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Path Integrals for Schrodinger Equation : A Kind of Operator-Valued Integration (Introduction to Path Integrals and Microlocal Analysis)

Kiyoko Furuya

Published Jan 1, 2011• 0 citations
OpenAlex

Abstract

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 $\subset$ can be defined on the powerset $\mathcal{P}(E)$ by subset inclusion.Define a filter $\mathcal{F}$ on $E$ as a subset of $\mathcal{P}(E)$ with the following properties:Definition 2.4.We say that filter base $\mathcal{B}$ converges to $x$ , denoted by $\mathcal{B}arrow x$ , if for every neighborhood $U$ of $x$ , there is a $B\in \mathcal{B}$ such that $B\subset U$ .In this case, $x$ is called a limit of $\mathcal{B}$ and $\mathcal{B}$ is called a convergent filter base.Lemma 2.5.For every neighbourhood base $\mathcal{U}(x)$ of $x$ , it follows that $\mathcal{U}(x)arrow x$ .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 $\mathbb{C}$ is called a locally convex linear topological space, or, in short, a locally convex space, if and only if its open sets $\ni O$ contains a convex, balanced and absorbing open set.Let $M\subset X$ .Then

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Feynman path integral of Riemann type

Kiyoko Furuya

In this paper, 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 the Schrödinger equation with singular potentials. Our class of potentials is wide enough: the real measurable potential U should be continuous except for a closed set of measure zero.

Journal of Mathematical Physics•Jul 1, 2006•0 citations

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

Mathematics of the Feynman path integral

Vassili Nikitich Kolokoltsov

(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

•Jan 1, 2001•2 citations

A dominated-type convergence theorem for the Feynman integral

G. W. Johnson, Jeong-Gyoo Kim

A dominated-type convergence theorem is proved for the operator-valued Feynman integral defined via the Trotter product formula. This definition is intimately related to the one of Feynman and the physical setting is the same as in his original paper. The convergence result given here provides the final piece of a unifying and rather satisfactory picture of three approaches to the Feynman integral.

Journal of Mathematical Physics•May 1, 2000•1 citation

Path integration with non-positive distributions and applications to the Schrödinger equation

Ekaterina S. Nathanson

In 1948, Richard Feynman published the first paper on his new approach to non-relativistic quantum mechanics. Before Feynman's work there were two mathematical formulations of quantum mechanics. Schrödinger's formulation was based on PDE (the Schrödinger equation) and states representation by wave functions, so it was in the framework of analysis and differential equations. The other formulation was Heisenberg's matrix algebra. Initially, they were thought to be competing. The proponents of one claimed that the other was “ wrong. ” Within a couple of years, John von Neumann had proved that they are equivalent.\nAlthough Feynman's theory was not fundamentally new, it nonetheless offered an entirely fresh and different perspective: via a precise formulation of Bohr's correspondence principle, it made quantum mechanics similar to classical mechanics in a precise sense. In addition, Feynman's approach made it possible to explain physical experiments, and, via diagrams, link them directly to computations. What resulted was a very powerful device for computing energies and scattering amplitudes - the famous Feynman's diagrams.\nIn his formulation, Feynman aimed at representing the solution to the non-relativistic Schrödinger equation in the form of an “ average ” over histories or paths of a particle. This solution is commonly known as the Feynman path integral. It plays an important role in the theory but appears as a postulate based on intuition coming from physics rather than a justified mathematical object. This is why Feynman's vision has caught the attention of many mathematicians as well as physicists.\nThe papers of Gelfand, Cameron, and Nelson are among the first, and more substantial, attempts to supply Feynman's theory with a rigorous mathematical foundation. These attempts were followed by many others, but unfortunately all of them were not quite satisfactory. The difficulty comes from a need to define a measure on an infinite-dimensional space of continuous functions that represent all possible paths of a particle. This Feynman's measure has to produce an integral with the properties requested by Feynman. In particular, the expression for the Feynman measure has to involve the non-absolutely integrable Fresnel integrands. The non-absolute integrability of the Fresnel integrands makes the measure fail to be positive and to have the countably additive property. Thus, a well-defined measure in the case of the Feynman path integral does not exist.\nExtensive research has been done on the methods of relating the Feynman path integral to the integral with respect to the Wiener measure. The method of analytic continuation in mass defines the Feynman path integral as a certain limit of the Wiener integrals. Unfortunately, this method can be used as definition for only almost all values of the mass parameter in the Schrödinger equation. For physicists, this is not a satisfactory result and needs to be improved.\nIn this work we examine those questions which originally led to the Feynman path integral. By now we know that Feynman's “ dream ” cannot be realized as a positive and countably additive measure on the path-space. Here, we offer a new way out by modifying Feynman's question, and thereby achieving a solution to the Schrödinger equation via a different kind of averages in the path-space.\nWe give our version of the question that Feynman “ should have asked ” in order to realize the elusive path integral. In our formulation, we get a Feynman path integral as a limit of linear functionals, as opposed to the more familiar inductive limits of positive measures, traditionally used for constructing the Wiener measure, and related Gaussian families. We adapt here an approach pioneered by Patrick Muldowney. In it, Muldowney suggested a Henstock integration technique in order to deal with the non-absolute integrability of the kind of Fresnel integrals which we need in our solution to Feynman's question. By applying Henstock's theory to Fresnel integrals, we construct a complex-valued “ probability distribution functions ” on the path-space. Then we use this “ probability ” distribution function to define the Feynman path integral as an inductive limit. This establishes a mathematically rigorous Feynman limit, and at the same time, preserves Feynman's intuitive idea in resulting functional. In addition, our definition, and our solution, do not place any restrictions on any of the parameters in the Schrödinger equation, and have a potential to offer useful computational experiments, and other theoretical insights.

