Mayer-Vietoris type formula for determinants of elliptic differential operators.

*(English)*Zbl 0759.58043The paper consists of four sections and an appendix. Section 1 is an introduction. In Section 2 the definitions and auxiliary results are given. The main results are formulated and proved in Section 3: For a closed codimension one submanifold \(\Gamma\) of a compact manifold \(M\), let \(M\) be the manifold with boundary obtained by cutting \(M\) along \(\Gamma\). Let \(A\) be an elliptic differential operator on \(M\) and \(B\) and \(C\) be two complementary boundary conditions on \(\Gamma\). If \((A,B)\) is an elliptic boundary value problem on \(M_ \Gamma\), then one defines an elliptic pseudodifferential operator \(R\) of Neumann type on \(\Gamma\) and proves the following factorization formula for the \(\zeta\)-regularized determinants: \(\text{Det }A/\text{Det}(A,B)=K\text{ Det }R\), with \(K\) a local quantity depending only on the jets of the symbols of \(A\), \(B\) and \(C\) along \(\Gamma\).

In Section 4 the particular case when \(M\) has dimension 2, \(A\) is the Laplace-Beltrami operator, and \(B\) resp. \(C\) is the Dirichlet resp. Neumann boundary condition is considered.

In the appendix the asymptotics of determinants of elliptic pseudodifferential operators with parameter is considered.

In Section 4 the particular case when \(M\) has dimension 2, \(A\) is the Laplace-Beltrami operator, and \(B\) resp. \(C\) is the Dirichlet resp. Neumann boundary condition is considered.

In the appendix the asymptotics of determinants of elliptic pseudodifferential operators with parameter is considered.

Reviewer: S.Bajzaev (Dushanbe)

##### MSC:

58J05 | Elliptic equations on manifolds, general theory |

58J40 | Pseudodifferential and Fourier integral operators on manifolds |

58J52 | Determinants and determinant bundles, analytic torsion |

35S15 | Boundary value problems for PDEs with pseudodifferential operators |

##### Keywords:

elliptic differential operator; Laplace-Beltrami operator; determinants; elliptic pseudodifferential operators
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\textit{D. Burghelea} et al., J. Funct. Anal. 107, No. 1, 34--65 (1992; Zbl 0759.58043)

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