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The purpose of this paper is to prove that if the polynomial      P ( ξ  ) P(\xi )       associated with a partial differential operator      P P       on      L p ( R n ) {L^p}({R^n})      , with constant coefficients, has the growth property,      | P ( ξ  ) | −  1 = O ( | ξ  | −  r ) , | ξ  | →  ∞  |P(\xi ){|^{ - 1}} = O(|\xi {|^{ - r}}),|\xi | \to \infty       for some      r &gt; 0 r &gt; 0      , then the spectrum of      P P       is either the whole complex plane or it is the numerical range of      P ( ξ  ) P(\xi )      ; and if      P ( ξ  ) P(\xi )       has some additional property (all the coefficients of      P ( ξ  ) P(\xi )       being real, for example), then the spectrum of      P P       is the numerical range for those      p p       sufficiently close to 2. It is further shown that the growth property alone is not sufficient to ensure that the spectrum of      P P       is the numerical range of      P ( ξ  ) P(\xi )      .

The spectrum of partial differential operators on 𝐿^{𝑝}(𝑅ⁿ)