Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Journal of Mathematical Physics

Convergence of Nelson diffusions with time-dependent electromagnetic potentials

Andrea Posilicano, Stefania Ugolini

Published Nov 1, 1993• 6 citations
Publisher

Abstract

Some recent results on the convergence of Nelson diffusions are extended to the case of Schrödinger operators with time-dependent electromagnetic potentials. It is proven that the sequence {P n}n≥1 of measures on the canonical space of physical trajectories associated to the solutions of Schrödinger equations in Nelson’s scheme, corresponding to the sequence {(Vn,An)}n≥1 ⊆C1(R;ℛ×L2(R3)), converges in the total variation norm under the assumptions that for every fixed t the scalar potentials Vn(t) converge in ℛ, the space of Rollnik class potentials, and the vector potentials An(t) converge in Lloc∞(R;L2(R3)). In order to prove these results conditions are given under which solutions of Schrödinger equations are continuous in the (time-dependent electromagnetic) potentials in the norm of the Sobolev space H1(R3).

DOI
10.1063/1.530339

Related papers

On the dynamics of diffusions and the related general electromagnetic potentials

Laura M. Morato

In this paper the Nelson’s stochastic mechanics is extended to general diffusion motions. A representation theorem is proved which gives a one-to-one correspondence between solutions of certain Schrödinger equations and diffusion processes satisfying appropriate regularity conditions. Exploiting results of stochastic mechanics on Riemannian manifolds it is shown that the real part of the Schrödinger equations corresponding to the considered diffusions can be interpreted as Newton’s second law where the force is produced by generalized electromagnetic potentials.

Journal of Mathematical Physics•Jun 1, 1982•23 citations

On Nelson processes with boundary condition

Minoru W. Yoshida

In Nelson [10], it is shown that for each solution of a Schr\"odinger equation $\phi(x, t),$ $x\in R^{d},$ $t\geqq 0$ , with $\Vert\phi\Vert=1$ , there exists a diffusion process which has ... probability density at time $t$ given byMore precisely the following is shown in [10]: suppose that we are given a real valued function $V(x),$ $x\in R^{d}$ , and a complex valued function $\phi_{0}(x),$ $x\in R^{d}$ , with $\Vert\phi_{0}\Vert=1$ , and consider the Schrodinger equation Then, under the assumption that $\phi$ and $\tilde{b}$ are sufficiently regular, there exists a diffusion process $\{X_{t}\},$ $t\geqq 0$ , with initial density $|\phi_{0}(x)|^{2}$ and generator $(1/2)\Delta+\tilde{b}(x, t)\cdot\nabla$ such that the probability density of this diffusion at time $t\geqq 0$ is $|\phi(x, t)|^{2}$ .In addition, in [10] it is also shown that the diffusion process $\{X_{t}\},$ $t\geqq 0$ , solves the second order stochastic differential equation $(1/2)(D_{*}D+DD_{*})X_{t}=-\nabla V(X_{t})$ , where $D$ and $D_{*}$ are Nelson's forward and backward stochastic derivatives.Generally sPeaking, the functions $\phi$ and $\tilde{b}$ may be singular, and the existence of the corresponding diffusion process must be studied carefully.Carlen $[1, 2]$ , Meyer and Zheng [8] and Nagasawa [9] considered rigorously the construction of a diffusion process with a given initial density and a generator.$ln$ order to construct a Markovian propagator for the diffusion process, Carlen $[1, 2]$ used partial differential equation methods.Meyer and Zheng [8] restricted themselves to the case when $\phi(x, t)=\phi_{0}(x),$ $\forall_{t\geqq 0}$ and considered the construction of the diffusion process through the relationships between Markov processes and Dirichlet forms.We may consult Nagasawa [9] about recent developments of this area.In Carlen [1], it is assumed that $V(x),$ $x\in R^{d}$ , is a Rellich class potential and $\Vert\nabla\phi_{0}\Vert^{2}<\infty$ , and shown that there exists a diffusion process $\{X_{t}\},$ $t\geqq 0$ , which has the probability density $|\phi(x, t)|^{2}$ at time $t\geqq 0$ and admits the representation

Journal of the Mathematical Society of Japan•Apr 1, 1990•0 citations

On the convergence of the Feynman path integral for a certain class of potentials

Vassilis G. Papanicolaou

A direct proof is given of the convergence of the discretized Feynman path integral to the fundamental solution of the (time-dependent) Schrödinger equation, for potentials which are bounded, integrable and continuous on the real line. No further smoothness assumptions are required.

