### Abstract

An approach to perturbation theory is considered based on the formula exp (—*iB*) *A* exp (*iB*) = *A* exp (*iB*]), where [*AB*] = *AB* − *BA* is the commutator. The operators *A* constitute a linear space and the operators *B* considered are such that *B*] take into itself. The present discussion considers the case where is finite dimensional with coordinates *c* _{1}, ···, *c*_{n} ; i.e., is isomorphic with the set of *n*‐dimensional vectors **c.** Under these circumstances sufficient conditions for the basic formula are given in terms of the ``analytic vectors'' of Nelson. The set, , of *B*'s available can be considered closed under the processes of taking linear combinations and forming the commutator. Thus is a Lie algebra. Exponentiation leads to a Lie group of operators *U,* and *A* and *A′* are said to be *B* equivalent if *A′* = *U* ^{−1} *AU.* For finite dimensional each *B* is associated with an *n × n* matrix, *b,* which specifies the operation *B*] relative to the vectors **c.** Effectively then, is finite dimensional. The matrices *b* form a Lie algebra with a corresponding Lie group of matrices *u* such that *A* and *A′* are *B* equivalent if and only if the corresponding two vectors **c** and **c′** are *u* images; i.e., **c′** = *u* **c.** Computationally, therefore, the set of *A′* equivalent to a given *A* is obtained by considering the orbit of a given vector **c** under the Lie group; i.e., the set of *u* **c.** A neighborhood of a given *A* consists of those operators *A′* in the form *A* + δ*A* where δ*A* is arbitrary except for a restriction on the size of its coordinates, *c*_{i}. Given *A,* a neighborhood can be found for which one can obtain by a well‐known construction on the orbits a set of functionally complete and functionally independent invariants for *B* equivalence. Computationally global and rational invariants are desirable and these can be obtained in the form of similarity invariants, provided that *a* can be mapped onto a set of *n × n* matrices *a* in such a way that if **c** corresponds to *a,* then *b* **c** corresponds to *a′* = [*ab*]. If the sets and are identical such a mapping is immediately available and if, in addition, the corresponding Lie algebra is semisimple, in general, given *A,* the *A*'s in some neighborhood are each equivalent to an *A″* which is a function of *A.* This corresponds to a case in which a very simple perturbation of *A* levels occurs. Two examples are discussed.

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