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This lecture covers the derivation of the fixed energy propagator and the integrated density of states, focusing on the Van Vleck - Pauli - Morete formula. The instructor explains the semiclassical propagator, barrier penetration, and the analytical structure of the propagator. The lecture also delves into the explicit formula for the propagator and its application in solving the Schrödinger equation. Key concepts include Wronskian, retarded propagator, and the relationship between eigenfunctions and eigenvalues of the Hamiltonian.