU^ S^) h = -h^ + V h, 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 V = Q + \frac The differential operator can thus always be reduced to one of the form Df = - f'' + qf.


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 vector ...
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 vector ...
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 :''f'' = ''h'' + ''T'' ''f'' with ''T'' the bounded linear map on ''C''( 'a'',''b'' E) defined by Tf(x) = \int_c^x K(x,y)f(y) \, dy, 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 A Neumann series is a mathematical series of the form : \sum_^\infty T^k where T is an operator and T^k := T^\circ its k times repeated application. This generalizes the geometric series. The series is named after the mathematician Carl Neuman ...
:''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 Charles is a masculine given name predominantly found in English and French speaking countries. It is from the French form ''Charles'' of the Proto-Germanic name (in runic alphabet) or ''*karilaz'' (in Latin alphabet), whose meaning was ...
.


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, th ...
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 denote ...
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 deriv ...
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 deriv ...
s of λ.


Green's formula

If ''f'' and ''g'' are ''C''2 functions on (''a'', ''b''), the
Wronskian In mathematics, the Wronskian (or Wrońskian) is a determinant introduced by and named by . It is used in the study of differential equations, where it can sometimes show linear independence in a set of solutions. Definition The Wronskian o ...
''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'') \int_x^y (Df) g - f (Dg) \, dt = W(f,g)(y) - W(f,g)(x). 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 W(f,g) =0, 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 C2 functions ''f'' on 'a'', ''b''satisfying the Robin boundary conditions \cos \alpha \,f(a) - \sin \alpha \,f^\prime(a)=0, \qquad \cos\beta \,f(b) - \sin \beta\, f^\prime(b)=0, 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 ...
(f,g) = \int_a^b f(x) \overline \, dx. 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 differential ...
''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 Df=-f^ + qf 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, th ...
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 denote ...
. 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 st ...
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 denote ...
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 RayleighRitz (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: :*(Df, f) \ge M (f, f) for some finite (possibly negative) constant M. In fact integrating by parts (Df,f)= f^\prime \overlinea^b + \int , f^\prime, ^2 + \int q , f, ^2. 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 C1
, 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 φλ(x), χλ(x) such that * ''D'' φλ = λ φλ, φλ(''a'') = sin α, φλ'(''a'') = cos α * ''D'' χλ = λ χλ, χλ(''b'') = sin β, χλ'(''b'') = cos β which at each point, together with their first derivatives, depend holomorphically on λ. Let :ω(λ) = W(φλ, χλ), 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 c ...
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'') = φλ (''x'') χλ(''y'') / ω(λ) for ''x'' ≥ ''y'' and χλ(''x'') φλ (''y'') / ω(λ) for ''y'' ≥ ''x''. This kernel defines an operator on the inner product space C 'a'',''b''via (G_\lambda f)(x) =\int_a^b G_\lambda(x,y) f(y)\, dy. 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\, ...
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 λ1 < λ2 < ··· 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 Minimax (sometimes MinMax, MM or saddle point) is a decision rule used in artificial intelligence, decision theory, game theory, statistics, and philosophy for ''mini''mizing the possible loss for a worst case (''max''imum loss) scenario. When ...
'' \lambda_k = \max_ \, \min_ . ''In particular if q1 ≤ q2, then'' \lambda_k(D_1) \le \lambda_k(D_2). 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 be ...
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 \lambda(G) = \min_ , then λ(''G'')= λk 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 characterized ...
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'') ≤ λk.


Wronskian as a Fredholm determinant

For simplicity, suppose that ''m'' ≤ ''q''(''x'') ≤ ''M'' on with Dirichlet boundary conditions. The minimax principle shows that n^2 + m \le \lambda_n(D) \le n^2 + M. It follows that the resolvent (''D'' − λ)−1 is a trace-class operator whenever λ is not an eigenvalue of ''D'' and hence that the
Fredholm determinant In mathematics, the Fredholm determinant is a complex-valued function which generalizes the determinant of a finite dimensional linear operator. It is defined for bounded operators on a Hilbert space which differ from the identity operator by a tra ...
det I − μ(''D'' − λ)−1 is defined. The Dirichlet boundary conditions imply that :ω(λ)= φλ(''b''). Using Picard iteration, Titchmarsh showed that φλ(''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), φμ(''b'') = 0. Moreover, \psi(x)=\partial_\lambda \varphi_\lambda(x), _ satisfies (''D'' − μ)ψ = φμ. 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 :(φμ, φμ) = ((''D'' − μ)ψ, φμ) = (ψ, (''D'' − μ)φμ) = 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, it follows that * \omega(\lambda) = C \prod (1 -\lambda/\lambda_n), for some non-zero constant ''C''. Hence \det ( I - \mu(D - \lambda)^) = \prod \left( 1 - \right) = \prod = . In particular if 0 is not an eigenvalue of ''D'' \omega(\mu) = \omega(0) \cdot \det ( I - \mu D^) .


