Spectral theory of ordinary differential equations
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.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.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).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.Equally importantly the theory also laid the mathematical foundations for the analysis of the Schrödinger equation and scattering matrix in quantum mechanics.The following is a version of the classical Picard existence theorem for second order differential equations with values in a Banach space E.[2] Let α, β be arbitrary elements of E, A a bounded operator on E and q a continuous function on [a, b].If f is twice continuously differentiable (i.e. C2) on (a, b) satisfying Df = λf, then f is called an eigenfunction of D with eigenvalue λ.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.This function ω(λ) plays the role 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.Since Gλ(x,y) is continuous on [a, b] × [a, b], it defines a Hilbert–Schmidt operator on the Hilbert space completion H of C[a, b] = H1 (or equivalently of the dense subspace H0), taking values in H1.Theorem — The eigenvalues of D are real of multiplicity one and form an increasing sequence λ1 < λ2 < ⋯ tending to infinity.Then T defines a compact self-adjoint operator on the Hilbert space H. By the spectral theorem for compact self-adjoint operators, H has an orthonormal basis consisting of eigenvectors ψn of T with Tψn = μn ψn, where μn tends to zero.Using Picard iteration, Titchmarsh showed that φλ(b), and hence ω(λ), is an entire function of finite order 1/2:Every positive form μ extends uniquely to the linear span of non-negative bounded lower semicontinuous functions g by the formula[8]The support of μ = dρ 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].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 conditions there.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.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.Weyl's fundamental observation was that dλ f satisfies a second order ordinary differential equation taking values in E: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).a constant which vanishes precisely when Φλ and Χλ are proportional, i.e. λ is an eigenvalue of D for these boundary conditions.Stone had previously shown[18] how part of Weyl's work could be simplified using von Neumann's spectral theorem.)Mehler and Fock studied this differential operator because it arose as the radial component of the Laplacian on 2-dimensional hyperbolic space.