Related papers: Holomorphic Extensions of Laplacians and Their Det…

The manifold Helmholtzian (1-Laplacian) operator $\Delta_1$ elegantly generalizes the Laplace-Beltrami operator to vector fields on a manifold $\mathcal M$. In this work, we propose the estimation of the manifold Helmholtzian from point…

A second order self-adjoint operator $\Delta=S\partial^2+U$ is uniquely defined by its principal symbol $S$ and potential $U$ if it acts on half-densities. We analyse the potential $U$ as a compensating field (gauge field) in the sense that…

We consider the limiting behavior of Teichm\"uller geodesics in the universal Teichm\"uller space $T(\mathbb{H})$. Our main result states that the limits of the Teichm\"uller geodesics in the Thurston's boundary of $T(\mathbb{H})$ may…

It is well known that the matrix of a metaplectic operator with respect to phase-space shifts is concentrated along the graph of a linear symplectic map. We show that the algebra generated by metaplectic operators and by pseudodifferential…

We study the geometry of hyperconvex representations of surface groups in ${\rm PSL}(d,\mathbb{C})$ and their deformation spaces: We produce a natural holomorphic extension of the classical Ahlfors--Bers map to a product of Teichm\"uller…

Let $M$ denote a finite volume, non-compact Riemann surface without elliptic points, and let $B$ denote the Lax-Phillips scattering operator. Using the superzeta function approach due to Voros, we define a Hurwitz-type zeta function…

Let $M$ be a finite volume, non-compact hyperbolic Riemann surface, possibly with elliptic fixed points, and let $\chi$ denote a finite dimensional unitary representation of the fundamental group of $M$. Let $\Delta$ denote the hyperbolic…

This paper is the first of two papers constructing a calculus of pseudodifferential operators suitable for doing analysis on Q-rank 1 locally symmetric spaces and Riemannian manifolds generalizing these. This generalization is the interior…

An operator $T$ on a Hilbert space $\mathcal H$ is called expansive, if $\|Tx\|\geq \|x\|$ ($x\in\mathcal H$). Expansive operators $T$ quasisimilar to the unilateral shift $S_N$ of finite multiplicity $N$ are studied. It is proved that…

We provide a simple recipe for obtaining all self-adjoint extensions, together with their resolvent, of the symmetric operator $S$ obtained by restricting the self-adjoint operator $A:\D(A)\subseteq\H\to\H$ to the dense, closed with respect…

We introduce the logarithmic analogue of the Laplace-Beltrami operator on Ahlfors regular metric-measure spaces. This operator is intrinsically defined with spectral properties analogous to those of elliptic pseudo-differential operators on…

A 'holographic formula' expressing the functional determinant of the scattering operator in an asymptotically locally anti-de Sitter(ALAdS) space has been proposed in terms of a relative functional determinant of the scalar Laplacian in the…

We consider the asymptotics of the determinants of large discrete Schr\"odinger operators, i.e. "discrete Laplacian $+$ diagonal": \[T_n(f) = -[\delta_{j,j+1}+\delta_{j+1,j}] + \mbox{diag}\left(f\left(\frac{1}{n}\right),…

In this paper, we introduce a new variation of the Teichm\"{u}ller space, namely the deformation space of hyperbolic structures on a surface with both enhancement and decoration. We construct the parameterization of this deformation space,…

We study the Calder\'on problem for a logarithmic Schr\"odinger type operator of the form $L_{\Delta} +q$, where $L_{\Delta}$ denotes the logarithmic Laplacian, which arises as formal derivative $\frac{d}{ds} \big|_{s=0}(-\Delta)^s$ of the…

We study the Dirac and the Laplacian operators on orientable Riemann surfaces of arbitrary genus g. In particular we compute their determinants with twisted boundary conditions along the b-cycles. All the ingredients of the final results…

(Revised version, January 2006. S. Gouezel pointed out that, when 1<r<2, the proof in the previous version was incomplete. In fixing this gap, we simplified the argument in Section 6. In addition, there is a new appendix, with an…

We study a family of higher-derivative conformal operators $P_{2k}^{(2)}$ acting on transverse-traceless symmetric 2-tensors on generic Einstein spaces. They are a natural generalization of the well-known construction for scalars. We first…

We study Laplace-type operators on hybrid manifolds, i.e. on configurations consisting of closed two-dimensional manifolds and one-dimensional segments. Such an operator can be constructed by using the Laplace-Beltrami operators on each…

Let $G = N \rtimes A$, where $N$ is a stratified group and $A = \mathbb{R}$ acts on $N$ via automorphic dilations. Homogeneous sub-Laplacians on $N$ and $A$ can be lifted to left-invariant operators on $G$ and their sum is a sub-Laplacian…