Lecture 15, 18.376, Thu Apr 06, 2023.
Summary
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Continue with Laplace Transform.
A)
Correct misconception that continuum spectrum corresponds to branch
  cuts in the Laplace Transform. Some times this is true, but mostly
  (for unbounded domains) the culprit is a natural boundary, which
  occurs because the Green's function stops being a Green's function
  as s crosses some boundary which switches decay at infinity with
  growth.
  Example: u_tt - u_xx = 0, for x > 0, and u(0, t) = 0.
     then: s^2 U - U_xx = u(x, 0) + s u_t(x, 0) = gamma(x, s)
           and  U(0, s) = 0,
     where U is the LT in time of u.

  The Green's function for U is
      G = (1/s) sinh(s x) e^{-s y}    for 0 < x < y
      G = (1/s) sinh(s y) e^{-s x}    for 0 < y < x

  Note that there is no branch points (or cuts), no singularity at all
  in G. But across re(s) = 0, G stops being a Green's function, so that
  (generally) U = \int_0^\infty G(x, y, s) gamma(y, s) dy
  stops making sense [and cannot be extended to re(s) < 0].
  This is here the source of the continuous spectrum.

B)
Introduce notion of Quasi-Modes; then start with section 3
    "Wave equation in an interval: Dirichlet/Radiation BC (example 3)"
of the notes: Examples of pde with the Laplace Transform.
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