In this paper we shall introduce a generalized equi-continuity of a family of semigroups and prove a new type of Trotter-Kato Theorem, applicable to the weak convergence of semigroups.In [13], we prove the existence of non-unitary solutions to formally self-adjoint Schr\"odinger equations.In that paper, we need the Trotter-Kato Theorem for the weak convergence.However, various versions of the Trotter-Kato Theorem in locally convex spaces already published are not applicable to the weak convergence as far as the authors knows.Therefore we shall give a generalized form of the Trotter-Kato Theorem in Yosida [18].Next we shall define a kind of operator-valued integration and define the Feynman path integrals of Riemann integral type.It seems that it is one of the best possible conditions of the existence of the path integrals of Riemann integral type for Schr\"odinger equation with singular potentials.Our class of potentials is wide
enough;the real measurable potential
U should be continuous except a closed set of measure zero.それどころか，内積を取る相手が完全系を構成するかどうかについても，あまり厳密に考えて いません (実際に，そうした論文を読んだことがあります). 証明が厳密でなくても，物理的な 直観に照らして正当 (と感じられる) ならば充分であり，そのうち数学の得意な人がちゃんとし た証明をしてくれると期待している訳です．J \S 2. Trotter-Kato Theorem for Weak Covergence We shall introduce a generalized equicontinuity of a family of semigroups and prove a new type of Trotter-Kato Theorem, applicable to the weak convergence of semigroups.We begin by introducing some terminology and notation and present those aspects of the basic theory which are required in subsequent subsections.\S 2.1.Filter Definition 2.1.Given a set
E , a partial ordering
⊂ can be defined on the powerset
P(E) by subset inclusion.Define a filter
F on
E as a subset of
P(E) with the following properties:Definition 2.4.We say that filter base
B converges to
x , denoted by
Barrowx , if for every neighborhood
U of
x , there is a
B∈B such that
B⊂U .In this case,
x is called a limit of
B and
B is called a convergent filter base.Lemma 2.5.For every neighbourhood base
U(x) of
x , it follows that
U(x)arrowx .Lemma 2.6.
X is a Hausdorff space ifand only ifevery filter base on
X has at most one limit.For details concerning the filter, we refer to Bourbaki [1].\S 2.2.Locally Convex Topologies Definition2.7.A linear topological space
X over the complex number field
C is called a locally convex linear topological space, or, in short, a locally convex space, if and only if its open sets
∋O contains a convex, balanced and absorbing open set.Let
M⊂X .Then
Abstract. In this paper, we will show the existence and uniqueness of a real-time, time-sliced Feynman path integral for quantum systems with vector potential. Our formulation of the path integral will be derived on the L 2 transition probability amplitude via improper Riemann integrals. Our formulation will hold for vector potential Hamiltonian for which its potential and vector potential each carries at most a finite number of singularities and discontinuities. 1. Introduction. In his paper(see [22] footnote 13), Feynman observed that by using wave functions, ill-defined oscillatory integrals can be given rigorous meaning. We will use Feynman’s observation and use wave functions to provide a convergence factor in the derivation of a real time propagator that takes the form of an L 2 transition probability amplitude and the derivation of a real time, time sliced Feynman
(and its generalisations that include magnetic fields) can be roughly divided into two classes. In the aproaches of the first class that we shall not discuss here in detail the Feynman integral is not supposed to be a genuine integral, but is specified as some generalised functional on an appropriate space of functions, which can be defined, for example, as the limit of certain discrete approximations (see e.g. [16], [44] and more recent papers [24], [25], [34]), by means of analytical continuation (see e.g. [26], [23] and references therein), by extensions of Parceval’s identity and by related axiomatic definitions (see [2], [5], [30], [11], [42] and references therein) or by means of the white noise analysis (see [22]), see also [46] for the discussion of path integral applied to the Dirac equation. These approaches still cover only a very restrictive class of potentials, for example, singular potentials were considered only by white noise analysis approach but only in one-dimensional case (see e.g. [6] and references therein). In the approaches of the second class, one tries to define the infinite-dimensional Feynmann integral as a genuine integral over a bona fide σ-additive measure on an appropriate space of trajectories. The first attempt made in [19] to construct such a measure was erronious and led to understanding that there is no direct generalisation of Wiener measure that can give an analog of FeynmanKac formula for the case of Schrodinger operators. A correct construction of the Feynman integral in terms of the Wiener measure was proposed in [15] and was based on the idea of the rotation of the classical trajectories in complex
The Schrödinger equation is fundamental in quantum mechanics as it makes it possible to determine the wave function from energies and to use this function in the mean calculation of variables, for example, as the most likely position of a group of one or more massive particles. In this paper, we present a survey on some theories involving the Schrödinger equation and the Feynman path integral. We also consider a Feynman–Kac-type formula, as introduced by Patrick Muldowney, with the Henstock integral in the description of the expectation of random walks of a particle. It is well known that the non-absolute integral defined by R. Henstock “fixes” the defects of the Feynman integral. Possible applications where the potential in the Schrödinger equation can be highly oscillating, discontinuous or delayed are mentioned in the end of the paper.