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In a non-homogeneous material, there is no reason for the DOS to be spatially invariant. For instance, in a finite crystal, it is unwise to expect the DOS in the bulk to equal the DOS at the surface. So, instead of a macroscopically averaged density, n(E), we use a spatially dependent density n(E,x,y,z).
The LDOS is typically probed by an STM, where the tunnel current is a function of the LDOS just beneath the tip.
See also, Michael Marder, "Condensed Matter Physics", Ch. 8
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