## 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** [29][31]

**Envelope theory** [54]

**Pure powers** [45]

**Sums of potential terms** [26][48][65]

**Symmetric potentials as mixtures of square wells** [51]

**Screened-Coulomb and Yukawa potentials** [35][49][50][56][57]

**Quartic anharmonic oscillator** [33]

**Coulomb-plus-linear potential** [32]

**Relation to the large-N approximation** [36]

**Singular potentials** [62][63][66][69][70][72]

**Log potential as limit of |x|^q as q -> 0** [54]

**Dirac eigenvalues** [34][37][38]

**The N-body problem** [25][53][60]

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