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

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.

### 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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### References:

 [1] D. Burghelea, L. Friedlander, and T. Kappeler, On the determinant of elliptic differential and finite difference operators in vector bundles over \?\textonesuperior , Comm. Math. Phys. 138 (1991), no. 1, 1 – 18. · Zbl 0734.58043 [2] D. Burghelea, L. Friedlander, and T. Kappeler, Regularized determinants for pseudodifferential operators in vector bundles over \?\textonesuperior , Integral Equations Operator Theory 16 (1993), no. 4, 496 – 513. · Zbl 0784.35126 [3] D. Burghelea, L. Friedlander, and T. Kappeler, Meyer-Vietoris type formula for determinants of elliptic differential operators, J. Funct. Anal. 107 (1992), no. 1, 34 – 65. · Zbl 0759.58043 [4] Tommy Dreyfus and Harry Dym, Product formulas for the eigenvalues of a class of boundary value problems, Duke Math. J. 45 (1978), no. 1, 15 – 37. · Zbl 0387.34021 [5] Robin Forman, Functional determinants and geometry, Invent. Math. 88 (1987), no. 3, 447 – 493. · Zbl 0602.58044 [6] Robin Forman, Determinants, finite-difference operators and boundary value problems, Comm. Math. Phys. 147 (1992), no. 3, 485 – 526. · Zbl 0767.58043 [7] Leonid Friedlander, The asymptotics of the determinant function for a class of operators, Proc. Amer. Math. Soc. 107 (1989), no. 1, 169 – 178. · Zbl 0694.47036 [8] D. B. Ray and I. M. Singer, \?-torsion and the Laplacian on Riemannian manifolds, Advances in Math. 7 (1971), 145 – 210. · Zbl 0239.58014 [9] R. Seeley, The resolvent of an elliptic boundary problem, Amer. J. Math. 91 (1969), 889 – 920. · Zbl 0191.11801 [10] R. T. Seeley, Analytic extension of the trace associated with elliptic boundary problems, Amer. J. Math. 91 (1969), 963 – 983. · Zbl 0191.11901 [11] A. Voros, Spectral functions, special functions and the Selberg zeta function, Comm. Math. Phys. 110 (1987), no. 3, 439 – 465. · Zbl 0631.10025
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