Wilkinson, Michael
(2001).
Energy-shell bases.
*Journal of Physics A: Mathematical and General*, 34(49) pp. 11149–11156.

## Abstract

A new type of basis set for quantum mechanical problems is introduced. These basis states are adapted to describing the dynamics of a Hamiltonian Ĥ which is dependent upon a parameter X. A function f(E) is defined which is an analytic function of E, and which is negligibly small when |E|>>δE, where δE is large compared to the typical level separation. The energy-shell basis set consists of states |ξn(X)} which are derived by applying the operator f(Ĥ(X)-Ē_{n}(X)) to elements of a fixed basis set, where Ē_{n}(X) is an analytic approximation to an eigenvalue E_{n}(X). The energy-shell basis states are combinations of states close to energy E_{n}, but vary more slowly as a function of X than the eigenfunctions | ^{ø}_{n}(X) of Ĥ (X). This feature gives the energy-shell basis states some advantages in analysing solutions of the time-dependent Schrödinger equation.

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