×

zbMATH — the first resource for mathematics

Mayer-Vietoris type formula for determinants of elliptic differential operators. (English) Zbl 0759.58043
The 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.

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
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Agmon, S.; Douglis, A.; Nirenberg, L., Estimates near the boundary for solutions of elliptic partial differential equations, satisfying general boundary conditions I, Comm. pure appl. math., 12, 623-727, (1959) · Zbl 0093.10401
[2] Agranovich, M.; Dynin, A., General boundary problems for elliptic systems in n-dimensional domains, Soviet math. dokl., 3, 1321-1327, (1962)
[3] Burghelea, D.; Friedlander, L.; Kappeler, T., On the determinant of elliptic differential operators on vector bundles over S1, Comm. math. phys., 138, 1-18, (1991) · Zbl 0734.58043
[4] {\scD. Burghelea, L. Friedlander, and T. Kappeler}, Regularized determinants for pseudodifferential operators on vector bundles over S1, Integral Equations Operator Theory, to appear. · Zbl 0784.35126
[5] {\scD. Burghelea, L. Friedlander, and T. Kappeler}, Regularized determinants of elliptic boundary value problems on the line segment, in preparation. · Zbl 0848.34063
[6] {\scD. Burghelea, L. Friedlander, T. Kappeler, and P. McDonald}, On the functional log det on the space of closed curves on a sphere, O.S.U., preprint. · Zbl 0805.58062
[7] Forman, R., Functional determinants and geometry, Invent. math., 88, 447-493, (1987) · Zbl 0602.58044
[8] Friedlander, L., The asymptotic of the determinant function for a class of operators, (), 169-178 · Zbl 0694.47036
[9] Grubb, G., Singular Green operators and their spectral asymptotic, Duke math. J., 51, 477-528, (1979) · Zbl 0553.58034
[10] {\scG. Grubb}, Functional calculus of pseudo-differential boundary problems, Progress in Mathematics, Vol. 65, Birkhäuser, Boston. · Zbl 0622.35001
[11] Guillemin, V., A new proof of Weyl’s formula on the asymptotic distribution of eigenvalues, Adv. math., 55, 131-160, (1985) · Zbl 0559.58025
[12] Hörmander, L., ()
[13] Ray, D.; Singer, I, R-torsion and the Laplacian on Riemannian manifolds, Adv. math., 7, 145-210, (1971) · Zbl 0239.58014
[14] Seeley, R., Analytic extension of the trace associated with elliptic boundary problems, Amer. J. math., 91, 963-983, (1969) · Zbl 0191.11901
[15] {\scR. Seeley}, private communication.
[16] Seeley, R., The resolvent of an elliptic boundary problem, Amer. J. math., 91, 889-920, (1969) · Zbl 0191.11801
[17] Seeley, R., Complex powers of elliptic operators, (), 288-307
[18] Shubin, M.A., Pseudodifferential operators and spectral theory, (1985), Springer-Verlag Berlin/New York · Zbl 0574.39006
[19] Voros, A., Spectral function, special functions and Selberg zeta function, Comm. math. phys., 110, 439-465, (1987) · Zbl 0631.10025
[20] Wodzicki, M., Noncommutative residue in K-theory, ()
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.