## Spectral Geometry

When a Hamiltonian operator H depends on a set of parameters, it's spectrum does too. Thus a discrete eigenvalue becomes a map E = F(v) from the parameter space v into the reals, and thereby generates a curve or surface in a Euclidean space. The term spectral geometry is used here to describe the study of such manifolds, and approximation theories for them.

We study therefore not a single operator but a set of operators, and their corresponding discrete eigenvalues, as functions of v. We devise a representation of the eigenvalue problem in which the positive-definite kinetic energy operator -Delta is replaced by a real variable s > 0. This allows the eigenvalue to be written exactly as the minimum of a semi-classical form {s + v bar{f}(s)} in which the potential-energy term is a function of s called the `kinetic potential'. The envelope method concerns potentials which can be written as a smooth transformation f = g(h) of a soluble potential h: if the transformation g has definite convexity, then the tangential potentials so generated provide energy bounds; the computation of the best such bound is carried out most effectively by means of the kinetic-potential apparatus. This same theory also allows us to discuss sums of potential terms.

In a further development, a change of variables is made which allows the semi-classical potential term to be written as the potential itself; the kinetic energy then becomes a so-called K-function. This form of the theory partially disentangles the potential and supports the goal of geometric spectral inversion, the complete reconstruction of the potential shape f(x) from a given energy function F(v).

### Specific topics and the locations of their treatment in this work:

Kinetic potentials  

Envelope theory  

Pure powers  

Sums of potential terms  

Symmetric potentials as mixtures of square wells  

Screened-Coulomb and Yukawa potentials  

Quartic anharmonic oscillator  

Coulomb-plus-linear potential  

Relation to the large-N approximation  

Singular potentials  

Log potential as limit of |x|^q as q -> 0  

Dirac eigenvalues  

The N-body problem Return to Home Page.