Sturm-Liouville theory
Sturm–Liouville problems (cf. Sturm–Liouville problem) have continued to provide new ideas and interesting developments in the spectral theory of operators (cf. also Spectral theory).
Consider the Sturm–Liouville differential equation on the half-line , in its reduced form
![]() | (a1) |
where is the complex spectral parameter and the real-valued function
is assumed to be integrable over any finite subinterval of
. The time-independent Schrödinger equation, at energy
, for a particle having fixed angular momentum quantum numbers moving in a spherically symmetric potential, may be written in the form (a1) — hence there are numerous applications to quantum mechanics ([a1], [a2]).
Suppose the end-point is a limit point. This holds in almost all applications and is valid, for example, if either
is bounded or if
satisfies the inequality
for some positive constant
. Let
denote the second-order differential operator
, defined as a self-adjoint operator in
subject to the Dirichlet boundary condition
(cf. also Linear ordinary differential equation of the second order). (Other boundary conditions may be considered — in general, there is a one-parameter family of boundary conditions
![]() | (a2) |
with the real parameter varying over the interval
.)
The eigenvalues of the Sturm–Liouville operator may be characterized as those
for which the differential equation has a (non-trivial) solution
satisfying both the boundary condition
and the
condition
. The solution
will always be locally square-integrable, and the
condition is a restriction on the large-
asymptotic behaviour of this function. It follows, therefore, that the set of discrete points of the spectrum of
(cf. also Spectrum of an operator) is governed by the asymptotic behaviour of appropriate solutions of (a1). Such considerations, which link asymptotic behaviour of solutions
of (a1) to spectral properties of the Sturm–Liouville operator
, may be extended to other parts of the spectrum of
, and provide a powerful tool of spectral analysis.
As a preliminary straightforward application of this general idea, one may use a Weyl sequence of approximate eigenfunctions of to show that if
as
; then the entire positive line
belongs to the essential spectrum of
, the essential spectrum consisting of all points of the spectrum of
apart from isolated eigenvalues. In contrast, if
as
, then no
belongs to the essential spectrum.
In order to carry out a more detailed spectral analysis of the Sturm–Liouville operator , one has to consider the spectral measure
associated with
, as well as its spectral decomposition. One of the most convenient ways to do this is through the Weyl
-function for
([a3], [a4], [a5], [a6]), here denoted by
(cf. also Titchmarsh–Weyl
-function). The
-function
for the Dirichlet Sturm–Liouville operator
on the half-line is uniquely defined for
by the condition that
![]() | (a3) |
where ,
are solutions of (a1), subject, respectively, to the conditions
![]() | (a4) |
The function is an analytic function of
in the upper half-plane, and has strictly positive imaginary part. Such functions are called Herglotz functions, or Nevanlinna functions. (Corresponding to a general boundary condition, as given by (a2), one can define in a similar way an
-function
, which is again a Herglotz function, to which the theory outlined below applies with minor modifications.)
As a Herglotz function, has a representation of the form ([a7]; cf. also Herglotz formula)
![]() | (a5) |
valid for all with
. (Actually, for a general Herglotz function, a term
, linear in
with
, must be added on the right-hand side, but the asymptotics of
-functions imply that here
[a7].)
In (a6), is a positive constant and
is the spectral function for the problem (a1) with Dirichlet boundary condition at
. The spectral function may be taken to be non-decreasing and right continuous, in which case
is defined by (a5) up to an additive constant, for a given
-function
. The measure
, defined on Borel subsets of
, is called the spectral measure associated with the Dirichlet problem.
The Lebesgue decomposition theorem (cf. Lebesgue theorem) leads to a decomposition of the spectral measure into the sum of a part absolutely continuous with respect to Lebesgue measure and a singular part, i.e.
![]() | (a6) |
where may be further decomposed into its singular continuous and discrete components, thus
![]() | (a7) |
The Radon–Nikodým theorem implies that the absolutely continuous part of the spectral measure may be described by means of a density function
given at (Lebesgue) almost all
by
; thus, for Borel subsets
of
one then has
. The support of the singular component
will be a set
having Lebesgue measure zero. The discrete part
is supported on the set of eigenvalues of the Sturm–Liouville operator
(cf. also Eigen value). These may be characterized as the points
for which
, and alternatively as the points of discontinuity of the spectral function
.