Tools from abstract spectral theory


Functions of bounded variation

A function ρ(''x'') of bounded variation 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 ...
of the variations \sum_^ , \rho(x_) - \rho(x_r), 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 o ...
s a= x_0 < x_1 < \dots < x_k =b 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: \rho(x) = \rho_+(x) - \rho_-(x), 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 inst ...
with respect to ρ \int_a^b f(x)\, d\rho(x) is defined to be the limit of approximating sums \sum_^ f(x_r)(\rho(x_)-\rho(x_r)) 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, exp ...
of the dissection, given by sup , ''x''''r''+1 - ''x''''r'', , tends to zero. This integral satisfies \left, \int_a^b f(x)\, d\rho(x)\\le V(\rho)\cdot \, f\, _\infty and thus defines a
bounded linear functional 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 vector s ...
''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 In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), ...
, μ, defined for non-negative ''f'' by , \mu, (f) = \sup_ , \mu(g), . 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 \mu = , \mu, -(, \mu, -\mu)\equiv \mu_+-\mu_- 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 \mu(g) = \lim \mu(f_n), 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 \rho(x)=\mu(\chi_), 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 points ...
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 Measure (mathematics), measures, see Riesz–Markov–Kakutani representation theorem.'' The Riesz representation theorem, ...
. The
support Support may refer to: Arts, entertainment, and media * Supporting character Business and finance * Support (technical analysis) * Child support * Customer support * Income Support Construction * Support (structure), or lateral support, a ...
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 T 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 vector ...
on ''H'' with 0 \leq T \leq I , 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 colors ...
\sigma(T) of T is contained in ,1/math>. If p(t) is a complex polynomial, then by the
spectral mapping theorem In mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A over the real or complex numbers (or over a non-Archimedean complete normed field) that at the same time is also a Banach ...
\sigma (p(T)) = p (\sigma(T)) and hence \, p(T)\, \leq \, p\, _\infty where \, \, \, _\infty denotes the
uniform norm In mathematical analysis, the uniform norm (or ) assigns to real- or complex-valued bounded functions defined on a set the non-negative number :\, f\, _\infty = \, f\, _ = \sup\left\. This norm is also called the , the , the , or, when th ...
on C ,1/math>. By the Weierstrass approximation theorem, polynomials are uniformly dense in C ,1/math>. It follows that f(T) can be defined \forall f \in C ,1/math>, with \sigma (f(T)) = f(\sigma(T)) and \, f(T) \, \leq \, f\, _\infty. If 0 \leq g \leq 1 is a lower semicontinuous function on ,1/math>, for example the characteristic function \chi_ of a subinterval of ,1/math>, then g is a pointwise increasing limit of non-negative f_n \in C ,1/math>. According to Szőkefalvi-Nagy, if \xi is a vector in ''H'', then the vectors \eta_n=f_n(T)\xi form a
Cauchy sequence In mathematics, a Cauchy sequence (; ), named after Augustin-Louis Cauchy, is a sequence whose elements become arbitrarily close to each other as the sequence progresses. More precisely, given any small positive distance, all but a finite numbe ...
in ''H'', since, for n \geq m, \, \eta_n-\eta_m\, ^2 \le (\eta_n,\xi) - (\eta_m,\xi), and ( \eta_n, \xi) = (f_n(T)\xi, \xi) is bounded and increasing, so has a limit. It follows that g(T) can be defined by g(T)\xi= \lim f_n(T)\xi. If \xi and \eta are vectors in ''H'', then \mu_(f) = (f(T) \xi,\eta) defines a bounded linear form \mu_ on ''H''. By the Riesz representation theorem \mu_=d\rho_ for a unique normalised function \rho_ of bounded variation on ,1/math>. d\rho_ (or sometimes slightly incorrectly \rho_ itself) is called the spectral measure determined by \xi and \eta. The operator g(T) is accordingly uniquely characterised by the equation (g(T)\xi,\eta) = \mu_(g) = \int_0^1 g(\lambda) \, d\rho_(\lambda). The spectral projection E(\lambda) is defined by E(\lambda)=\chi_(T), so that \rho_(\lambda)=(E(\lambda)\xi,\eta). It follows that g(T) = \int_0^1 g(\lambda) \,dE(\lambda), which is understood in the sense that for any vectors \xi and \eta, (g(T)\xi,\eta) = \int_0^1 g(\lambda)\, d(E(\lambda)\xi,\eta) = \int_0^1 g(\lambda)\, d\rho_(\lambda). For a single vector \xi, \, \mu_ = \mu_ is a positive form on ,1/math> (in other words proportional to a
probability measure In mathematics, a probability measure is a real-valued function defined on a set of events in a probability space that satisfies measure properties such as ''countable additivity''. The difference between a probability measure and the more ge ...
on ,1/math>) and \rho_ = \rho_ is non-negative and non-decreasing. Polarisation shows that all the forms \mu_ can naturally be expressed in terms of such positive forms, since \mu_ = \frac\bigg(\mu_+i\mu_-\mu_-i\mu_\bigg) If the vector \xi is such that 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 characterized ...
of the vectors (T^n\xi) is dense in ''H'', i.e. \xi is a ''cyclic vector'' for T, then the map U defined by U(f) = f(T)\xi, \, C ,1\rightarrow H satisfies (Uf_1,Uf_2)= \int_0^1 f_1(\lambda) \overline \, d\rho_\xi(\lambda). Let L_2( ,1 d\rho_\xi) denote the Hilbert space completion of C ,1/math> associated with the possibly degenerate inner product on the right hand side. Thus U extends to a
unitary transformation In mathematics, a unitary transformation is a transformation that preserves the inner product: the inner product of two vectors before the transformation is equal to their inner product after the transformation. Formal definition More precisely, ...
of L_2( ,1 \rho_\xi) onto ''H''. UTU^\ast is then just multiplication by \lambda on L_2( ,1 d\rho_\xi); and more generally Uf(T)U^\ast is multiplication by f(\lambda). In this case, the support of d\rho_\xi is exactly \sigma(T), so that * ''the self-adjoint operator becomes a multiplication operator on the space of functions on its spectrum with inner product given by the spectral measure''.


