Editing Self-adjoint operator (section)

=== Example: structure of the Laplacian ===
The Laplacian on '''R'''<sup>''n''</sup> is the operator
: <math>\Delta = \sum_{i=1}^n \partial_{x_i}^2.</math>

As remarked above, the Laplacian is diagonalized by the Fourier transform. Actually it is more natural to consider the ''negative'' of the Laplacian −Δ since as an operator it is non-negative; (see [[elliptic operator]]).

{{math theorem|math_statement=If ''n'' = 1, then −Δ has uniform multiplicity <math>\text{mult} = 2</math>, otherwise −Δ has uniform multiplicity <math>\text{mult} = \omega</math>.  Moreover, the measure ''μ''<sub>'''mult'''</sub> may be taken to be Lebesgue measure on [0, ∞).}}