 Dirichlet half-line Green function for minus d2 plus 1

ID: dirichlet-half-line-green-function-for-minus-d2-plus-1

On the positive half-line, the Green function for

- d^{2} / d x^{2} + 1

with a zero Dirichlet boundary condition at the origin and decay at infinity is

G (x, ξ) = sinh (min (x, ξ)) e^{- m a x (x, ξ)} = \frac{1}{2} (e^{- ∣ x - ξ ∣} - e^{- (x + ξ)}) .

It is continuous, has derivative jump

G_{x} (ξ +, ξ) - G_{x} (ξ -, ξ) = - 1

, and is nonnegative. It is the negative of the Dirichlet half-line Green function for d2 minus 1; the sign of the leading differential term matters.

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