Journal of Mathematical Physics•Feb 1, 1990•2 citations

A new path integral representation for the solutions of the Schrödinger, heat and stochastic Schrödinger equations

Vassili Nikitich Kolokoltsov

Solutions to the Schrödinger, heat and stochastic Schrödinger equation with rather general potentials are represented, both in x- and p-representations, as integrals over the path space with respect to σ-finite measures. In the case of x-representation, the corresponding measure is concentrated on the Cameron–Martin Hilbert space of curves with L2-integrable derivatives. The case of the Schrödinger equation is treated by means of a regularization based on the introduction of either complex times or continuous non-demolition observations.

Mathematical Proceedings of the Cambridge Philosophical Society•Mar 1, 2002•12 citations

ON CONVERGENCE OF THE FEYNMAN PATH INTEGRAL FORMULATED THROUGH BROKEN LINE PATHS

Wataru Ichinose

We study the convergence of the Feynman path integral formulated through broken line paths in nonrelativistic quantum mechanics. The rigorous proof of its convergence had been given little except for special cases for a long time. In the preceding paper the author showed for a class of potentials that this path integral converges and gives the probability amplitude, i.e. the solution of the Schrödinger equation. In the present paper we generalize this result, showing the boundedness theorem on the weighted Sobolev spaces for some integral operators and joining this boundedness theorem to the method in the preceding paper. We note that the result obtained is gauge invariant.

Reviews in Mathematical Physics•Sep 1, 1999•24 citations

Density-potential mappings in quantum dynamics

Michael Ruggenthaler, Klaas J. H. Giesbertz, Markus Penz +1 more

In a recent paper [Europhys. Lett. 95, 13001 (2011)] the question of whether the density of a time-dependent quantum system determines its external potential was reformulated as a fixed-point problem. This idea was used to generalize the existence and uniqueness theorems underlying time-dependent density-functional theory. In this work we extend this proof to allow for more general norms and provide a numerical implementation of the fixed-point iteration scheme. We focus on the one-dimensional case because it allows for a more in-depth analysis using singular Sturm-Liouville theory and at the same time provides an easy visualization of the numerical applications in space and time. We give an explicit relation between the boundary conditions on the density and the convergence properties of the fixed-point procedure via the spectral properties of the associated Sturm-Liouville operator. We show precisely under which conditions discrete and continuous spectra arise and give explicit examples. These conditions are then used to show that, in the most physically relevant cases, the fixed-point procedure converges. This is further demonstrated with an example.

