Effective Hamiltonians for constrained quantum systems by Jakob Wachsmuth, Stefan Teufel

By Jakob Wachsmuth, Stefan Teufel

The authors reflect on the time-dependent Schrodinger equation on a Riemannian manifold A with a possible that localizes a undeniable subspace of states as regards to a set submanifold C. whilst the authors scale the aptitude within the instructions basic to C through a parameter e 1, the recommendations focus in an e -neighborhood of C. this case happens for instance in quantum wave courses and for the movement of nuclei in digital power surfaces in quantum molecular dynamics. The authors derive a good Schrodinger equation at the submanifold C and exhibit that its suggestions, definitely lifted to A , approximate the suggestions of the unique equation on A as much as mistakes of order e three |t| at time t. in addition, the authors turn out that the eigenvalues of the corresponding potent Hamiltonian under a definite power coincide as much as blunders of order e three with these of the whole Hamiltonian lower than average stipulations

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2. 2. 3. 15) • Pε 1, L(D(H εm )) l ε ν/ε P ν/ε ε j 1, L(H) ε [H , P ] L(D(H εm+1 ,D(H εm )) [H ε , P ε ] χ(H ε ) L(H,D(H εm ) U ε L(D(H εm ),D(Heff 1, εm )) ν/ε l P ε ν/ε j 1, L(D(H ε )) = O(ε), = O(ε3 ), U ε∗ m ε ),D(H εm )) L(D(Heff 1 for all j, l, m ∈ N0 and each Borel functions χ : R → [−1, 1] satisfying supp χ ⊂ (−∞, E]. The proof can be found at the end of this subsection. 3) and its adjoint. 4. 1. i) It holds A ∈ L L2 (A, dτ ), H with 1 Aψ L2 (N C,dμ) ≤ ψ L2 (A,dτ ) ∀ ψ ∈ L2 (A, dτ ). 2.

8 vi), has coefficients in Cb∞ (C). Hence, it (0) is bounded by (Heff )2 with a constant independent of ε because all derivatives carry ˜ an ε. We notice that χ(H ˜ ε ) L(H,D(H (0) m )) 1 for all m ∈ N0 because the support eff (0) ˜ of χ ˜ is bounded independently of ε. Thus we obtain that M χ(H ˜˜ eff ) is bounded. (0) (0) ˜ ˜ The same is true for χ(H ˜ eff )M 1 − χ(H ˜ eff ) because it is operator-bounded by (0) the adjoint of M χ(H ˜ eff ). 31) correspond to bounded operators. 8 vi) and whose derivatives carry (0) at least one ε each.

3. The following heuristic sketch of the construction should give ˜ε exist. 3. Let Ef be a constraint energy band. We search for Pε ∈ L(H) with i) Pε Pε = Pε , ii) [Hε , Pε ] χ(Hε ) = O(ε3 ) The former simply means that Pε is an orthogonal projection, while the latter says that Pε χ(Hε )H is invariant under the Hamiltonian Hε up to errors of order ε3 . Since the projector P0 associated with Ef is a spectral projection of Hf , we know that [Hf , P0 ] = 0, [Ef , P0 ] = 0, and Hf P0 = Ef P0 . 7 shows that Hε = H0 +O(ε) with H0 = −ε2 Δh +Hf .

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