arXiv subjects

D. Borthwick

Publications and source records attributed to D. Borthwick.

9 recordsLinked to original sources

arXiv2026

A semiclassical approach to spectral estimates for random Landau Schrodinger operators

We prove spectral properties for random Landau Schrödinger operators on $L^2(\mathbb{R}^2)$ with bounded, random potentials supported in a square $Λ_L \subset \mathbb{R}^2$ of side length $L>0$, using semiclassical pseudodifferential calculus. The semiclassical parameter $h$ is the inverse of the magnetic field strength $B > 0$. By means of the Grushin method, we are led to the analysis of an effective Hamiltonian on $L^2 (\mathbb{R})$, the principal term of which is a sum of certain compact, self-adjoint pseudodifferential operators. By analyzing these operators, we prove semiclassical Wegner and Minami estimates for the random Landau Schrodinger operator in energy intervals in the spectral bands around each Landau level.

math-ph↗

arXiv2010

Inverse scattering results for manifolds hyperbolic near infinity

We study the inverse resonance problem for conformally compact manifolds which are hyperbolic outside a compact set. Our results include compactness of isoresonant metrics in dimension two and of isophasal negatively curved metrics in dimension three. In dimensions four or higher we prove topological finiteness theorems under the negative curvature assumption.

math.SP↗

arXiv2010

Resonances for manifolds hyperbolic at infinity: optimal lower bounds on order of growth

Suppose that $(X, g)$ is a conformally compact $(n+1)$-dimensional manifold that is hyperbolic at infinity in the sense that outside of a compact set $K \subset X$ the sectional curvatures of $g$ are identically equal to minus one. We prove that the counting function for the resolvent resonances has maximal order of growth $(n+1)$ generically for such manifolds.

math.SP↗

arXiv2004

A local quantum version of the Kolmogorov theorem

Consider in $L^2 (\R^l)$ the operator family $H(ε):=P_0(\hbar,ω)+εQ_0$. $P_0$ is the quantum harmonic oscillator with diophantine frequency vector $\om$, $Q_0$ a bounded pseudodifferential operator with symbol holomorphic and decreasing to zero at infinity, and $\ep\in\R$. Then there exists $\ep^\ast >0$ with the property that if $|\ep|<\ep^\ast$ there is a diophantine frequency $\om(\ep)$ such that all eigenvalues $E_n(\hbar,\ep)$ of $H(\ep)$ near 0 are given by the quantization formula $E_α(\hbar,\ep)= {\cal E}(\hbar,\ep)+\la\om(\ep),α\ra\hbar +|\om(\ep)|\hbar/2 + \ep O(α\hbar)^2$, where $α$ is an $l$-multi-index.

math-ph↗

arXiv2003

Selberg's zeta function and the spectral geometry of geometrically finite hyperbolic surfaces

For hyperbolic Riemann surfaces of finite geometry, we study Selberg's zeta function and its relation to the relative scattering phase and the resonances of the Laplacian. As an application we show that the conjugacy class of a finitely generated, torsion-free, discrete subgroup of SL(2,R) is determined by its trace spectrum up to finitely many possibilities, thus generalizing results of McKean and Mueller to groups which are not necessarily cofinite.

math.DG↗

arXiv2001

Determinants of Laplacians and Isopolar Metrics on Surfaces of Infinite Area

We construct a determinant of the Laplacian for infinite-area surfaces which are hyperbolic near infinity and without cusps. In the case of a convex co-compact hyperbolic metric, the determinant can be related to the Selberg zeta function and thus shown to be an entire function of order two with zeros at the eigenvalues and resonances of the Laplacian. In the hyperbolic near infinity case the determinant is analyzed through the zeta-regularized relative determinant for a conformal metric perturbation. We establish that this relative determinant is a ratio of entire functions of order two with divisor corresponding to eigenvalues and resonances of the perturbed and unperturbed metrics. These results are applied to the problem of compactness in the smooth topology for the class of metrics with a given set of eigenvalues and resonances.

math.DG↗

arXiv1994

Matrix Cartan superdomains, super Toeplitz operators, and quantization

We present a general theory of non-perturbative quantization of a class of hermitian symmetric supermanifolds. The quantization scheme is based on the notion of a super Toeplitz operator on a suitable Z_2 -graded Hilbert space of superholomorphic functions. The quantized supermanifold arises as the C^* -algebra generated by all such operators. We prove that our quantization framework reproduces the invariant super Poisson structure on the classical supermanifold as Planck's constant tends to zero.

hep-th↗

arXiv1994

Supersymmetry and Fredholm modules over quantized spaces

The purpose of this paper is to apply the framework of non- commutative differential geometry to quantum deformations of a class of Kahler manifolds. For the examples of the Cartan domains of type I and flat space, we construct Fredholm modules over the quantized manifolds using the supercharges which arise in the quantization of supersymmetric generalizations of the manifolds. We compute the explicit formula for the Chern character on generators of the Toeplitz C^* -algebra.

hep-th↗

arXiv1994

Legendrian Distributions with Applications to Poincaré Series

Let $X$ be a compact Kahler manifold and $L\to X$ a quantizing holomorphic Hermitian line bundle. To immersed Lagrangian submanifolds $Λ$ of $X$ satisfying a Bohr-Sommerfeld condition we associate sequences $\{ |Λ, k\rangle \}_{k=1}^\infty$, where $\forall k$ $|Λ, k\rangle$ is a holomorphic section of $L^{\otimes k}$. The terms in each sequence concentrate on $Λ$, and a sequence itself has a symbol which is a half-form, $σ$, on $Λ$. We prove estimates, as $k\to\infty$, of the norm squares $\langle Λ, k|Λ, k\rangle$ in terms of $\int_Λσ\overlineσ$. More generally, we show that if $Λ_1$ and $Λ_2$ are two Bohr-Sommerfeld Lagrangian submanifolds intersecting cleanly, the inner products $\langleΛ_1, k|Λ_2, k\rangle$ have an asymptotic expansion as $k\to\infty$, the leading coefficient being an integral over the intersection $Λ_1\capΛ_2$. Our construction is a quantization scheme of Bohr-Sommerfeld Lagrangian submanifolds of $X$. We prove that the Poincaré series on hyperbolic surfaces are a particular case, and therefore obtain estimates of their norms and inner products.

hep-th↗