Fractional calculus for the one-dimensional Laplacian with Robin boundary condition

We study fractional powers of the one-dimensional Robin Laplacian $A_h=-d^2/d{x}^2$ on the half-line, subject to the boundary condition $u^{\prime }(0)=hu(0)$, in the non-negative regime $h\ge 0$. The analysis is based on a Robin-deformed cosine transform with kernel

$${\cos }_h(x,\lambda )=\cos (\lambda x)+\frac{h}{\lambda }\sin (\lambda x),$$

which diagonalizes $A_h$. We prove a product formula for these eigenfunctions and use it to define positive generalized translations and a commutative Robin convolution. We then derive an explicit Robin heat kernel and Gaussian bounds. Finally, we define the fractional powers $A_h^{s/2}$, $0<s<2$, by spectral calculus and obtain their Bochner representation and singular-integral formula in terms of the Robin translations and an associated Lévy kernel. The Caffarelli–Silvestre extension is formulated for an extension exponent $0<\sigma <1$; when it is applied to the operator $A_h^{s/2}$, one takes $\sigma =s/2$. The case $h=0$ recovers the fractional Neumann Laplacian on the half-line.