EFFECTIVE HAMILTONIANS FOR THIN DIRICHLET TUBES WITH VARYING CROSS-SECTION

Abstract. We show how to translate recent results on effective Hamiltoni-ans for quantum systems constrained to a submanifold by a sharply peaked potential to quantum systems on thin Dirichlet tubes. While the structure of the problem and the form of the effective Hamiltonian stays the same, the difficulties in the proofs are different. The question whether a Schrödinger Hamiltonian, which localizes states close to a submanifold of the configuration space by large forces, may be replaced by an effective operator on the submanifold is studied extensively and in various different settings in the literature (see e.g. [1, 2, 3, 4, 5, 6, 7, 8, 9]). It is well-known that restricting the classical Hamiltonian system to the sub-manifold and then using Dirac’s approach to quantizing constrained Hamiltonian systems [10] is too restricted. For there are lots of cases where the extrinsic cur-vature of the submanifold, which never shows up in Dirac’s approach, plays a role. Therefore two other approaches have been investigated: • Soft constraints: A rapidly increasing potential is used to localize solutions

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