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Using Nottale's theory of scale relativity, we derive a generalizedSchr\"odinger equation applying to dark matter halos. This equation involves alogarithmic nonlinearity associated with an effective temperature and a sourceof dissipation. Fundamentally, this wave equation arises from thenondifferentiability of the trajectories of the dark matter particles whoseorigin may be due to ordinary quantum mechanics, classical ergodic (or almostergodic) chaos, or to the fractal nature of spacetime at the cosmic scale. Thegeneralized Schr\"odinger equation involves a coefficient ${\cal D}$, possiblydifferent from $\hbar/2m$, whose value for dark matter halos is ${\calD}=1.02\times 10^{23}\, {\rm m^2/s}$. We suggest that the cold dark mattercrisis may be solved by the fractal (nondifferentiable) structure of spacetimeat the cosmic scale, or by the chaotic motion of the particles on a very longtimescale, instead of ordinary quantum mechanics. The equilibrium states of thegeneralized Schr\"odinger equation correspond to configurations with acore-halo structure. The quantumlike potential generates a solitonic core thatsolves the cusp problem of the classical cold dark matter model. Thelogarithmic nonlinearity accounts for the presence of an isothermal halo thatleads to flat rotation curves. The damping term ensures that the system relaxestowards an equilibrium state. This property is guaranteed by an $H$-theoremsatisfied by a Boltzmann-like free energy functional. In our approach, thetemperature and the friction arise from a single formalism. They correspond tothe real and imaginary parts of the complex friction coefficient present in thescale covariant equation of dynamics that is at the basis of Nottale's theoryof scale relativity.

Derivation of a generalized Schr\"odinger equation for dark matter halos from the theory of scale relativity