ENVELOPE THEORY IN SPECTRAL GEOMETRY

It is shown that the discrete spectrum of Schrodinger Hamiltonians of the form H = -DELTA + upsilonf may be represented by the semiclassical expression E(nl) = min(r>0) {K(nl)(f)(r) + upsilonf(r)}. The K functions are found to be invariant with respect to coupling and shifts: K(Af+B) = K(f). For pure power laws, f(r) = sgn(q)r(q), and the log potential, they are also invariant with respect to scale, and have the simple forms (P(nl)(q)/r)2 and (L(nl)/r)2, respectively. K functions are also derived for sech-squared and Hulthen potentials. If f = g(h), where g is a smooth transformation, then the envelope approximation is expressed in terms of K by the relation K(f) congruent-to K(h). When the transformation g has definite convexity, then the approximation immediately yields eigenvalue bounds for all n and l. The theory is used to prove the log-power theorem L(nl) = P(nl)(0), which, in turn, generates a simple eigenvalue formula for the log potential.