By Jakob Wachsmuth, Stefan Teufel
The authors give some thought to the time-dependent Schrodinger equation on a Riemannian manifold A with a possible that localizes a undeniable subspace of states on the subject of a hard and fast submanifold C. whilst the authors scale the capability within the instructions general to C by means of a parameter e 1, the options focus in an e -neighborhood of C. this case happens for instance in quantum wave publications and for the movement of nuclei in digital strength surfaces in quantum molecular dynamics. The authors derive a good Schrodinger equation at the submanifold C and express that its ideas, definitely lifted to A , approximate the recommendations of the unique equation on A as much as mistakes of order e three |t| at time t. moreover, the authors end up that the eigenvalues of the corresponding potent Hamiltonian less than a undeniable strength coincide as much as blunders of order e three with these of the total Hamiltonian less than average stipulations
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Extra info for Effective Hamiltonians for constrained quantum systems
4 because the spectral calculus also implies χ(Heﬀ ∗ we have AA = 1. 2 implies that the second diﬀerence is of order ε3 |t|. So it suﬃces to estimate the ﬁrst ε diﬀerence. 12). 16) and χ(H ˜ ε ) L(H,D(H ε )) 1. The latter holds because H ε is bounded from below and the support of χ ˜ is bounded from above, both independent of ε. 18) and A 1 we have shown that ε )U ε A e−iHA t − A∗ U ε∗ e−iHeff t U ε A A∗ U ε∗ χ(Heﬀ ε ε L(L2 (A,dτ )) ≤ C ε3 |t|, which was the claim. 5. g. chapter 2 in ) to χ1 .
1). 5. For any ﬁxed sociated with geﬀ ˜ε for all ε below a certain ε0 . 2 yields some unitary U we assumed that the eigenspace bundle associated with Ef is trivializable, there ˜ε . 5. 2 we have Uε Uε = Uε U0 U0 Uε = Uε∗ P0 U and ˜ε U ˜ε∗ U0∗ = U0 U0∗ = 1. 1, we next set U ε := Mρ˜∗ Uε Dε∗ Mρ∗ with ρ := dμ dμεeff dμ dμ⊗dν ε ε∗ and . 9), the unitarity of Mρ˜, Mρ , and Dε implies U U = 1. ρ˜ := Furthermore, we simply deﬁne P ε by P ε := U ε∗ U ε . Then U ε is unitary from P ε H to L2 (C, dμεeﬀ ).
Eff ) (2) ε ε holds (Heﬀ − Heﬀ ) U χ(H ε )U ε∗ L(Heff ) ε Heﬀ Step 3: It = O(ε3 ). ε This step contains the central order-by-order calculation of Heﬀ and is therefore χ ˜ by far the longest one. For any ψ we set ψ := Mρ˜ψ, ψ := U ε χ(H ε )U ε∗ ψ, and ˜ Of course, we have ψ χ = ψ˜χ , ψ˜ L2 (C,dμ) = ψ H , and ψ˜χ := Uε χ(Hε )Uε∗ ψ. eff ˜ ˜ ψχ L2 (C,dμ) ≤ ψ L2 (C,dμ) for all ψ ∈ Heﬀ . (2) We ﬁrst explain why the cut oﬀ in the deﬁnition of Heﬀ does not matter here. 3. 3. 5 b) with the same δ by Step 2.
Effective Hamiltonians for constrained quantum systems by Jakob Wachsmuth, Stefan Teufel
Categories: Mathematical Physics