of this equation (for some non-zero wave function belonging to L 2 (R 3 ), and form a countable increasing sequence -E 0 ; -E 1 ; -E 2 ; : : : ; having zero as a limit point. These are the possible energy levels of bound states of the Hydrogen atom, and they belong to the discrete spectrum of the Hamiltonian operator.
The second type of state of the Hydrogen atom corresponds to a so-called scattering state. An electron in a scattering state follows an evolution in which the electron is far from the central nucleus at both large positive and negative times, and undergoes a scattering process at intermediate times. Such scattering states have positive total energy, but in this case rather than discrete eigenvalues the Hamiltonian H has continuous spectrum. The energy of a scattering state belongs to a continuum rather than a discrete set of values.
The mathematical description and analysis of discrete and continuous spectrum for di erential operators, particularly those with relevance to problems in Mathematical Physics, and the further reÿnement and development of these ideas, has been one of the main (possible the main) tasks of spectral theory in the latter-half of the 20th century. Research into this area has brought into play (and in turn has stimulated further mathematical progress, in a remarkable cross-fertilisation of ideas) a whole range of mathematical tools drawn from real and complex analysis, functional analysis and operator theory, as well as the theory of ordinary and partial di erential equations.
In focussing in this article on spectral theory for di erential operators in one dimension, I shall refer almost entirely to developments in our understanding of spectral theory for ordinary rather than partial di erential operators. For a broader view of advances in the theory, covering aspects of Schr odinger operators, both in one dimension and in higher dimensions, see the review article by Simon [3]. In any case, these two aspects of the theory cannot, and should not, be entirely separated, and there are many ways in which the study of ordinary di erential operators is leading to new approaches to higher-dimensional problems.
The transition from three dimensions to one is easily accomplished in the case of the Hamiltonian for the Hydrogen atom, or indeed in the case of any (non-relativistic) Hamiltonian for a single particle moving in a potential V = V (|r |) which is spherically symmetric. From the point of view of quantum mechanics, this transition is made possible by the spherical symmetry of the problem, by the fact that the operator H for total energy commutes with operators of angular momentum. We can then deÿne subspaces H l; m of the Hilbert space L 2 (R 3 ); labelled by quantum numbers l; m of angular momentum (with l = 0; 1; 2; : : : ; and m = -l; -(l -1); : : : (l -1); l) which are invariant subspaces of the operator H . The subspace H l; m is spanned by wave functions of the form f(r)Y lm (Â; )=r; where r = |r |; r; Â; are spherical polar co-ordinates, and Y lm is a spherical harmonic. The condition of ÿnite norm for the wave function in L 2 (R 3 ) then becomes ∞ 0 |f(r)| 2 dr¡∞; that is, a condition of ÿnite norm for f in the Hilbert space L 2 (0; ∞). The action of the Hamiltonian operator H on f then becomes H : f → -d 2 f=dr 2 + V (r)f + (l(l + 1)=r 2 )f; with, in the case of the Hydrogen atom, V (r) = -1=r, the Coulomb potential.
The study of the quantum Hamiltonian for a particle moving in three dimensions, in a ÿeld of force determined by a spherically symmetric potential function V (r); is therefore dependent on the solution of a related family of one-dimensional problems (corresponding to potentials respectively V (r)+l(l+1)=r 2 ; l = 0; 1; 2; : : :). Note here that we cannot quite say that the study of the three-dimensional problem is reduced to the solution of the one-dimensional problems. For example, in scattering theory one is concerned with asymptotic localization of quantum states for large positive and negative times, and the full three-dimensional picture has to be