Quantum scattering by Chebyshev particles

Abstract Biological cells, aerosols and dust grains are corrugated rather than spherical, yet quantum scattering theory is developed almost entirely for smooth spherical targets. We lift that restriction for a controlled family of corrugated quasi-spheres and identify the matter-wave observables that carry a readable signature of the surface structure. We study the non--relativistic scattering of a plane wave by a three-dimensional Chebyshev particle, a sphere whose radius is modulated by a Chebyshev polynomial. The surface is modeled as a boundary--wall delta-potential and the Lippmann--Schwinger equation is solved through position-dependent reflection and transmission coefficients, yielding the probability density, the differential and total cross-sections, and the quantum refractive index. The β-fold azimuthal symmetry imposes a selection rule m = m' + qβ on the coupling between partial waves; in the varepsilon → 0 limit the formalism recovers the analytical spherical delta-shell result, which we use as a benchmark. The total cross-section shows transparency windows at the zeros of j_ell(kR) for either sign of γ, which we interpret as a three-dimensional Ramsauer--Townsend effect; the attractive barrier adds sharper resonances associated with quasi-bound states. Switching on the polar undulation α breaks separability in the polar angle and couples partial waves, producing narrow resonances whose positions shift with small changes in the deformation. For attractive particles the refractive index has a negative real part and a positive imaginary part tied to the total cross-section; for repulsive particles it is smooth and monotonic. In both regimes the index approaches its vacuum value n→1 at high energy, where the particle becomes transparent.