Weyl–Titchmarsh–Kodaira theory

The eigenfunction expansion associated with singular differential operators of the form Df = -(pf')' + qf on an open interval (''a'', ''b'') requires an initial analysis of the behaviour of the fundamental eigenfunctions near the endpoints ''a'' and ''b'' to determine possible
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 th ...
s there. Unlike the regular Sturm–Liouville case, in some circumstances spectral values of ''D'' can have multiplicity 2. In the development outlined below standard assumptions will be imposed on ''p'' and ''q'' that guarantee that the spectrum of ''D'' has multiplicity one everywhere and is bounded below. This includes almost all important applications; modifications required for the more general case will be discussed later. Having chosen the boundary conditions, as in the classical theory the resolvent of ''D'', (''D'' + ''R'' )−1 for ''R '' large and positive, is given by an operator ''T'' corresponding to a Green's function constructed from two fundamental eigenfunctions. In the classical case ''T'' was a compact self-adjoint operator; in this case ''T'' is just a self-adjoint bounded operator with 0 ≤ ''T'' ≤ I. The abstract theory of spectral measure can therefore be applied to ''T'' to give the eigenfunction expansion for ''D''. The central idea in the proof of Weyl and Kodaira can be explained informally as follows. Assume that the spectrum of ''D'' lies in _E(\lambda)_=\chi_(T) be_the_spectral_projection_of_''D''_corresponding_to_the_interval_[1,λ.html" ;"title=",∞) and that ''T'' =''D''−1 and let E(\lambda) =\chi_(T) be the spectral projection of ''D'' corresponding to the interval [1,λ">,∞) and that ''T'' =''D''−1 and let E(\lambda) =\chi_(T) be the spectral projection of ''D'' corresponding to the interval [1,λ For an arbitrary function ''f'' define f(x,\lambda)= (E(\lambda)f)(x). ''f''(''x'',λ) may be regarded as a differentiable map into the space of functions of bounded variation ρ; or equivalently as a differentiable map x\mapsto (d_\lambda f)(x) into the Banach space E of bounded linear functionals ''d''ρ on C �,βwhenever �,βis a compact subinterval of [1, ∞). Weyl's fundamental observation was that ''d''λ ''f'' satisfies a second order ordinary differential equation taking values in E: : D (d_\lambda f) = \lambda \cdot d_\lambda f. After imposing initial conditions on the first two derivatives at a fixed point ''c'', this equation can be solved explicitly in terms of the two fundamental eigenfunctions and the "initial value" functionals (d_\lambda f)(c)= d_\lambda f(c,\cdot), \quad (d_\lambda f)^\prime(c)= d_\lambda f_x(c,\cdot). This point of view may now be turned on its head: ''f''(''c'',λ) and ''f''''x''(''c'',λ) may be written as f(c,\lambda)=(f,\xi_1(\lambda)), \quad f_x(c,\lambda)=(f,\xi_2(\lambda)), where ξ1(λ) and ξ2(λ) are given purely in terms of the fundamental eigenfunctions. The functions of bounded variation \sigma_(\lambda) = (\xi_i(\lambda),\xi_j(\lambda)) determine a spectral measure on the spectrum of ''D'' and can be computed explicitly from the behaviour of the fundamental eigenfunctions (the Titchmarsh–Kodaira formula).


