[Paper Review] Schrödinger equations with time-dependent strong magnetic fields

Daisuke Aiba, Kenji Yajima|arXiv (Cornell University)|Feb 22, 2013

Advanced Mathematical Physics Problems7 references3 citations

TL;DR

This paper establishes the existence and uniqueness of unitary propagators for time-dependent Schrödinger equations with time-dependent vector potentials $ A(t,x) $ and electric potentials $ V(t,x) $ that are almost critically singular, particularly when magnetic fields $ B(t,x) $ are strong at infinity. Under suitable conditions on the time derivatives of $ V $ and $ A $, the authors prove that the system generates a unique unitary evolution in $ L^2(\mathbb{R}^d) $, extending results on essential selfadjointness to time-dependent, strongly singular magnetic fields.

ABSTRACT

We consider d-dimensional time dependent Schrödinger equations on the Hilbert space of square integrable functions. We assume magnetic and scalar potentials are almost critically singular with respect to spatial variables both locally and at infinity for the fixed time Schrödinger operator H(t) to be essentially self-adjoint on the compactly supported smooth functions. In particular, if magnetic field B(t,x) is very strong at infinity, the scalar potential can explode to negative infinity faster than quadratic functions. We show that equations uniquely generate unitary propagators under suitable conditions on the size and singularities of time derivatives of potentials. Basic tools are Kato's abstract theory for evolution equations, Iwatsuka's identity which rewrites H(t) to an elliptic differential operator in which B(t,x) appears explicitly, and a new diamagnetic like inequality.

Motivation & Objective

• To establish the existence and uniqueness of unitary propagators for time-dependent Schrödinger equations with time-dependent vector and scalar potentials that are almost critically singular.
• To extend the theory of essential selfadjointness to time-dependent Hamiltonians with strong magnetic fields that grow rapidly at infinity.
• To characterize conditions on the time derivatives of $ V(t,x) $ and $ A(t,x) $ that ensure the existence of a unique unitary evolution operator in $ L^2(\mathbb{R}^d) $.
• To provide a rigorous framework for quantum dynamics in the presence of time-dependent, strongly singular electromagnetic potentials.

Proposed method

• The authors use a scale of Hilbert spaces $ \mathcal{X} \subset \mathcal{H} \subset \mathcal{Y} $, where $ \mathcal{Y} $ is a dense subspace and $ \mathcal{X} $ is its dual, to handle the singularities of $ V $ and $ A $.
• They define a time-dependent quadratic form $ q_0(t) $ associated with the magnetic kinetic energy and use it to construct a reference selfadjoint operator $ \tilde{H}_0(t) $.
• A conjugation transformation $ G(t) $ is introduced to absorb time-dependent parts of the potential, transforming the original problem into one with a time-dependent but norm-continuous Hamiltonian in the scale of Hilbert spaces.
• The existence of the unitary propagator is established via an application of a general theorem on time-dependent Hamiltonians in a scale of Hilbert spaces, requiring norm continuity and regularity of the time dependence of the Hamiltonian.
• The proof relies on the closed graph theorem to show equivalence of norms in $ \mathcal{Y}_t $, and uses duality to ensure the dual space $ \mathcal{X}_t $ is independent of time as a set.
• The unitarity of the propagator is verified by showing that the $ L^2 $-norm is preserved along the evolution, using the coupling between $ \mathcal{X} $ and $ \mathcal{Y} $.

Experimental results

Research questions

• RQ1 Under what conditions on time-dependent vector and scalar potentials does the Schrödinger equation with strong magnetic fields generate a unique unitary propagator?
• RQ2 How can the essential selfadjointness of time-dependent Schrödinger operators be extended to cases where the magnetic field grows rapidly at infinity?
• RQ3 What role do the time derivatives of $ V(t,x) $ and $ A(t,x) $ play in ensuring the existence of a unitary evolution?
• RQ4 Can the dynamics of Schrödinger equations with almost critically singular potentials be rigorously defined in $ L^2(\mathbb{R}^d) $ under time-dependent magnetic fields?

Key findings

• The paper proves that under suitable conditions on the time derivatives of $ V(t,x) $ and $ A(t,x) $, the Schrödinger equation $ i\partial_t u = H(t)u $ generates a unique unitary propagator $ U(t,s) $ on $ L^2(\mathbb{R}^d) $.
• The unitary propagator satisfies strong continuity and the group property $ U(t,s)U(s,r) = U(t,r) $, with $ U(t,t) = \mathbf{1} $.
• The $ L^2 $-norm of solutions is preserved, ensuring unitarity of the evolution operator for all initial data in $ L^2(\mathbb{R}^d) $.
• The method relies on a scale of Hilbert spaces $ \mathcal{X} \subset \mathcal{H} \subset \mathcal{Y} $, with $ \mathcal{Y} $ equipped with a time-dependent quadratic form $ q_0(t) $, to control singularities.
• The time dependence of the Hamiltonian is shown to be norm continuous in $ \mathcal{B}(\mathcal{Y}, \mathcal{X}) $, which is essential for the application of the abstract existence theorem.
• The construction ensures that the propagator $ U(t,s) $ is unitary and strongly continuous on $ \mathcal{H} $, even when $ V(t,x) $ and $ A(t,x) $ are singular and $ |B(t,x)| $ grows rapidly at infinity.