For many physical applications of the Sturm–Liouville problem (a1), the spectrum of the associated differential operator with Dirichlet boundary condition is either purely discrete (e.g. if ), or purely absolutely continuous (e.g. if
and
), or a combination of discrete and absolutely continuous spectrum (e.g. if
) and
is not a positive operator). However, solution of the inverse Sturm–Liouville problem (cf. also Sturm–Liouville problem, inverse), which leads to the determination of a function
from its spectral measure
shows that other types of spectra, including for example combinations of absolutely continuous, singular continuous and discrete spectra, are possible. In view of the generality of types of spectral behaviour, mathematicians have sought ways of further characterizing the spectral properties of Sturm–Liouville operators which will apply to a wide range of cases.
The supports of the various components of the spectral measure may be characterized in terms of the boundary behaviour of the -function
.
For (Lebesgue) almost-all , define the boundary value function
by
![]() | (a8) |
Here exists as a finite limit for almost-all
, and one defines
whenever
. Then:
i) the set of all at which
exists and is real and finite, has zero
-measure;
ii) is supported on the set of all
at which
exists finitely, with
; the density function for the measure
is then
;
iii) is supported on the set of all
at which
. These supports can also be characterized in terms of large-
asymptotics of solutions of (a1), by using the notion of subordinacy. A non-trivial solution
, for given
, is said to be subordinate if the norm of
in
is much smaller, in the limit
, than that of any other solution of (a1) that is not a constant multiple of
. That is,
is subordinate if, for any other solution
linearly independent of
, one has (see [a8], and [a9] for extensions to operators with two singular end-points)
![]() |
Then the following result holds, linking subordinacy with boundary behaviour of the -function, and thereby to the spectral analysis of Sturm–Liouville operators: A subordinate solution of (a1) exists, at real spectral parameter
if and only if either
a) exists and is real and finite (in which case the solution
is subordinate); or
b) (in which case
is subordinate). In particular, the singular component
of the spectral measure is concentrated on the set of
at which the solution
is subordinate, and
is concentrated on the set of
at which there is no subordinate solution.
Recent developments (as of 2000) of the idea of subordinacy have led [a10] to further refinements of the analysis of singular spectra, in which the Hausdorff dimension of the spectral support plays a significant role.
The use of subordinacy and other techniques of spectral analysis have led to a deeper understanding of spectral properties for Sturm–Liouville operators in terms of the large- behaviour of solutions of the Sturm–Liouville equation (a1). Of course there still remains the problem of analyzing the large-
asymptotics of solutions of (a1). However, advances in asymptotic analysis have led to the successful treatment of an ever widening class of Sturm–Liouville spectral problems. Examples of some of the most significant classes of function
that can be handled in this way are as follows.
integrable plus function of bounded variation.
(For this case plus a more general treatment of asymptotics of solutions of systems of differential equations, see [a11].)
Suppose one can write , where
is continuous and of bounded variation, and
. Suppose also, for simplicity, that
as
. Then, for
, the WKB method leads to solutions
of (a1), having the asymptotic behaviour, as
,
![]() |
Asymptotics for lead to exponential growth or decay of solutions. The spectrum is purely absolutely continuous for
and purely discrete for
.
Example of eigenvalues embedded in continuous spectrum.
([a12]) In (a1), with , let
![]() |
A simple calculation then shows that
![]() |
as . This solution
is an eigenfunction of the Dirichlet operator
, with eigenvalue
. One may verify that the interval
belongs to the absolutely continuous spectrum in this example.
periodic.
([a13]) Suppose that satisfies
for some
. Then the absolutely continuous spectrum consists of a sequence of disjoint intervals. The detailed location of these intervals is dependent on the particular function
, though general results can be obtained regarding the asymptotic separation of the intervals for large
.
almost periodic or random.
There is an extensive literature (see, for example, [a14]) on the spectral properties of with
either almost periodic or
a random function. Such problems can give rise to a singular continuous spectrum, or to a pure point spectrum which is dense in an interval. As an example of the latter phenomenon, on each interval
with
, set
, where the
are constant and distributed independently for different
, with (say) uniform probability distribution over the interval
. Then, with probability
, the Sturm–Liouville operator
will have eigenvalues dense in the interval
.
slowly oscillating.
([a15]) A typical function of this type is given by , where
is a constant. The function
oscillates more and more slowly as
increases. One can show that, for almost all
,
has eigenvalues dense in the interval
.
a sparse function.
([a16]) A typical function of this type may be defined by , where
has compact support and the sequence
is strongly divergent as
. Such a function
will give rise to a singular continuous spectrum provided
diverges sufficiently rapidly.
slowly decaying.
