By Ivan G. Avramidi
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Extra info for Analytic and Geometric Methods for Heat Kernel Applications in Finance
METHODS FOR SOLUTION OF PDE complement of the resolvent, that is, the set of complex numbers λ which are not regular, is called the spectrum of the operator A. Note that every eigenvalue belongs to the spectrum but not all points in the spectrum are eigenvalues. The set of all eigenvalues is called the point spectrum. The remaining part of the spectrum is called the continuous spectrum. The continuous spectrum has far reacher and more complicated structure than the point spectrum. It contains, in general, a singular part and an absolutely continuous part.
Let αij be a real nondegenerate constant symmetric matrix, β j be a real constant vector and γ be a real constant. 154) CHAPTER 2. METHODS FOR SOLUTION OF PDE 40 n αjk pj pk . 155) j,k=1 Since the operator L is elliptic the leading symbol σL (p) is positive definite for any real p = 0, that is, n αjk pj pk > 0 σL (p) = for any real p = 0. 156) j,k=1 The operator L is self-adjoint if β i = 0. 157) and the boundary condition that f goes to zero at x → ±∞. Let gˆ(p) be Fourier transform of the function g.
307) n=1 is called the remainder term of the asymptotic series. The condition RN (x) = o(ϕN (x)) means, in particular, that for any fixed N lim RN (x) = 0 . 309) then the asymptotic series diverges. In general, there are three possibilities: a) asymptotic series converges to the original function; b) asymptotic series converges to a different function; c) asymptotic series diverges. One can also show that the asymptotic expansion of a function with respect to an asymptotic sequence is unique. However, two different functions can have the same asymptotic expansion.