Investigation of the \(\zeta\)-function of operators corresponding to a class of nonlocal elliptic problems (Q2703937)

For a certain generalized class of elliptic pseudodifferential operators, contributions of additional terms to the \(\zeta\)-function are calculated. The class of operators treated in this paper were introduced by \textit{B. Yu. Sternin} and \textit{V. E. Shalatov} [Russ. Acad. Sci., Sb., Math. 81, No. 2, 363-396 (1995); translation from Mat. Sb. 185, No. 3, 117-159 (1994; Zbl 0840.35132)]. Precisely, let \(\pi: E\to B\) be a smooth bundle with the distinguished section \(i: B\to E\), \(\widehat L\), \(\widehat C\) be pseudodifferential operators of order \(2p\) and \(2l\), respectively, on \(E\) and \(\widehat D\) a pseudodifferential operator of order \(2k\) on \(B\). Let \(i^*\), \(\pi^*\) be the induced operators of \(i\), \(\pi\) and \(i_*\), \(\pi_*\) their duals. Then NEWLINE\[NEWLINE\widehat\Phi_{11}=\widehat Li_*\widehat Di^*\widehat C,\quad \widehat\Phi_{12}= \widehat Li_*\widehat D\pi_*\widehat C,\quad \widehat\Phi_{21}=\widehat L\pi^*\widehat Di^*\widehat C,\quad\widehat L\pi^*\widehat D\pi_*\widehat C,NEWLINE\]NEWLINE are Fourier integral operators and the \(\zeta\)-function \(\zeta_{\mathcal A}\) of \({\mathcal A}=\widehat A+ \sum\widehat\Phi_{ij}\) can be defined if it satisfies the Sternin-Shatalov ellipticity condition. Here \(\widehat A\) is an elliptic pseudodifferential operator of order \(2m\) on \(E\). The order of each \(\widehat\Phi_{ij}\) is also \(2m\). Then it is shown \(\zeta_{\Phi_{ij}+A}- \zeta_A\) have poles at the points \(z_i= (n- k)/(2m)\), \(k> (3/2)\nu\), \(\nu\) is the dimension of the fibre of \(E\) (Theorem 1). Precisely NEWLINE\[NEWLINE\zeta_{\Phi_{ij}+A}- \zeta_A= \sum^{n-2m}_{k=-N} {C_k\over 1+ k/(2m)- z}+ A(z),NEWLINE\]NEWLINE where \(n\) is the dimension of \(E\) and \(A(z)\) is an entire function (Theorem 2). Exact calculations of the coefficients at the leading poles and final poles are also given. After giving the proof for the case \(\zeta_{\Phi_{12}+A}- \zeta_A\) (Section 2-4), a detailed calculus for the case \(A\) is the Laplacian is given in the final Section 5. The answer shows explicit dependence of the \(\zeta\)-function of the operator on the Riemannian volumes of \(B\) and \(E\).