•Jul 1, 2014•5 citations

The Wiener integral and perturbation theory of the Schrödinger operator

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.

Bulletin of the American Mathematical Society•Jan 1, 1964•8 citations
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;theenough;theenough;the real measurable potential UUU 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 EEE , a partial ordering ⊂\subset⊂ can be defined on the powerset P(E)\mathcal{P}(E)P(E) by subset inclusion.Define a filter F\mathcal{F}F on EEE as a subset of P(E)\mathcal{P}(E)P(E) with the following properties:Definition 2.4.We say that filter base B\mathcal{B}B converges to xxx , denoted by Barrowx\mathcal{B}arrow xBarrowx , if for every neighborhood UUU of xxx , there is a B∈BB\in \mathcal{B}B∈B such that B⊂UB\subset UB⊂U .In this case, xxx is called a limit of B\mathcal{B}B and B\mathcal{B}B is called a convergent filter base.Lemma 2.5.For every neighbourhood base U(x)\mathcal{U}(x)U(x) of xxx , it follows that U(x)arrowx\mathcal{U}(x)arrow xU(x)arrowx .Lemma 2.6.XXX is a Hausdorff space ifand only ifevery filter base on XXX 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 XXX over the complex number field C\mathbb{C}C is called a locally convex linear topological space, or, in short, a locally convex space, if and only if its open sets ∋O\ni O∋O contains a convex, balanced and absorbing open set.Let M⊂XM\subset XM⊂X .Then
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.
(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
In 1948, Richard Feynman published the first paper on his new approach to non-relativistic quantum mechanics. Before Feynman's work there were two mathematical formulations of quantum mechanics. Schrödinger's formulation was based on PDE (the Schrödinger equation) and states representation by wave functions, so it was in the framework of analysis and differential equations. The other formulation was Heisenberg's matrix algebra. Initially, they were thought to be competing. The proponents of one claimed that the other was “ wrong. ” Within a couple of years, John von Neumann had proved that they are equivalent.\nAlthough Feynman's theory was not fundamentally new, it nonetheless offered an entirely fresh and different perspective: via a precise formulation of Bohr's correspondence principle, it made quantum mechanics similar to classical mechanics in a precise sense. In addition, Feynman's approach made it possible to explain physical experiments, and, via diagrams, link them directly to computations. What resulted was a very powerful device for computing energies and scattering amplitudes - the famous Feynman's diagrams.\nIn his formulation, Feynman aimed at representing the solution to the non-relativistic Schrödinger equation in the form of an “ average ” over histories or paths of a particle. This solution is commonly known as the Feynman path integral. It plays an important role in the theory but appears as a postulate based on intuition coming from physics rather than a justified mathematical object. This is why Feynman's vision has caught the attention of many mathematicians as well as physicists.\nThe papers of Gelfand, Cameron, and Nelson are among the first, and more substantial, attempts to supply Feynman's theory with a rigorous mathematical foundation. These attempts were followed by many others, but unfortunately all of them were not quite satisfactory. The difficulty comes from a need to define a measure on an infinite-dimensional space of continuous functions that represent all possible paths of a particle. This Feynman's measure has to produce an integral with the properties requested by Feynman. In particular, the expression for the Feynman measure has to involve the non-absolutely integrable Fresnel integrands. The non-absolute integrability of the Fresnel integrands makes the measure fail to be positive and to have the countably additive property. Thus, a well-defined measure in the case of the Feynman path integral does not exist.\nExtensive research has been done on the methods of relating the Feynman path integral to the integral with respect to the Wiener measure. The method of analytic continuation in mass defines the Feynman path integral as a certain limit of the Wiener integrals. Unfortunately, this method can be used as definition for only almost all values of the mass parameter in the Schrödinger equation. For physicists, this is not a satisfactory result and needs to be improved.\nIn this work we examine those questions which originally led to the Feynman path integral. By now we know that Feynman's “ dream ” cannot be realized as a positive and countably additive measure on the path-space. Here, we offer a new way out by modifying Feynman's question, and thereby achieving a solution to the Schrödinger equation via a different kind of averages in the path-space.\nWe give our version of the question that Feynman “ should have asked ” in order to realize the elusive path integral. In our formulation, we get a Feynman path integral as a limit of linear functionals, as opposed to the more familiar inductive limits of positive measures, traditionally used for constructing the Wiener measure, and related Gaussian families. We adapt here an approach pioneered by Patrick Muldowney. In it, Muldowney suggested a Henstock integration technique in order to deal with the non-absolute integrability of the kind of Fresnel integrals which we need in our solution to Feynman's question. By applying Henstock's theory to Fresnel integrals, we construct a complex-valued “ probability distribution functions ” on the path-space. Then we use this “ probability ” distribution function to define the Feynman path integral as an inductive limit. This establishes a mathematically rigorous Feynman limit, and at the same time, preserves Feynman's intuitive idea in resulting functional. In addition, our definition, and our solution, do not place any restrictions on any of the parameters in the Schrödinger equation, and have a potential to offer useful computational experiments, and other theoretical insights.
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.