Spectral theory of ordinary differential equations

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 dissertation Hermann Weyl generalized the classical Sturm–Liouville theory on a finite closed interval 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, 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. It has had important applications in quantum mechanics, operator theory and harmonic analysis on semisimple Lie groups.
Introduction.
Spectral theory for second order ordinary differential equations on a compact interval was developed by Jacques Charles François Sturm and Joseph Liouville in the nineteenth century and is now known as Sturm–Liouville theory. In modern language it is an application of the spectral theorem for compact operators due to David Hilbert. In his dissertation, published in 1910, Hermann Weyl 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 conditions in terms of his celebrated dichotomy between "limit points" and "limit circles".
In the 1920s John von Neumann established a general spectral theorem for unbounded self-adjoint operators, which Kunihiko Kodaira 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 and the United Kingdom had been interrupted by World War II). Titchmarsh had followed the method of the German mathematician Emil Hilb, who derived the eigenfunction expansions using complex function theory instead of operator theory. 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 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'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 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. More generally, the Plancherel theorem for SL(2,R) of Harish Chandra and Gelfand–Naimark can be deduced from Weyl's theory for the hypergeometric equation, as can the theory of spherical functions for the isometry groups of higher dimensional hyperbolic spaces. Harish Chandra's later development of the Plancherel theorem for general real semisimple Lie groups 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 and scattering matrix in quantum mechanics.
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 defined by
then
and
Hence,
where
and
The term in "g' " can be removed using an Euler integrating factor. If "S' "/"S" = −"R"/2, then "h" = "Sg"
satisfies 
where the potential "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 E.
Let α, β be arbitrary elements of E, "A" a bounded operator on "E" and "q" a continuous function on ["a","b"].
Then, for "c" = "a" or "b",
the differential equation
has a unique solution "f" in "C"2(["a","b"],E) satisfying the initial conditions
In fact a solution of the differential equation with these initial conditions is equivalent to a solution
of the integral equation
with "T" the bounded linear map on "C"(["a","b"], E) defined by
where "K" is the Volterra kernel
and
Since ||"T"k|| tends to 0, this integral equation has a unique solution given by the Neumann series
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 of "L" with eigenvalue λ.
Green's formula.
If "f" and "g" are "C"2 functions on ("a", "b"), the Wronskian "W"("f", "g") is defined by
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 C2 functions "f" on ["a", "b"] satisfying the Robin boundary conditions
with inner product
In practise usually one of the two standard boundary conditions:
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 of "D" (for the above choice of boundary values) if "Df" = λ "f" for some complex number λ, the corresponding eigenvalue.
By Green's formula, "D" is formally self-adjoint on "H"0, since the Wronskian "W(f,g)" vanishes if both "f,g" satisfy the boundary conditions:
As a consequence, exactly as for a self-adjoint matrix in finite dimensions,
It turns out that the eigenvalues can be described by the maximum-minimum principle of Rayleigh–Ritz (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:
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:
In fact, since
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
which at each point, together with their first derivatives, depend holomorphically on λ. Let
be an entire holomorphic function.
This function ω(λ) plays the rôle of the characteristic polynomial 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 by
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 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,
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"