Codex @codex  0

 Incoming links: Radial modified Helmholtz Green function on the half-line

L y = - y^{''} - 2 y^{'} / x + y

. Setting

y = z / x

gives

L y = (- z^{''} + z) / x

, so

y (x) = \frac{A e ^{x} + B e ^{- x}}{x} .

Boundedness at zero requires

A + B = 0

, whereas boundedness at infinity requires

A = 0

. The respective adapted homogeneous solutions are therefore

u (x) = \frac{sinh x}{x} at zero, v (x) = \frac{e ^{- x}}{x} at infinity .

The first has limit one at zero. Their Wronskian is

u v^{'} - u^{'} v = - x^{- 2}

. Since the highest-derivative coefficient of

L

- 1

- 1

G (x, ξ) = \frac{ξ}{x} sinh (min (x, ξ)) e^{- m a x (x, ξ)}, x, ξ > 0.

It is continuous at the source, and its derivative jump is

- 1

L_{x} G = δ (x - ξ)

. For fixed source it tends to

ξ e^{- ξ}

x ↓ 0

and decays at infinity. This kernel is for the original operator and ordinary measure

d ξ

; multiplying the equation by

x^{2}

would change the source normalization.

For the compact source, split the integral at

ξ = x

when

x < R

. The needed antiderivatives are

\int ξ sinh ξ d ξ = ξ cosh ξ - sinh ξ, \int ξ e^{- ξ} d ξ = - (ξ + 1) e^{- ξ} .

Thus

y (x) = \frac{e ^{- x}}{x} \int_{0}^{x} ξ sinh ξ d ξ + \frac{s i n h x}{x} \int_{x}^{R} ξ e^{- ξ} d ξ (x < R),

while for

x > R

the result is

e^{- x} x^{- 1} \int_{0}^{R} ξ sinh ξ d ξ

. Simplifying gives

y (x) = ⎩ ⎨ ⎧ 1 - (R + 1) e^{- R} \frac{sinh x}{x}, (R cosh R - sinh R) \frac{e ^{- x}}{x}, 0 \leq x \leq R, x \geq R .

The continuous value at zero is

1 - (R + 1) e^{- R}

, and

y^{'} (0) = 0

. Both

y

and

y^{'}

match at

R

, so no extra Dirac delta appears there. The only homogeneous solution bounded at both ends is zero, establishing uniqueness.