The sharp Weyl formula
The sharp Weyl formula
This chapter considers the sharp Weyl formula using the tools provided in the previous chapter. It attempts to prove the sharp Weyl formula which says that there is a constant c, depending on (M,g) in a natural way, so that N(λ) = cλⁿ + O(λsuperscript n minus 1). The chapter then details the sup-norm estimates for eigenfunctions and spectral clusters. Next, this chapter proves the sharp Weyl formula and in doing so, outlines a number of theorems, the first of which the chapter focuses on in establishing its sharpness and in obtaining improved bounds for its Weyl formula's error term. Finally, the chapter shows that improved bounds are also available for the remainder term in the Weyl formula when (M,g) has nonpositive sectional curvature.
Keywords: sharp Weyl formula, Weyl formula, Hadamard parametrix, sup-norm estimates, spectral asymptotics, spherical harmonics, torus, nonpositive curvature
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