  We propose an "extended Schmidt law" with explicit dependence of the star
formation efficiency (SFE=SFR/Mgas) on the stellar mass surface density. This
relation has a power-law index of 0.48+-0.04 and an 1-sigma observed scatter on
the SFE of 0.4 dex, which holds over 5 orders of magnitude in the stellar
density for individual global galaxies including various types especially the
low-surface-brightness (LSB) galaxies that deviate significantly from the
Kennicutt-Schmidt law. When applying it to regions at sub-kpc resolution of a
sample of 12 spiral galaxies, the extended Schmidt law not only holds for LSB
regions but also shows significantly smaller scatters both within and across
galaxies compared to the Kennicutt-Schmidt law. We argue that this new relation
points to the role of existing stars in regulating the SFE, thus encoding
better the star formation physics. Comparison with physical models of star
formation recipes shows that the extended Schmidt law can be reproduced by some
models including gas free-fall in a stellar-gravitational potential and
pressure-supported star formation. By implementing this new law into the
analytic model of gas accretion in Lambda CDM, we show that it can re-produce
the observed main sequence of star-forming galaxies (a relation between the SFR
and stellar mass) from z=0 up to z=2.
