By Jakob Wachsmuth, Stefan Teufel
The authors think of the time-dependent Schrodinger equation on a Riemannian manifold A with a possible that localizes a definite subspace of states on the subject of a set submanifold C. while the authors scale the aptitude within the instructions common to C via a parameter e 1, the ideas focus in an e -neighborhood of C. this example happens for instance in quantum wave courses and for the movement of nuclei in digital capability surfaces in quantum molecular dynamics. The authors derive an efficient Schrodinger equation at the submanifold C and express that its suggestions, definitely lifted to A , approximate the suggestions of the unique equation on A as much as error of order e three |t| at time t. in addition, the authors turn out that the eigenvalues of the corresponding potent Hamiltonian lower than a definite strength coincide as much as mistakes of order e three with these of the entire Hamiltonian below moderate stipulations
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Extra info for Effective Hamiltonians for constrained quantum systems
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1). 3) γ (∇⊥ = ∂xi nγ + Γγiα nα . ∂x v) i 49 50 4. THE WHOLE STORY Now let (w, v) ∈ C 1 ([−1, 1], N Ωj ) with w(0) = x, w(0) ˙ = ∂xi , & v(0) = n, ∇⊥ w˙ v = 0. 3) and the choice of the curve v in the last step. 6. 3). Let τ, τ1 , τ2 ∈ Γ(T C) and ψ, ψ1 , ψ2 ∈ C 2 C, Hf (q) . ,k as above. e. d ψ1 |ψ2 Hf (τ ) = ∇hτ ψ1 |ψ2 Hf + ψ1 |∇hτ ψ2 Hf . Since ∇⊥ is a metric connection, Γγiα is anti-symmetric in α and γ, in particular Γα iα = 0 for all α. Therefore an integration by parts yields that Γγiα nα ∂nγ ψ1 ψ2 Hf + ψ1 Γγiα nα ∂nγ ψ2 Hf = 0.
BB ∗ = 1 implies that BA = B, AB ∗ = B ∗ and A2 = A. By definition H Therefore RH˜ (z) − BRH (z)B ∗ = RH˜ (z) 1 − (BHB ∗ − z)BRH (z)B ∗ = RH˜ (z) 1 − B(H − z)ARH (z)B ∗ = RH˜ (z) 1 − BAB ∗ − B[H, A]RH (z)B ∗ = −RH˜ (z) B[H, A]RH (z)B ∗ . 22) For the first part of c) we compute B ∗ RH˜ (z) − BRH (z)B ∗ B χ22 (H) = −B ∗ RH˜ (z) B[H, A]RH (z)Aχ2 (H)χ2 (H) = −B ∗ RH˜ (z) B[H, A]RH (z) χ2 (H)A + [A, χ2 (H)] χ2 (H). 23) We will write CB for a constant depending only on the norms B L(D(H l ),D(H˜ l )) and B ∗ L(D(H˜ l ),D(H l )) for l ≤ m.
However, one should think of all the operators applied to φ as the adjoint applied to the corresponding term containing ψ. 6 implies 2 Re ϕf |∇h ϕf Hf = ∇hτ ϕf |ϕf Hf + ϕf |∇hτ ϕf Hf = d ϕf |ϕf Hf (τ ) = 0. 3. 25) = φ˜∗ ϕf |Hf ϕf C Hf Hf . 39) C ˜ ∗ , ψ˜χ (r1 + r2 ) + g φ˜∗ (r1 + r2 )∗ , −iεdψ˜χ dμ g (−iεdφ) with VBH = Nq C ε geff (∇h ϕ∗f , (1 − P0 )∇h ϕf ) dν, = − iεdψ − Im ε ϕf |∇h ϕf pεeff ψ Hf − ε2 Nq C + ε2 ϕf 2 W( . ) − ϕf | W( . )ϕf 2 3 ϕ∗f R ∇v ϕf , ν ν dν Hf ∇h ϕf Hf ψ, as well as r1 := Im R1 for R1 := ϕf 2 W( .