In many papers dealing with the Schrodinger equation with magnetic potential $$u_t=i(\nabla+iA(t,x))^2u$$ the authors say that this equation can be studied with Kato's methods for abstract evolution equations. Is there someone who can suggest some reference in which this approach is used?
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1$\begingroup$ I guess you mean $u_t = i(\nabla+iA)^2 u$ (otherwise the equation is not self-adjoint). $\endgroup$– Carlo BeenakkerCommented Jan 7, 2015 at 21:17
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$\begingroup$ Do you mean operator splitting? $\endgroup$– Steve HuntsmanCommented Jan 7, 2015 at 22:15
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$\begingroup$ No I mean methods for the study of abstract evolution equation (like stable families of generator of continuous semigroups) $\endgroup$– SamCommented Jan 7, 2015 at 22:26
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$\begingroup$ What exactly are you interested in? The operator on the RHS is skew-adjoint (provided you have "good" boundary conditions and/or reasonable magnetic potential), hence it generates a unitary group on $L^2$ by Stone's Theorem. What is usually referred to as "Kato's theory" is a collection of much deeper results on admissible scalar potentials. $\endgroup$– Delio MugnoloCommented Jan 8, 2015 at 8:43
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2$\begingroup$ Quite frequently (though I cannot guarantee it) in this context the reference to "Kato's methods" refers to the developments centered around his two papers ams.org/mathscinet-getitem?mr=279626 and ams.org/mathscinet-getitem?mr=326483 The theory is strong enough that oftentimes authors just refers to it as a blackbox guaranteeing the existence of "evolution" for the linear operator. But you can probably find something interesting if you look at MathSciNet references to those two papers. $\endgroup$– Willie WongCommented Jan 8, 2015 at 8:53
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1 Answer
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Maybe the following references will be helpful: http://link.springer.com/article/10.1007%2FBF01682741 (Remarks on schrödinger operators with vector potentials, by Tosio Kato), https://projecteuclid.org/euclid.dmj/1077313102 (Schrödinger operators with magnetic fields. I. general interactions, by J. Avron, I. Herbst, and B. Simon).
Many relevant references can be found through these two more recent articles: http://arxiv.org/abs/math-ph/0510055 (Recent developments in quantum mechanics with magnetic fields, by Laszlo Erdos) and http://arxiv.org/abs/1410.8210 (Magnetic Schrödinger operators and Manes critical value, by Peter Herbrich).
answered Jan 8, 2015 at 12:58