By Alexey V. Shchepetilov
The current monograph offers a brief and concise advent to classical and quantum mechanics on two-point homogenous Riemannian areas, with empahsis on areas with consistent curvature. bankruptcy 1-4 give you the simple notations from differential geometry for learning two-body dynamics in those areas. bankruptcy five bargains with the matter of discovering explicitly invariant expressions for the two-body quantum Hamiltonian. bankruptcy 6 addresses one-body difficulties in a valuable strength. bankruptcy 7 stories the classical counterpart of the quantum process of bankruptcy five. bankruptcy eight investigates a few purposes within the quantum realm, particularly for the coulomb and oscillator potentials.
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Additional info for Calculus and mechanics on two-point homogenous Riemannian spaces
This section contains basic facts, connected with the notion of 38 2 Diﬀerential Operators on Smooth Manifolds the self-adjointness (for details see [44, 144, 145]), which is used below for diﬀerential operators on Riemannian manifolds. t. t. the second one. Let T : H ⊃ Dom(T ) → H be a linear operator deﬁned on a linear subspace5 Dom(T ) ⊂ H , dense in H . Let Dom(T ∗ ) be a linear subspace of the space H , consisting of all ϕ ∈ H such that the map ψ → T ψ, ϕ H is a bounded linear functional on Dom(T ) .
Em ) is a G-invariant element from S(V ) and this is a one-to-one correspondence between polynomial invariants in V and invariant elements in S(V ). Proof. If Ag is a matrix of an action of g ∈ G in the base x1 , . . , xm of the space V ∗ , then ATg is the matrix of g-action in the base e1 , . . , em . Due to the orthogonality of G-action in V one has ATg = A−1 = Ag−1 . This completes g the proof. ,il ) ai1 · . . · ail , P (a1 , . . ,il ) be an ordered polynomial, where (i1 , . . il ) are selections from the set (1, .
Let also Kx0 ⊂ G be the stationary subgroup of a point x0 ∈ M and kx0 ⊂ g ≡ Te G be the corresponding Lie subalgebra, where e ∈ G is the unit element. Choose a subspace px0 ⊂ g such that g = px0 ⊕ kx0 (a direct sum of linear spaces). From now the point x0 is ﬁxed in this section and we omit the index x0 in notations for Kx0 , kx0 , px0 and so on. Identify the space M with the factor space G/K of left cosets. Let π1 : G → G/K be the natural projection. Denote by Lq : q1 → qq1 , Rq : q1 → q1 q, q, q1 ∈ G the left and the right shifts on the group G and by τq : x → qx, q ∈ G, x∈M the action of an element q ∈ G on M .