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Transactions of the American Mathematical Society

The Wiener integral and the Schrödinger operator

Donald G. Babbitt

Published Jan 1, 1965• 6 citations
Publisher

Abstract

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.

DOI
10.1090/s0002-9947-1965-0186926-9

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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.
Introduction.It has been known for a long time that the Wiener integral can be effectively used in studying various properties of the Schrödinger operator A-V(x) where A = d 2 /dxl + ' ' ' +d 2 /dx 2 n , the Laplacian in m-dimensional Euclidean space E m and V(x) is a realvalued function on E m satisfying certain regularity conditions.For example, see works by Kac [6], Getoor [2; 3], Ray [9] and Nelson [8].Getoor [2] was the first to point out the connection between a Wiener type integral and the perturbation theory of certain operators of the form Q -V(x) where 0 is the infinitesimal generator of a homogeneous Markov process on a locally compact metric space X and V(x) is a real-valued Borel measurable function on X, bounded on bounded sets and bounded below everywhere.When X -E m and the Markov process is a suitably normed Wiener process, 2 12= A and his results apply to the Schrödinger operator A-V(x).In this paper we will only consider the case X = E m and Q -A but V(x) will be allowed to have certain singularities precluded by Getoor's conditions on V.For example, we can consider the perturbation of A+e/r, where X = E Z and r= (x 2 -\-y 2 +z 2 ) 112 , i.e., the Schrödinger operator with attractive Coulomb potential.In particular the essential condition on V(x) is that A -V is semi-bounded above on a certain dense domain D in L 2 (£ w ), the Hubert space of complexvalued square summable function on E m .z Main results.In this section we will introduce some notation, state our theorems and corollaries, sketch one proof and remark on the remaining proofs.Co(E m ) will denote the space of infinitely differentiable, complexvalued functions on E m with compact support.Let L l ™(E m ) denote 1 The preparation of this paper was sponsored in part by the National Science Foundation.2 In particular we assume a 2 = 2 where a 2 is the variance per unit of time of the Wiener process.See [6]. DEFINITION. A symmetric operator A with domain DÇ-L2(E m) is said to be semibounded from above if there exists a real number k< oo such that (A^, ^)^k{\p, \j/) s&||^|| 2 for all ^££> and where (</>, xl/)=f<f>^dx.
One of the key elements of Feynman’s formulation of non-relativistic quantum mechanics is a so-called Feynman path integral. It plays an important role in the theory, but it appears as a postulate based on intuition, rather than a well-defined object. All previous attempts to supply Feynman’s theory with rigorous mathematics underpinning, based on the physical requirements, have not been satisfactory. The difficulty comes from the need to define a measure on the infinite dimensional space of paths and to create an integral that would possess all of the properties requested by Feynman. In the present paper, we consider a new approach to defining the Feynman path integral, based on the theory developed by Muldowney [A Modern Theory of Random Variable: With Applications in Stochastic Calcolus, Financial Mathematics, and Feynman Integration (John Wiley & Sons, Inc., New Jersey, 2012)]. Muldowney uses the Henstock integration technique and deals with non-absolute integrability of the Fresnel integrals, in order to obtain a representation of the Feynman path integral as a functional. This approach offers a mathematically rigorous definition supporting Feynman’s intuitive derivations. But in his work, Muldowney gives only local in space-time solutions. A physical solution to the non-relativistic Schrödinger equation must be global, and it must be given in the form of a unitary one-parameter group in L2(ℝn). The purpose of this paper is to show that a system of one-dimensional local Muldowney’s solutions may be extended to yield a global solution. Moreover, the global extension can be represented by a unitary one-parameter group acting in L2(ℝn).
Abstract. We investigate a class of operators resulting from a quantization scheme attributed to Berezin. These so-called Berezin-Toeplitz operators are defined on a Hilbert space of square-integrable holomorphic sections in a line bundle over the classical phase space. In this paper we consider self-adjoint Berezin-Toeplitz operators associated with semibounded quadratic forms. Following a concept of Daubechies and Klauder, the semigroups generated by these operators may under certain conditions be represented in the form of Wiener-regularized path integrals. More explicitly, the integration is taken over Brownian-motion paths in phase space in the ultra-diffusive limit. All results are the consequence of a relation between Berezin-Toeplitz operators and Schrödinger operators defined via certain quadratic forms. The probabilistic representation is derived in conjunction with a version of the Feynman-Kac formula. 1.