相关论文: Geometry of determinants of elliptic operators

Determinants of invertible pseudo-differential operators (PDOs) close to positive self-adjoint ones are defined throughthe zeta-function regularization. We define a multiplicative anomaly as the ratio $\det(AB)/(\det(A)\det(B))$ considered…

Consider an elliptic self-adjoint pseudodifferential operator $A$ acting on $m$-columns of half-densities on a closed manifold $M$, whose principal symbol is assumed to have simple eigenvalues. Relying on a basis of pseudodifferential…

Consider an elliptic self-adjoint pseudodifferential operator $A$ acting on $m$-columns of half-densities on a closed manifold $M$, whose principal symbol is assumed to have simple eigenvalues. We show existence and uniqueness of $m$…

Zeta-regularized determinants are well-known to fail to be multiplicative. Hence one is lead to study the n-fold multiplicative anomaly M_n(A_1,...,A_n) :=\frac{\det_\zeta\Big(\prod_{i=1}^n A_i\Big)}{\prod_{i=1}^n \det_\zeta(A_i)} attached…

Let $A$ be an elliptic pseudodifferential operator of positive order on a compact closed manifold, and let $T$ be a pseudodifferential operator of negative order such that $T^m$ is of trace class. We compute $\log\det(A(I+T))-\log\det…

The multiplicative anomaly related to the functional regularized determinants involving products of elliptic operators is introduced and some of its properties discussed. Its relevance concerning the mathematical consistency is stressed.…

Observing that the logarithm of a product of two elliptic operators differs from the sum of the logarithms by a finite sum of operator brackets, we infer that regularised traces of this difference are local as finite sums of noncommutative…

We demonstrate a method of associating the principal symbol at a $K$-point with a linear differential operator acting between modules over a commutative algebra, and we use it to define the ellipticity of a linear differential operator in a…

Consider an h-pseudodifferential operator P, whose symbol extends holomorphically to a tubular neighborhood of the real phase space and converges sufficiently fast to 1, so that the determinant of P is well-defined. We show that the modulus…

We construct a parametrix of a resolvent of elliptic differential operators acting on half-densities on manifolds with ends. The construction is carried out by introducing suitable pseudodifferential operators compatible with the end…

The Teichmueller space Teich(S) of a surface S in genus g>1 is a totally real submanifold of the quasifuchsian space QF(S). We show that the determinant of the Laplacian det'(\Delta) on Teich(S) has a unique holomorphic extension to QF(S).…

The well-known formula $det(A\cdot B)=\det A \cdot \det B$ can be easily proved for finite dimensional matrices but it may be incorrect for the functional determinants of differential operators, including the ones which are relevant for…

We study closed extensions A of an elliptic differential operator on a manifold with conical singularities, acting as an unbounded operator on a weighted L_p-space. Under suitable conditions we show that the resolvent (\lambda-A)^{-1}…

We study the zeta-regularized determinant of a non self-adjoint elliptic operator on a closed odd-dimensional manifold. We show that, if the spectrum of the operator is symmetric with respect to the imaginary axis, then the determinant is…

The functional determinant multiplicative anomaly, or defect, is more closely investigated and explicit forms for products of linear operators are produced. I also present formulae for the defect of products of second order operators in…

We study differential operators on an elliptic curve of order higher than 2 which are algebraically integrable (i.e., finite gap). We discuss classification of such operators of order 3 with one pole, discovering exotic operators on special…

Consider an elliptic self-adjoint pseudodifferential operator $A$ acting on $m$-columns of half-densities on a closed manifold $M$, whose principal symbol is assumed to have simple eigenvalues. We show that the spectrum of $A$ decomposes,…

As it is shown in previous works, discrete periodic operators with defects are unitarily equivalent to the operators of the form $$ {\mathcal A}{\bf u}={\bf A}_0{\bf u}+{\bf A}_1\int_0^1dk_1{\bf B}_1{\bf u}+...+{\bf…

Let $\mathsf m$ be any conical (or smooth) metric of finite volume on the Riemann sphere $\Bbb CP^1$. On a compact Riemann surface $X$ of genus $g$ consider a meromorphic funciton $f: X\to {\Bbb C}P^1$ such that all poles and critical…

We study a linear elliptic differential operator of the form $\mathcal{P}=\Delta + V - \lambda$ on a quasi-asymptotically conical manifold $(M, g)$, where $g$ is a polyhomogeneous metric and $V$ is a $b$-vector field that is unbounded with…