Limit circle and limit point for singular equations

Let ''q''(''x'') be a continuous real-valued function on (0,∞) and let ''D'' be the second order differential operator Df= -f'' + qf on (0,∞). Fix a point ''c'' in (0,∞) and, for λ complex, let \varphi_\lambda, \theta_\lambda be the unique fundamental eigenfunctions of ''D'' on (0,∞) satisfying (D-\lambda)\varphi_\lambda = 0, \quad (D-\lambda)\theta_\lambda =0 together with the initial conditions at ''c'' \varphi_\lambda(c)=1,\, \varphi_\lambda'(c)=0, \, \theta_\lambda(c)=0, \, \theta_\lambda'(c)=1. Then their Wronskian satisfies W(\varphi_\lambda,\theta_\lambda) = \varphi_\lambda\theta_\lambda'- \theta_\lambda \varphi_\lambda' \equiv 1, since it is constant and equal to 1 at ''c''. Let λ be non-real and 0 < ''x'' < ∞. If the complex number \mu is such that f=\varphi +\mu \theta satisfies the boundary condition \cos\beta\, f(x) - \sin\beta\, f'(x) = 0 for some \beta (or, equivalently, f'(x) / f(x) is real) then, using integration by parts, one obtains (\lambda) \int_c^x , \varphi +\mu \theta, ^2 =(\mu). Therefore, the set of \mu satisfying this equation is not empty. This set is a
circle A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is cons ...
in the complex \mu-plane. Points \mu in its interior are characterized by \int_c^x , \varphi +\mu \theta, ^2 < if ''x'' > ''c'' and by \int_x^c , \varphi +\mu \theta, ^2 < if ''x'' < ''c''. Let ''D''''x'' be the closed disc enclosed by the circle. By definition these closed discs are nested and decrease as ''x'' approaches 0 or ∞. So in the limit, the circles tend either to a limit circle or a limit point at each end. If \mu is a limit point or a point on the limit circle at 0 or ∞, then f=\varphi + \mu\theta is square integrable (L2) near 0 or ∞, since \mu lies in ''D''''x'' for all ''x>c'' (in the ∞ case) and so \int_c^x , \varphi +\mu \theta, ^2 < is bounded independent of ''x''. In particular:. * ''there are always non-zero solutions of Df = λf which are square integrable near 0 resp. ∞''; * ''in the limit circle case all solutions of Df = λf are square integrable near 0 resp. ∞''. The radius of the disc ''D''''x'' can be calculated to be : \left, \ and this implies that in the limit point case \theta cannot be square integrable near 0 resp. ∞. Therefore, we have a converse to the second statement above: * ''in the limit point case there is exactly one non-zero solution (up to scalar multiples) of Df = λf which is square integrable near 0 resp. ∞''. On the other hand, if ''Dg'' = λ' ''g'' for another value λ', then h(x) = g(x) -(\lambda^\prime-\lambda) \int_c^x (\varphi_\lambda(x) \theta_\lambda(y) - \theta_\lambda(x)\varphi_\lambda(y))g(y)\, dy satisfies ''Dh'' = λ''h'', so that g(x)=c_1 \varphi_\lambda + c_2 \theta_\lambda + (\lambda^\prime-\lambda) \int_c^x (\varphi_\lambda(x) \theta_\lambda(y) - \theta_\lambda(x)\varphi_\lambda(y))g(y)\, dy. This formula may also be obtained directly by the variation of constant method from (D-λ)g = (λ'-λ)g. Using this to estimate ''g'', it follows that * ''the limit point/limit circle behaviour at 0 or ∞ is independent of the choice of λ''. More generally if ''Dg''= (λ – ''r'') ''g'' for some function ''r''(''x''), then g(x)=c_1 \varphi_\lambda + c_2 \theta_\lambda - \int_c^x (\varphi_\lambda(x) \theta_\lambda(y) - \theta_\lambda(x)\varphi_\lambda(y))r(y)g(y)\, dy. From this it follows that * ''if r is continuous at 0, then D + r is limit point or limit circle at 0 precisely when D is'', so that in particular * ''if q(x)- a/x2 is continuous at 0, then D is limit point at 0 if and only if a ≥ ¾''. Similarly * ''if r has a finite limit at ∞, then D + r is limit point or limit circle at ∞ precisely when D is'', so that in particular * ''if q has a finite limit at ∞, then D is limit point at ∞''. Many more elaborate criteria to be limit point or limit circle can be found in the mathematical literature.


