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On the asymptotic eigenvalue distribution of a pseudo-differential operator
1Department of Mathematics, Princeton University, Princeton, New Jersey 08544.
Summary
This study characterizes the number of eigenvalues for pseudo-differential operators. It relates eigenvalue distribution to the geometric properties of the operator
Area of Science:
- Mathematical analysis
- Spectral theory
- Partial differential equations
Background:
- Pseudo-differential operators are crucial in analyzing differential equations.
- Understanding the distribution of eigenvalues provides insights into operator behavior.
Purpose of the Study:
- To provide a description of the number of eigenvalues less than K for a pseudo-differential operator with a positive symbol.
- To establish a relationship between eigenvalue count and phase space geometry.
Main Methods:
- The study describes the number of eigenvalues N(K) less than K.
- This is achieved by counting unit cubes within a specific region of phase space defined by the operator's symbol.
Main Results:
- A formula is presented relating N(K) to the number of unit cubes where the symbol is less than CK.
- The order of magnitude of N(K) is determined for elliptic symbols.
Conclusions:
- The findings offer a new perspective on spectral properties of pseudo-differential operators.
- The established connection between spectral distribution and geometric measure is significant for theoretical advancements.
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