ABSTRACT
Similar to B in Section 9.3, the original rescaled “adjoint” operator B∗ is defined by the convolution (80). For the purpose of applications, we will need its restriction, defined in terms of its spectral decomposition, obtained via the semigroup representation (80). Comparing semigroups (80) and (44), we see that the only difference is in the argument of the rescaled kernel F (·). Therefore, instead of (59), we have to use the following expansion:
F ( (y e−
1 2m τ − z)(1− e−τ )− 12m )
= ∑
( ye−
1 2m τ − z)γ (1 − e−τ )− |γ|2m ,
where ( y e−
1 2m τ − z)γ = ∑(0≤δ≤γ) Cδγ e−
|γ−δ| 2m τ yγ−δ (−z)δ.