Green's function (singular case)

Consider the differential operator D_0 f = -(p_0f')' + q_0f on (0,∞) with ''q''0 positive and continuous on (0,∞) and ''p''0 continuously differentiable in [0,∞), positive in (0,∞) and ''p''0(0)=0. Moreover, assume that after reduction to standard form ''D''0 becomes the equivalent operator Df= -f'' + qf on (0,∞) where ''q'' has a finite limit at ∞. Thus *''D is limit point at ∞''. At 0, ''D'' may be either limit circle or limit point. In either case there is an eigenfunction Φ0 with ''D''Φ0=0 and Φ0 square integrable near 0. In the limit circle case, Φ0 determines a
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 th ...
at 0: W(f,\Phi_0)(0)=0. For λ complex, let Φλ and Χλ satisfy * (''D'' – λ)Φλ = 0, (''D'' – λ)Χλ = 0 * Χλ square integrable near infinity * Φλ square integrable at 0 if 0 is ''limit point'' * Φλ satisfies the boundary condition above if 0 is ''limit circle''. Let \omega(\lambda) = W(\Phi_\lambda,\Chi_\lambda), a constant which vanishes precisely when Φλ and Χλ are proportional, i.e. λ is an
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 denote ...
of ''D'' for these boundary conditions. On the other hand, this cannot occur if Im λ ≠ 0 or if λ is negative. Indeed, if ''D f''= λ''f'' with ''q''0 – λ ≥ δ >0, then by Green's formula (''Df'',''f'') = (''f'',''Df''), since ''W''(''f'',''f''*) is constant. So λ must be real. If ''f'' is taken to be real-valued in the ''D''0 realization, then for 0 < ''x'' < ''y'' [p_0 f f']_x^y = \int_x^y (q_0 -\lambda), f, ^2 + p_0 (f')^2 . Since ''p''0(0) = 0 and ''f'' is integrable near 0, ''p''0''f'' ''f'' ' must vanish at 0. Setting ''x'' = 0, it follows that ''f''(''y'') ''f'' '(''y'') >0, so that ''f''2 is increasing, contradicting the square integrability of ''f'' near ∞. Thus, adding a positive scalar to ''q'', it may be assumed that :''ω(λ) ≠ 0 when λ is not in [1,∞)''. If ω(λ) ≠ 0, 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 ...
''G''λ(''x'',''y'') at λ is defined by G_\lambda(x,y) = \Phi_\lambda(x)\Chi_\lambda(y)/\omega(\lambda) \,\, (x\le y), \,\,\,\, \Chi_\lambda(x)\Phi_\lambda(y)/\omega(\lambda) \,\, (x\ge y). and is independent of the choice of λ and Χλ. In the examples there will be a third "bad" eigenfunction Ψλ defined and holomorphic for λ not in [1, ∞) such that Ψλ satisfies the boundary conditions at neither 0 nor ∞. This means that for λ not in [1, ∞) * ''W''(Φλλ) is nowhere vanishing; * ''W''(Χλλ) is nowhere vanishing. In this case Χλ is proportional to Φλ + ''m''(λ) Ψλ, where * ''m''(λ) = – ''W''(Φλλ) / ''W''(Ψλλ). Let ''H''1 be the space of square integrable continuous functions on (0,∞) and let ''H''0 be * the space of C2 functions ''f'' on (0,∞) of compact support if ''D'' is limit point at 0 * the space of C2 functions ''f'' on (0,∞) with ''W''(''f'',Φ0)=0 at 0 and with ''f'' = 0 near ∞ if ''D'' is limit circle at 0. Define ''T'' = ''G''0 by (Tf)(x) =\int_0^\infty G_0(x,y)f(y) \, dy. Then ''T'' ''D'' = ''I'' on ''H''0, ''D'' ''T'' = ''I'' on ''H''1 and the operator ''D'' is bounded below on ''H''0: (Df,f) \ge (f,f). Thus ''T'' is a self-adjoint bounded operator with 0 ≤ ''T'' ≤ ''I''. Formally ''T'' = ''D''−1. The corresponding operators ''G''λ defined for λ not in (D-\lambda)^=T(I-\lambda_T)^ and_satisfy_''G''λ_(''D''_–_λ)_=_''I''_on_''H''0,_(''D''_–_λ)''G''λ_=_''I''_on_''H''1.