([a17]) A challenging problem in the spectral theory of Sturm–Liouville equations has been the analysis of the Dirichlet operator under the hypothesis that
satisfy a bound for sufficiently large
, of the form
, for some
. If additional conditions are imposed, for example appropriate bounds on the derivative of
(assuming
to be differentiable), then such functions
would fall under the category "integrable plus function of bounded variation" considered above, for which a spectral analysis can be carried out. However, in the absence of further conditions on
, it is already clear from the example of an eigenvalue in the continuous spectrum above that one cannot prove absolute continuity of the spectrum for
. In fact, for various
, a dense point spectrum or singular continuous spectrum may be present. A major advance in understanding this problem has been the proof [a17] that, under the hypothesis of
locally integrable and
(
), the entire semi-interval
is contained in the absolutely continuous spectrum. Any subinterval
of
will satisfy
; this does not exclude the possibility of a subset
having Lebesgue measure zero with
, and results have been obtained which further characterize the support of
, for given
. Further extensions of some of these results to the more general case of
square integrable have been obtained (see [a19]).
Numerical approaches.
(See, for example, [a18] and references contained therein.)
Sophisticated software capable of treating an increasingly wide class of spectral problems has been developed. These numerical approaches, often incorporating the use of interval analysis and leading to guaranteed error bounds for eigenvalues, have been used to investigate a variety of limit point and limit circle problems, and to estimate the -function and spectral density function for a range of values of
.
References
[a1] | R.G. Newton, "Scattering theory of waves and particles" , Springer (1982) |
[a2] | E. Prugovečki, "Quantum mechanics in Hilbert space" , Acad. Press (1981) |
[a3] | E.C. Titchmarsh, "Eigenfunction expansions, Part 1" , Oxford Univ. Press (1962) |
[a4] | E.A. Coddington, N. Levinson, "Theory of ordinary differential equations" , McGraw-Hill (1955) |
[a5] | J. Chaudhury, W.N. Everitt, "On the spectrum of ordinary second order differential operators" Proc. Royal Soc. Edinburgh A , 68 (1968) pp. 95–115 |
[a6] | M.S.P. Eastham, H. Kalf, "Schrödinger-type operators with continuous spectra" , Pitman (1982) |
[a7] | N.I. Akhiezer, I.M. Glazman, "Theory of linear operators in Hilbert space" , Pitman (1981) |
[a8] | D.J. Gilbert, D.B. Pearson, "On subordinacy and analysis of the spectrum of one-dimensional Schrödinger operators" J. Math. Anal. Appl. , 128 (1987) pp. 30–56 |
[a9] | D.J. Gilbert, "On subordinacy and analysis of the spectrum of Schrödinger operators with two singular endpoints" Proc. Royal Soc. Edinburgh A , 112 (1989) pp. 213–229 |
[a10] | S. Jitomirskaya, Y. Last, "Dimensional Hausdorff properties of singular continuous spectra" Phys. Rev. Lett. , 76 : 11 (1996) pp. 1765–1769 |
[a11] | M.S.P. Eastham, "The asymptotic solution of linear differential systems" , Oxford Univ. Press (1989) |
[a12] | M. Reed, B. Simon, "Methods of modern mathematical physics: Analysis of operators" , IV , Acad. Press (1978) |
[a13] | M.S.P. Eastham, "The spectral theory of periodic differential operators" , Scottish Acad. Press (1973) |
[a14] | L. Pastur, A. Figotin, "Spectra of random and almost periodic operators" , Springer (1991) |
[a15] | G. Stolz, "Spectral theory for slowly oscillating potentials: Schrödinger operators" Math. Nachr. , 183 (1997) pp. 275–294 |
[a16] | B. Simon, G. Stolz, "Operators with singular continuous spectrum: sparse potentials" Proc. Amer. Math. Soc. , 124 : 7 (1996) pp. 2073–2080 |
[a17] | A. Kiselev, "Absolutely continuous spectrum of one-dimensional Schrödinger operators with slowly decreasing potentials" Comm. Math. Phys. , 179 (1996) pp. 377–400 |
[a18] | "Spectral theory and computational methods of Sturm–Liouville problems" D. Hinton (ed.) P.W. Schaefer (ed.) , M. Dekker (1997) |
[a19] | P. Deift, R. Killip, "On the absolutely continuous spectrum of one-dimensional Schrödinger operators with square-summable potentials" Comm. Math. Phys. , 203 (1999) pp. 341–347 |
Sturm-Liouville theory. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Sturm-Liouville_theory&oldid=23061