arXiv:1203.5894•Physical Review A•May 9, 2012•31 citations
Some recent results on the convergence of Nelson diffusions are extended to the case of Schrödinger operators with time-dependent electromagnetic potentials. It is proven that the sequence {P n}n≥1 of measures on the canonical space of physical trajectories associated to the solutions of Schrödinger equations in Nelson’s scheme, corresponding to the sequence {(Vn,An)}n≥1 ⊆C1(R;ℛ×L2(R3)), converges in the total variation norm under the assumptions that for every fixed t the scalar potentials Vn(t) converge in ℛ, the space of Rollnik class potentials, and the vector potentials An(t) converge in Lloc∞(R;L2(R3)). In order to prove these results conditions are given under which solutions of Schrödinger equations are continuous in the (time-dependent electromagnetic) potentials in the norm of the Sobolev space H1(R3).
In this paper the Nelson’s stochastic mechanics is extended to general diffusion motions. A representation theorem is proved which gives a one-to-one correspondence between solutions of certain Schrödinger equations and diffusion processes satisfying appropriate regularity conditions. Exploiting results of stochastic mechanics on Riemannian manifolds it is shown that the real part of the Schrödinger equations corresponding to the considered diffusions can be interpreted as Newton’s second law where the force is produced by generalized electromagnetic potentials.
In Nelson [10], it is shown that for each solution of a Schr\"odinger equation ϕ(x,t),\phi(x, t),ϕ(x,t), x∈Rd,x\in R^{d},x∈Rd, t≧0t\geqq 0t≧0 , with ∥ϕ∥=1\Vert\phi\Vert=1∥ϕ∥=1 , there exists a diffusion process which has ... probability density at time ttt given byMore precisely the following is shown in [10]: suppose that we are given a real valued function V(x),V(x),V(x), x∈Rdx\in R^{d}x∈Rd , and a complex valued function ϕ0(x),\phi_{0}(x),ϕ0​(x), x∈Rdx\in R^{d}x∈Rd , with ∥ϕ0∥=1\Vert\phi_{0}\Vert=1∥ϕ0​∥=1 , and consider the Schrodinger equation Then, under the assumption that ϕ\phiϕ and b~\tilde{b}b~ are sufficiently regular, there exists a diffusion process {Xt},\{X_{t}\},{Xt​}, t≧0t\geqq 0t≧0 , with initial density ∣ϕ0(x)∣2|\phi_{0}(x)|^{2}∣ϕ0​(x)∣2 and generator (1/2)Δ+b~(x,t)⋅∇(1/2)\Delta+\tilde{b}(x, t)\cdot\nabla(1/2)Δ+b~(x,t)⋅∇ such that the probability density of this diffusion at time t≧0t\geqq 0t≧0 is ∣ϕ(x,t)∣2|\phi(x, t)|^{2}∣ϕ(x,t)∣2 .In addition, in [10] it is also shown that the diffusion process {Xt},\{X_{t}\},{Xt​}, t≧0t\geqq 0t≧0 , solves the second order stochastic differential equation (1/2)(D∗D+DD∗)Xt=−∇V(Xt)(1/2)(D_{*}D+DD_{*})X_{t}=-\nabla V(X_{t})(1/2)(D∗​D+DD∗​)Xt​=−∇V(Xt​) , where DDD and D∗D_{*}D∗​ are Nelson's forward and backward stochastic derivatives.Generally sPeaking, the functions ϕ\phiϕ and b~\tilde{b}b~ may be singular, and the existence of the corresponding diffusion process must be studied carefully.Carlen [1,2][1, 2][1,2] , Meyer and Zheng [8] and Nagasawa [9] considered rigorously the construction of a diffusion process with a given initial density and a generator.lnlnln order to construct a Markovian propagator for the diffusion process, Carlen [1,2][1, 2][1,2] used partial differential equation methods.Meyer and Zheng [8] restricted themselves to the case when ϕ(x,t)=ϕ0(x),\phi(x, t)=\phi_{0}(x),ϕ(x,t)=ϕ0​(x), ∀t≧0\forall_{t\geqq 0}∀t≧0​ and considered the construction of the diffusion process through the relationships between Markov processes and Dirichlet forms.We may consult Nagasawa [9] about recent developments of this area.In Carlen [1], it is assumed that V(x),V(x),V(x), x∈Rdx\in R^{d}x∈Rd , is a Rellich class potential and ∥∇ϕ0∥2<∞\Vert\nabla\phi_{0}\Vert^{2}<\infty∥∇ϕ0​∥2<∞ , and shown that there exists a diffusion process {Xt},\{X_{t}\},{Xt​}, t≧0t\geqq 0t≧0 , which has the probability density ∣ϕ(x,t)∣2|\phi(x, t)|^{2}∣ϕ(x,t)∣2 at time t≧0t\geqq 0t≧0 and admits the representation
Solutions to the Schrödinger, heat and stochastic Schrödinger equation with rather general potentials are represented, both in x- and p-representations, as integrals over the path space with respect to σ-finite measures. In the case of x-representation, the corresponding measure is concentrated on the Cameron–Martin Hilbert space of curves with L2-integrable derivatives. The case of the Schrödinger equation is treated by means of a regularization based on the introduction of either complex times or continuous non-demolition observations.
We study the convergence of the Feynman path integral formulated through broken line paths in nonrelativistic quantum mechanics. The rigorous proof of its convergence had been given little except for special cases for a long time. In the preceding paper the author showed for a class of potentials that this path integral converges and gives the probability amplitude, i.e. the solution of the Schrödinger equation. In the present paper we generalize this result, showing the boundedness theorem on the weighted Sobolev spaces for some integral operators and joining this boundedness theorem to the method in the preceding paper. We note that the result obtained is gauge invariant.
In a recent paper [Europhys. Lett. 95, 13001 (2011)] the question of whether the density of a time-dependent quantum system determines its external potential was reformulated as a fixed-point problem. This idea was used to generalize the existence and uniqueness theorems underlying time-dependent density-functional theory. In this work we extend this proof to allow for more general norms and provide a numerical implementation of the fixed-point iteration scheme. We focus on the one-dimensional case because it allows for a more in-depth analysis using singular Sturm-Liouville theory and at the same time provides an easy visualization of the numerical applications in space and time. We give an explicit relation between the boundary conditions on the density and the convergence properties of the fixed-point procedure via the spectral properties of the associated Sturm-Liouville operator. We show precisely under which conditions discrete and continuous spectra arise and give explicit examples. These conditions are then used to show that, in the most physically relevant cases, the fixed-point procedure converges. This is further demonstrated with an example.