__Spectral_theorem_and_Titchmarsh–Kodaira_formula_

Theorem._''For_every_real_number_λ_let_ρ(λ)_be_defined_by_the''_Titchmarsh–Kodaira_formula: _\rho(\lambda)_=_\lim__\lim___\int_\delta^_\,_m(t_+_i\varepsilon)_\,_dt. ''Then_ρ(λ)_is_a_lower_semicontinuous_non-decreasing_function_of_λ_and_if'' _(Uf)(\lambda)_=_\int_0^\infty_f(x)_\Phi(x,\lambda)_\,_dx, ''then_U_defines_a_unitary_transformation_of_L2(0,∞)_onto_L2([1,∞),_dρ)_such_that''_UDU−1_''corresponds_to_multiplication_by_λ._'' ''The_inverse_transformation_U−1_is_given_by'' _(U^g)(x)_=_\int_1^\infty_g(\lambda)_\Phi(x,\lambda)_\,_d\rho(\lambda). ''The_spectrum_of_D_equals_the_support_of_dρ.'' Kodaira_gave_a_streamlined_version_of_Weyl's_original_proof._(Marshall_Harvey_Stone.html" ;"title=",∞) can be formally identified with (D-\lambda)^=T(I-\lambda T)^ and satisfy ''G''λ (''D'' – λ) = ''I'' on ''H''0, (''D'' – λ)''G''λ = ''I'' on ''H''1.


