On the determinant of elliptic boundary value problems on a line segment. (English) Zbl 0848.34063

Burghelea, D.; Friedlander, L.; Kappeler, T.

Proc. Am. Math. Soc. 123, No. 10, 3027-3038 (1995).

The differential operator \(\sum^{2n}_{k = 0} a_k (x) (- i {d \over dx})^k\) is considered on \([0,T]\), where the \(a_k\) are complex-valued \(r \times r\) matrices depending smoothly on \(x\) and \(a_{2n}\) is nonsingular, together with boundary conditions \(\sum_{k = 0}^{ \alpha_j} b_{jk} ({d \over dx})^k u(T) = 0\), \(\sum^{\beta_j}_{k = 0} c_{jk} ({d \over dx})^k u(0) = 0\), where \(b_{jk}\), \(c_{jk}\) are \(r \times r\) matrices with \(b_{j, \alpha_j} = c_{j, \beta_j} = \text{Id}\) and \(0 \leq \alpha_1 < \alpha_2 < \dots \alpha_n \leq 2n - 1\), \(0 \leq \beta_1 < \beta_2 < \dots \beta_n \leq 2n - 1\). For separated boundary conditions \(2nr \times 2nr\) matrices \(B\) and \(C\) are constructed from \(b_{jk}\) and \(c_{jk}\). If the boundary value problem \(A\) has no nontrivial solution, \(\text{Det}_\theta A\) \((\theta\) a suitable angle) is defined by \(\log \text{Det}_\theta A : = - {d \over ds} |_{s = 0} \zeta_{A, \theta} (s)\), where \(\zeta_{A, \theta} (s)\) is the meromorphic extension of \(\zeta_{A, \theta} (s) = \sum_{j \geq 1} \lambda_1^{- s}\), \(\lambda_j\) being the eigenvalues of \(A\). The main result of the paper is the formula \[ \text{Det}_\theta A = K_\theta \exp \left\{ {i \over 2} \int^T_0 \text{tr} \bigl( a^{-1}_{2n} (x) a_{2n - 1} (a) \bigr) dx \right\} \text{det} \bigl( BY(T) - C \bigr), \] where \(Y\) is the fundamental matrix of the differential operator with \(Y(0) = \text{Id}\), and \(K_\theta\) can be explicitly calculated from the given data.

Reviewer: M.Möller (Regensburg)

MSC:

34L05	General spectral theory of ordinary differential operators
35S05	Pseudodifferential operators as generalizations of partial differential operators

Keywords:

differential operator; boundary value problem; eigenvalues

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