The Wiener Integral and the Schrodinger Operator

Felix Alexandrovich Berezin

Elliott W. Montroll

Ian M Davies, Aubrey Truman

Donald G. Babbitt

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.

Helge Holden, Arne Jensen

Chandra Halim, Muhammad Farchani Rosyid

The implementation of Lévy path integral generated by Lévy stochastic process on fractional Schrödinger equation has been investigated in the framework of fractional quantum mechanics. As the comparison, the implementation of Feynmann path integral generated by Wiener stochastic process on Schrödinger equation also has been investigated in the framework of standard quantum mechanics. There are two stochastic processes. There are Lévy stochastic and Wiener stochastic process. Both of them are able to produce fractal. In fractal’s concept, there is a value known as fractal dimension. The implementation of fractal dimension is the diffusion equation obtained by using Fokker Planck equation. In this paper, Lévy and Wiener fractal dimension have been obtained. There are for Lévy and 2 for Wiener/Brown fractal dimension. Fractional quantum mechanics is generalization of standard quantum mechanics. A fractional quantum mechanics state is represented by wave function from fractional Schrödinger equation. Fractional Schrödinger equation is obtained by using kernel of Lévy path integral generated by Lévy stochastic process. Otherwise, standard quantum mechanics state is represented by wave function from standard Schrödinger equation. Standard Schrödinger equation is obtained by using kernel of Feynmann path integral generated by Wiener/Brown stochastic process. Both Lévy and Feynmann Kernel have been investigated and the outputs are the Fourier Integral momentum phase of those kernels. We find that the forms of those kernels have similiraty. Therefore, we obtain Schrödinger equation from Lévy and Feynmann Kernel and also the comparison of Lévy energy in fractional quantum mechanics and particle energy in standard quantum mechanics.