Direct and Inverse Eigenvalue Problems — a gentle introduction
Most people working with scientific problems are familiar with the ideas of eigenvalues from linear algebra, but what about eigenvalue problems arising in physics? What are the so-called “spectral problems” and how are they associated with ordinary or partial differential equations and boundary valued problems? Furthermore, how direct and inverse problems are defined for eigenvalue problems? In this article, we will try to give some brief ideas and examples.
Eigenvalue problem for a square matrix — the finite dimensional case:
Let’s start with the vary basics from linear algebra. The standard matrix eigenvalue problem can be represented as:
where A is a square matrix, x is the eigenvector and λ the eigenvalue. The direct problem is to find the eigenvalues and the corresponding eigenvectors for a given matrix A. The corresponding inverse problem is to find a matrix that has those eigenvalues (and possibly eigenvectors).
Eigenvalue problems for differential operators — the infinite dimensional case:
Now instead of assuming a square matrix, we define the eigenvalue problem in an infinite dimensional space, by using an operator problem:
where D is the (differential) operator and (λ,u) the corresponding eigenpair.
In general, spectral/eigenvalue problems are associated with the investigation of vibrations of physical systems like strings or membranes, mechanical systems, atoms or molecules etc.
The direct spectral problem consists on finding the frequencies and the mode shapes of the vibrations of a system for which all physical properties are known. The inverse spectral problem is to determine some properties of the system from a knowledge of its natural frequencies and modes of vibration.
From a mathematical perspective, inverse spectral problems concern the recovery of coefficients of a differential equation from the knowledge of the eigenvalues, i.e. the spectral values of the corresponding differential operator.
The first study on the spectral properties of the differential operator
was performed by Sturm and Liouville in 1836 and the corresponding inverse problem was firstly considered by Ambartsumyan in 1929.
Example 1: The Sturm-Liouville eigenvalue problem
We consider a string of length L and mass density ρ = ρ(x) > 0, 0<x<L which is fixed at the endpoints x = 0 and x = L. If an external force sets the string into motion, tones are produced due to vibrations.
Let u(x, t) be the transverse displacement at x and time t. It obeys the wave equation:
with boundary conditions:
A periodic vibration of the form :
with frequency k is called pure tone. Thus, u solves the boundary value problem if and only if y and k satisfy the Sturm-Liouville eigenvalue problem:
The above is a boundary value problem for an ordinary differential equation of the second order, with homogeneous (Dirichlet) boundary conditions.
For the direct problem we assume that ρ(x) is known and we want to compute the eigenfrequencies k and the corresponding eigenfunctions. The inverse problem is to recover the density ρ(x) from a set of measured frequencies k.
Example 2: The potential scattering eigenvalue problem
We consider the scattering of a non-relativistic particle by a fixed force and we use the time-dependent Schrödinger equation:
where V(x,t) is the potential, Ψ the wave function, h is the Plank´s constant and m the mass of the particle.
We restrict ourselves to the time-independent Schrödinger equation in one space dimension
where E corresponds to the energy in the particular mode of the state u(x) which represents the amplitude at the point x.
For the finite interval case (particle in a box), the values of constant E are allowed to form a discrete sequence {En}, which means that not all energy levels are possible. The direct problem is given the potential V, to find the energy levels and the corresponding wavefunctions. The inverse spectral problem is given the energy levels of the quantum mechanical system to reconstruct the unknown potential, and hence recover the qualitative characteristics of the underlying force.
Spectral problems can be challenging to solve. Research has been focused both on theoretical aspects (existence and properties of the spectrum, geometrical properties of the eigenfunctions, uniqueness of the inverse problem e.a.) as well as the numerical solution (estimation of eigenvalues/eigenfunctions, convergence, reconstructions e.a.).
Inverse spectral problems are more interesting, because their solution is associated with applications like non-destructive testing of materials, medical imaging and seismic exploration among others. Unfortunately, their nature is more complicated and usually they are “ill-posed”. This could mean i.e. that the knowledge of the spectrum can not uniquely determine the unknown function we want to estimate (e.g. the density of the string or the scattering potential).
Stay tuned for the next part of this article, for more inverse spectral problems, numerical solutions and (non)-uniqueness issues…
References:
- Chadan K., Colton D., Päivärinta L. and Rundell W. (1997), An Introduction to Inverse Scattering and Inverse Spectral Problems, (Philadelphia: SIAM Publications)
- Kirsch A. (2011), An Introduction to the Mathematical Theory of Inverse Problems, 2nd edn (New York: Springer)
