The Wiener integral and the Schrödinger operator
Donald G. Babbitt
where x = (x1, * , xn) C Rn, bj(x), j = 1, * , n and V(x) are real-valued functions on Rn, aj = d/ (dx1), i = V 1 and CZ Z), a dense linear subset of L2(Rn). i) is called the domain of A (b; V; t) and is denoted by t (A). The domain ?) and/or (b; V) will be omitted from A(b; V; Z) when no confusion should arise. In practice, the spectral resolution of a certain self-adjoint extension, A, of A is important. One important technique used in studying pertinent properties of the spectral resolution of A(b; V;Z) is to represent e Q-L2(Rn), by the Wiener integral of a certain functional F(b; V; O) and then to translate the (at least formally) transparent properties of this integral into less transparent but important properties of A. For example, see [14] and [ 18]. This functional integral representation has been shown to be rigorously valid for the case where bj(x) = 0 (j = 1, * *,*n), and where V(x) is Borel measurable and bounded from below. Unfortunately, A is only given implicitly. See [ 8], [ 9], and [ 10] . It is the purpose of this paper to extend the validity of the functional integral representation of e-At% to the case where b(x) = (b1(x) , .., bn(x)) is a sufficiently smooth vector field which doesn't grow too rapidly at infinity, and where V(x) can have certain negative singularities. Moreover, A will be given explicitly(). The Schrodinger operator containing (normal) Zeeman effect and Coulomb interaction terms will be included. ??2, 3, and 4 will give a brief account of functional integration, semigroup theory and quadratic forms on a Hilbert space, respectively. They will be needed for the main body of the paper consisting of ??5 and 6. ?5 will give the statement of the theorems along with certain remarks. ?6 will contain the proofs of the theorems.
Transactions of the American Mathematical Society•Jan 1, 1965•6 citations