Weyl's law is an asymptotic formula for the growth of the number of eigenvalues of the operator 
on a bounded region, where 
is the Laplacian.
In Cartesian coordinates, 
. Let 
be a nonempty bounded region with smooth boundary,
and let 
be
the eigenvalues of 
for Dirichlet
boundary conditions, counted with multiplicity.
The minus sign is used so that these eigenvalues are
positive. Define the counting function by
 |
(1)
|
Then, as 
,
 |
(2)
|
where 
is the 
-dimensional volume of 
and 
is the volume
of the unit ball in 
, with 
the gamma function (Weyl
1912, Ivrii 2016). Equivalently,
 |
(3)
|
In two dimensions the formula becomes 
, where 
is the area of the region. In three
dimensions it becomes 
.
Thus the full sequence of eigenvalues, including multiplicities, determines the dimension
and volume of the region. It need not determine its shape,
as illustrated by isospectral manifolds.
A sharper two-term formula holds if the periodic trajectories of the associated billiards form a set of measure
zero among unit-speed initial conditions, with respect to the measure induced
by Liouville measure on that energy surface
in phase space. For smooth boundary and 
, it is
 |
(4)
|
where 
is the 
-dimensional measure of the
boundary (Ivrii 1980, 2016). For Neumann
boundary conditions, the boundary term has a plus sign. In three dimensions the
boundary term is therefore 
for Dirichlet boundary conditions
and 
for Neumann
boundary conditions.
Further geometric information is encoded by the heat trace, which smooths the counting function. Its asymptotic
expansion does not by itself imply a corresponding pointwise expansion of 
with the same number of terms.
