Effective Hamiltonians for Constrained Quantum Systems
Description:... The authors consider the time-dependent Schrödinger equation on a Riemannian manifold img style="width:14px;height:12px" src="http://cdn.mathjax.org/mathjax/2.3-latest/fonts/HTML-CSS/TeX/png/Caligraphic/Regular/120/0041.png" with a potential that localizes a certain subspace of states close to a fixed submanifold img style="width:9px;height:12px" src="http://cdn.mathjax.org/mathjax/2.3-latest/fonts/HTML-CSS/TeX/png/Caligraphic/Regular/120/0043.png" . When the authors scale the potential in the directions normal to img style="width:9px;height:12px" src="http://cdn.mathjax.org/mathjax/2.3-latest/fonts/HTML-CSS/TeX/png/Caligraphic/Regular/120/0043.png" by a parameter img style="width:8px;height:7px;margin-right:0.04em" src="http://cdn.mathjax.org/mathjax/2.3-latest/fonts/HTML-CSS/TeX/png/Math/Italic/120/03B5.png"img style="width:16px;height:12px;vertical-align:-2px;margin-right:0.059em" src="http://cdn.mathjax.org/mathjax/2.3-latest/fonts/HTML-CSS/TeX/png/Main/Regular/120/226A.png"img style="width:8px;height:12px;margin-right:0.076em" src="http://cdn.mathjax.org/mathjax/2.3-latest/fonts/HTML-CSS/TeX/png/Main/Regular/120/0031.png" , the solutions concentrate in an img style="width:8px;height:7px;margin-right:0.04em" src="http://cdn.mathjax.org/mathjax/2.3-latest/fonts/HTML-CSS/TeX/png/Math/Italic/120/03B5.png" -neighborhood of img style="width:9px;height:12px" src="http://cdn.mathjax.org/mathjax/2.3-latest/fonts/HTML-CSS/TeX/png/Caligraphic/Regular/120/0043.png" .This situation occurs for example in quantum wave guides and for the
motion of nuclei in electronic potential surfaces in quantum molecular
dynamics. The authors derive an effective Schrödinger equation on the
submanifold img style="width:9px;height:12px" src="http://cdn.mathjax.org/mathjax/2.3-latest/fonts/HTML-CSS/TeX/png/Caligraphic/Regular/120/0043.png" and show that its solutions, suitably lifted to img style="width:14px;height:12px" src="http://cdn.mathjax.org/mathjax/2.3-latest/fonts/HTML-CSS/TeX/png/Caligraphic/Regular/120/0041.png" , approximate the solutions of the original equation on img style="width:14px;height:12px" src="http://cdn.mathjax.org/mathjax/2.3-latest/fonts/HTML-CSS/TeX/png/Caligraphic/Regular/120/0041.png" up to errors of order img style="width:8px;height:7px;margin-right:0.04em" src="http://cdn.mathjax.org/mathjax/2.3-latest/fonts/HTML-CSS/TeX/png/Math/Italic/120/03B5.png" img style="width:6px;height:8px;margin-right:0.045em" src="http://cdn.mathjax.org/mathjax/2.3-latest/fonts/HTML-CSS/TeX/png/Main/Regular/085/0033.png" img style="width:3px;height:18px;vertical-align:-4px;margin-right:0.124em" src="http://cdn.mathjax.org/mathjax/2.3-latest/fonts/HTML-CSS/TeX/png/Main/Regular/120/007C.png"img style="width:6px;height:10px;margin-right:0.032em" src="http://cdn.mathjax.org/mathjax/2.3-latest/fonts/HTML-CSS/TeX/png/Math/Italic/120/0074.png"img style="width:3px;height:18px;vertical-align:-4px;margin-right:0.124em" src="http://cdn.mathjax.org/mathjax/2.3-latest/fonts/HTML-CSS/TeX/png/Main/Regular/120/007C.png" at time img style="width:6px;height:10px;margin-right:0.032em" src="http://cdn.mathjax.org/mathjax/2.3-latest/fonts/HTML-CSS/TeX/png/Math/Italic/120/0074.png" .Furthermore, the authors prove that the eigenvalues of the
corresponding effective Hamiltonian below a certain energy coincide up
to errors of order img style="width:8px;height:7px;margin-right:0.04em" src="http://cdn.mathjax.org/mathjax/2.3-latest/fonts/HTML-CSS/TeX/png/Math/Italic/120/03B5.png" img style="width:6px;height:8px;margin-right:0.045em" src="http://cdn.mathjax.org/mathjax/2.3-latest/fonts/HTML-CSS/TeX/png/Main/Regular/085/0033.png" with those of the full Hamiltonian under reasonable conditions.
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