Spectral theorem and Titchmarsh–Kodaira formula

Theorem. ''For every real number λ let ρ(λ) be defined by the'' Titchmarsh–Kodaira formula: \rho(\lambda) = \lim_ \lim_ \int_\delta^ \, m(t + i\varepsilon) \, dt. ''Then ρ(λ) is a lower semicontinuous non-decreasing function of λ and if'' (Uf)(\lambda) = \int_0^\infty f(x) \Phi(x,\lambda) \, dx, ''then U defines a unitary transformation of L2(0,∞) onto L2([1,∞), dρ) such that'' UDU−1 ''corresponds to multiplication by λ. '' ''The inverse transformation U−1 is given by'' (U^g)(x) = \int_1^\infty g(\lambda) \Phi(x,\lambda) \, d\rho(\lambda). ''The spectrum of D equals the support of dρ.'' Kodaira gave a streamlined version of Weyl's original proof. (Marshall Harvey Stone">M.H. Stone
had previously shown how part of Weyl's work could be simplified using von Neumann's spectral theorem.) In fact for ''T'' =''D''−1 with 0 ≤ ''T'' ≤ ''I'', the spectral projection ''E''(λ) of ''T'' is defined by E(\lambda) =\chi_(T) It is also the spectral projection of ''D'' corresponding to the interval [1,λ]. For ''f'' in ''H''1 define f(x,\lambda)= (E(\lambda)f)(x). ''f''(''x'',λ) may be regarded as a differentiable map into the space of functions ρ of bounded variation; or equivalently as a differentiable map x\mapsto (d_\lambda f)(x) into the Banach space E of bounded linear functionals ''d''ρ on C �,βfor any compact subinterval �,βof [1, ∞). The functionals (or measures) ''d''λ ''f''(''x'') satisfies the following E-valued second order ordinary differential equation: D (d_\lambda f) = \lambda \cdot d_\lambda f, with initial conditions at ''c'' in (0,∞) (d_\lambda f)(c)= d_\lambda f(c,\cdot)=\mu^, \quad (d_\lambda f)^\prime(c)= d_\lambda f_x(c,\cdot)=\mu^. If φλ and χλ are the special eigenfunctions adapted to ''c'', then d_\lambda f (x) = \varphi_\lambda(x) \mu^ + \chi_\lambda(x) \mu^. Moreover, \mu^= d_\lambda (f,\xi^_\lambda), where \xi^_\lambda = D E(\lambda) \eta^, with \eta_z^(y) = G_z(c,y), \,\,\,\, \eta_z^(x)=\partial_x G_z(c,y), \,\,\,\, (z \notin [1,\infty)). (As the notation suggests, ξλ(0) and ξλ(1) do not depend on the choice of ''z''.) Setting \sigma_(\lambda) = (\xi^_\lambda, \xi^_\lambda), it follows that d_\lambda (E(\lambda)\eta_z^,\eta_z^) = , \lambda - z, ^ \cdot d_\lambda \sigma_(\lambda). On the other hand, there are holomorphic functions ''a''(λ), ''b''(λ) such that * φλ + ''a''(λ) χλ is proportional to Φλ; * φλ + ''b''(λ) χλ is proportional to Χλ. Since ''W''(φλλ) = 1, the Green's function is given by G_\lambda(x,y) = \,\, (x\le y), \,\,\,\, \,\, (y\le x). Direct calculation shows that (\eta_z^,\eta_z^) = \, M_(z)/ \, z, where the so-called ''characteristic matrix'' ''M''''ij''(''z'') is given by M_(z)= ,\,\, M_(z)=M_(z)=, \,\, M_(z)= . Hence \int_^\infty (\, z)\cdot, \lambda-z, ^\, d\sigma_(\lambda) = M_(z), which immediately implies \sigma_(\lambda) = \lim_ \lim_ \int_\delta^\, M_(t +i\varepsilon)\, dt. (This is a special case of the Stieltjes transformation, "Stieltjes inversion formula".) Setting ψλ(0)λ and ψλ(1)λ, it follows that (E(\mu)f)(x)= \sum_\int_0^\mu \int_0^\infty\psi^_\lambda(x)\psi^_\lambda(y) f(y)\, dy \,d\sigma_(\lambda) = \int_0^\mu \int_0^\infty\Phi_\lambda(x) \Phi_\lambda(y) f(y)\, dy \, d\rho(\lambda). This identity is equivalent to the spectral theorem and Titchmarsh–Kodaira formula.


Application to the hypergeometric equation

The Mehler–Fock transform concerns the eigenfunction expansion associated with the Legendre differential operator ''D'' Df = -((x^2-1) f')' =-(x^2-1)f'' -2x f' on (1,∞). The eigenfunctions are the
Legendre function In physical science and mathematics, the Legendre functions , and associated Legendre functions , , and Legendre functions of the second kind, , are all solutions of Legendre's differential equation. The Legendre polynomials and the associated L ...
s P_(\cosh r) = \int_0^ \left( \right)^\, d\theta with eigenvalue λ ≥ 0. The two Mehler–Fock transformations are Uf(\lambda)=\int_1^\infty f(x)\, P_(x) \, dx and U^g(x)=\int_0^\infty g(\lambda) \, \tanh \pi \sqrt\,d\lambda. (Often this is written in terms of the variable τ = .) Mehler and Fock studied this differential operator because it arose as the radial component of the Laplacian on 2-dimensional hyperbolic space. More generally, consider the group ''G'' = SU(1,1) consisting of complex matrices of the form \left(\begin\alpha & \beta\\ \overline & \overline\end\right) with determinant , ''α'', 2 − , ''β'', 2 = 1.


Application to the hydrogen atom


Generalisations and alternative approaches

A Weyl function can be defined at a singular endpoint a giving rise to a singular version of Weyl–Titchmarsh–Kodaira theory. this applies for example to the case of radial Schrödinger operators Df = -f'' + \frac f+ V(x) f, \qquad x\in(0,\infty) The whole theory can also be extended to the case where the coefficients are allowed to be measures.


Gelfand–Levitan theory


Notes


References

* * * * * * * * * * * * * * * * * * * * * * * {{SpectralTheory Ordinary differential equations Operator theory Spectral theory