Inverse Eigenvalue Problems

The classic problem in inverse Sturm-Liouville theory is to determine (one of) the coefficients p, q or r in the second order equation
-(p(t)y'n(t))' + q(t) yn(t) = λn r(t) yn(t)
from knowledge of the eigenvalues λn. We must impose boundary conditions, and we assume these have the form y(0) = 0, y'(1) + βy(1)= 0. The eigenvectors yn(t) can be normalised by either ||y²||2 = 1 or the initial condition y'(0) = 1. It is usual to use the latter condition. A good reference to inverse Sturm-Liouville problems can be found in [CCPR] have shown that if

The Liouville transform t->x, y->u allows the above to be put into canonical form

-u"n(x) + Q(x) un(x) = λn un(x)
and leaves the spectral values λn and the boundary conditions unchanged. Here Q(x) is a function of p, q, r.

The fundamental question is: Can Q(x) be determined by the sequence {λn}?

The classical result of Borg (1946) is the following:

Theorem. Let {λn} be the eigenvalues corresponding to a boundary parameter β1 and let {µn} be the eigenvalues corresponding to a boundary parameter β2. Then if β1 differs from β2, {λn , µn} uniquely determines Q.

Other data is possible. If ρn := ||u²||2, then the sequence pair {λn , ρn} uniquely determines Q. If Q is symmetric about the mid-point, Q(x) = Q(1-x), then a single (Dirichlet) spectrum is sufficient to determine Q.

There are a variety of ways to prove the above theorem, including ones that lead to constructive algorithms. We ( Paul Sacks and I) are fond of the method in [RS1, RS2]

Our current interest is in the case of singular potentials.

Singular inverse Sturm-Liouville Problems

Here we have for each integer l = 1, 2, . . the equation
-un"(x) + l(l+1)x-2 un(x) + q(x)un(x) = λn un(x)
with the boundary conditions, u(x) bounded as x->0, u(1) = 0. (Alternatively we can use either the Neumann condition at x=1 or the impedance condition u'(1) + βu(1) = 0. )

This problem comes from the radial solution obtained by separating the variables in the three dimensional Laplacian. The number l is the angular momentum quantum number. A particular application we have in mind is in helioseismology; characterizing the interior of the sun from its eigenvalues; specifically the "five minute" oscillations. These are the natural frequencies coming from an equation modeling normal modes of acoustic vibration and in the simplest case takes the form of the above singular equation.

A natural question arises; does the (double) sequence of spectral values {λn} for n = 1, 2, . . . and l=1, 2, . . uniquely determine q?

A simple operation count would indicate that this problem would be overposed and that some subset of the data would suffice, although as far as we are aware no general result of this fact is known.

Motivated by Borg's result, the conjecture is

    If {λn} is the complete spectrum corresponding to the value l=l1 and {µn} is the complete spectrum corresponding to the value l=l2, then {λn , µn} uniquely determines q(x).

Carlson and Shubin [CS] have shown that if l1-l2 is odd then q(x) is uniquely determined up to a finite dimensional subspace of L²(0,1). In work in progress Paul sacks and I have shown a local uniqueness result for l values less than or equal to three.

Theorem. Let {λn} and {µn} be the complete spectra corresponding to distinct values l1 and l2 with l1 , l2 ≦ 3. Then there is at most one potential in a neighbourhood of q = 0 corresponding to the spectral pair {λn , µn}.

There is much left to do. Of course we would like to remove the locality of the result and to find a proof (or counterexample: there is a delicate balance in resolving even the cases to date) that is valid for all values of l. There are also other subsets of the complete double spectral sequence that can be used in the reconstruction of q. The actual helioseismological application has an equation with two unknown coefficients, the mass density ρ and the propagation speed for acoustic waves c. We would like to determine both these unknown coefficients. Which subset of the spectral data (if any) would allow this?

References

  • An introduction to Inverse Scattering and Inverse Spectral Problems, Khosrow Chadan, David Colton, Lassi Päivärinta and William Rundell, SIAM, Philadelphia, 1997.

  • Robert Carlson and Carol Shubin, Spectral rigidity for radial Schrödinger operators. J. Differential Equations 113, (1994), no. 2, 338-354.

  • William Rundell and Paul E. Sacks "Reconstruction techniques for classical inverse Sturm-Liouville problems", Mathematics of Computation, 58, 197, (1992), 161-183.

      Abstract: This paper gives constructive algorithms for the classical inverse Sturm-Liouville problem. It is shown that many of the formulations of this problem are equivalent to solving an overdetermined boundary value problem for a certain hyperbolic operator. Two methods of solving this latter problem are then provided, and numerical examples are presented.
      AMS Classification: Primary 34B25 35R30, Secondary 42A70

  • William Rundell and Paul E. Sacks "The reconstruction of inverse Sturm-Liouville operators", Inverse Problems, 8, (1992), 457-482.

      Abstract: This paper gives a constructive algorithm for the inverse Sturm-Liouville operator in non-potential form; that is, L u = -(p u')'= λ r u where either p(x) or r(x) has to be determined. We also consider the important case of an unknown impedance, a(x)=p(x)=r(x). It is shown that many of the formulations of this problem are equivalent to solving an overdetermined boundary value problem for a certain hyperbolic operator. An iterative procedure for solving this latter problem is provided and numerical examples are presented.
      AMS Classification: Primary 34B25 35R30, Secondary 42A70