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Effective hamiltonians for constrained quantum systems
"... We consider the timedependent Schrödinger equation on a Riemannian manifold A with a potential that localizes a certain class of states close to a fixed submanifold C. When we scale the potential in the directions normal to C by a parameter ε ≪ 1, the solutions concentrate in an εneighborhood of C ..."
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We consider the timedependent Schrödinger equation on a Riemannian manifold A with a potential that localizes a certain class of states close to a fixed submanifold C. When we scale the potential in the directions normal to C by a parameter ε ≪ 1, the solutions concentrate in an εneighborhood of C. This situation occurs for example in quantum wave guides and for the motion of nuclei in electronic potential surfaces in quantum molecular dynamics. We derive an effective Schrödinger equation on the submanifold C and show that its solutions, suitably lifted to A, approximate the solutions of the original equation on A up to errors of order ε 3 t  at time t. Furthermore, we prove that the eigenvalues of the corresponding effective Hamiltonian below a certain energy coincide up to errors of order ε 3 with those of the full Hamiltonian under reasonable conditions. Our results hold in the situation where tangential and normal energies are of the same order, and where exchange between these energies occurs. In earlier results tangential energies were assumed to be small compared to normal energies, and rather restrictive assumptions were needed, to ensure that the separation of energies is maintained during the time evolution. Most importantly, we can allow for constraining potentials that change their shape along the submanifold, which is the typical situation in the applications mentioned above. Since we consider a very general situation, our effective Hamiltonian contains many nontrivial terms of different origin. In particular, the geometry of the normal bundle of C and a generalized Berry connection on an eigenspace bundle over C play a crucial role. In order to explain the meaning and the relevance of some of the terms in the effective Hamiltonian, we analyze in some detail the application to quantum wave guides, where C is a curve in A = R 3. This allows us to generalize two recent results on spectra of such wave guides.
GEOMETRIC VERSUS SPECTRAL CONVERGENCE FOR THE NEUMANN LAPLACIAN UNDER EXTERIOR PERTURBATIONS OF THE DOMAIN
, 901
"... Abstract. We analyze the behavior of the eigenvalues and eigenfunctions of the Laplace operator with homogeneous Neumann boundary conditions when the domain is perturbed. We show that if Ω0 ⊂ Ωǫ are bounded domains (although not necessarily uniformly bounded) and we know that the eigenvalues and eig ..."
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Abstract. We analyze the behavior of the eigenvalues and eigenfunctions of the Laplace operator with homogeneous Neumann boundary conditions when the domain is perturbed. We show that if Ω0 ⊂ Ωǫ are bounded domains (although not necessarily uniformly bounded) and we know that the eigenvalues and eigenfunctions with Neumann boundary condition in Ωǫ converge to the ones in Ω0, then necessarily we have that Ωǫ \ Ω0  → 0 while it is not necessarily true that dist(Ωǫ, Ω0) ǫ→0 − → 0. As a matter of fact we will construct an example of a perturbation where the spectra behave continuously but dist(Ωǫ,Ω0) ǫ→0 1.
Effective Dynamics for Constrained Quantum Systems
, 2009
"... We consider the time dependent Schrödinger equation on a Riemannian manifold A with a potential that localizes a certain class of states close to a fixed submanifold C, the constraint manifold. When we scale the potential in the directions normal to C by a parameter ε ≪ 1, the solutions concentrate ..."
Abstract

Cited by 1 (0 self)
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We consider the time dependent Schrödinger equation on a Riemannian manifold A with a potential that localizes a certain class of states close to a fixed submanifold C, the constraint manifold. When we scale the potential in the directions normal to C by a parameter ε ≪ 1, the solutions concentrate in an εneighborhood of the submanifold. We derive an effective Schrödinger equation on the submanifold C and show that its solutions, suitably lifted to A, approximate the solutions of the original equation on A up to errors of order ε 3 t  at time t. Our result holds in the situation where tangential and normal energies are of the same order, and where exchange between normal and tangential energies occurs. In earlier results tangential energies were assumed to be small compared to normal energies, and rather restrictive assumptions were needed, to insure that the separation of energies is maintained during the time evolution. Most importantly, we can now allow for constraining potentials that change their shape along the submanifold, which is the typical situation in applications like molecular dynamics and quantum waveguides.