Study magnetic Schrödinger operators in Euclidean space.
problem Semiclassical spectral analysis of magnetic Schrödinger operators.
method Spectral problems for Bochner-Schrödinger operator on manifolds.
result Survey and describe ideas of proofs for magnetic Schrödinger operators.
New method eliminates domain size restrictions for X-ray transform inversion.
problem Injectivity and stability of X-ray transform in convex domains.
method Semiclassical analysis to invert X-ray transform without small domain assumptions.
result Elimination of domain size restrictions for injectivity and stability.
Combines noncommutative geometry and spectral theory for new Weyl laws.
problem Developing new Weyl laws for noncommutative manifolds.
method Functional analysis, spectral theory, and Tauberian conditions.
result Generalizes and simplifies recent results on Weyl laws and integration formulas.
Researchers extend asymptotic analysis to Bergman projections with Gevrey weights.
problem Analyzing Bergman projections with Gevrey weights.
method Extending direct approach to semiclassical asymptotics to Gevrey weights using Fourier integral operators.
result Gevrey symbol amplitude of asymptotic Bergman projection with Gevrey weights and Gevrey-type growth rate.
Study eigenvalues of Toeplitz operators on symplectic manifolds with discrete wells.
problem Understanding eigenvalues of Toeplitz operators on symplectic manifolds.
method Semiclassical analysis of Toeplitz operators on high tensor powers of a positive line bundle.
result Upper bounds for low-lying eigenvalues of the Bochner-Laplacian in the semiclassical limit.
Study eigenvalues of Bochner Laplacian on symplectic manifolds.
problem Understanding low-lying eigenvalues of Bochner Laplacian on symplectic manifolds.
method Analyzes high tensor powers of positive line bundles on symplectic manifolds.
result Asymptotic expansions for low-lying eigenvalues.
Magnitude study on manifolds using fractional Laplacian.
problem Magnitude invariant of compact metric spaces via fractional Laplacian.
method Semiclassical analysis of nonlocal boundary value problem related to fractional Laplacian.
result Asymptotic expansion of magnitude in terms of curvature invariants.
Analyzes magnetic Laplacian on hyperbolic surfaces, highlighting key quantum phenomena.
problem Understanding quantum phenomena on hyperbolic surfaces with magnetic fields.
method Semiclassical analysis and mathematical modeling of the magnetic Laplacian.
result Discovers new insights into quantum behavior on hyperbolic surfaces with magnetic fields.
The geometry of supermanifolds provided with Q-structure (i.e. with odd vector field Q satisfying {Q,Q}=0), P-structure (odd symplectic structure ) and S-structure (volume element) or with various combinations of these structures is studied. The results are applied to the analysis of Batalin-Vilkovisky ap…
Develops geometric Weyl calculus for curved spacetimes.
problem Quantum field theory on curved spacetimes.
method Geometric framework for Weyl quantization on pseudo-Riemannian manifolds.
result Explicit computations of Weyl symbols for various operators.
The paper proves a new semiclassical limit result for abstract Schrödinger operators.
problem Abstract Schrödinger operators on locally compact spaces.
method Probabilistic proof using the principle of not feeling the boundary.
result A new semiclassical limit result for partition functions.
Study on quantum state entanglement using Kaehler manifolds.
problem Quantum state entanglement on Kaehler manifolds.
method Semiclassical asymptotics and pure states on spheres.
result Entropy analysis of quantum states on spheres.
We treat the Witten operator on the de Rham complex with semiclassical heat kernel methods to derive the Poincaré-Hopf theorem and degenerate generalizations of it. Thereby, we see how the semiclassical asymptotics of the Witten heat kernel are related to approaches using the Thom form of Mathai and Quillen.
Study semiclassical measures on complex hyperbolic quotients, identifying measure supports.
problem Understanding Laplacian eigenfunctions on complex hyperbolic quotients.
method Combining fractal uncertainty principle and Ratner theory to analyze measure supports.
result Semiclassical measures support is either cosphere bundle or a compact submanifold.
Study the Bochner-Schrödinger operator's trace in semiclassical limit.
problem Trace formula for Bochner-Schrödinger operator on tensor powers of line and vector bundles.
method Semiclassical analysis of the Bochner-Schrödinger operator Hp on tensor powers of a Hermitian line bundle and vector bundle. result Complete asymptotic expansion of the trace of φ(Hp) in the semiclassical limit po∞. These lectures are an introduction to formal semiclassical quantization of classical field theory. First we develop the Hamiltonian formalism for classical field theories on space time with boundary. It does not have to be a cylinder as in the usual Hamiltonian framework. Then we outline formal semiclassical quantizati…
We consider a magnetic Schrödinger operator Hh, depending on a semiclassical parameter h>0, on a compact Riemannian manifold. We assume that there is no electric field. We suppose that the minimal value b0 of the intensity of the magnetic field b is strictly positive. We give a survey of the results on asympt…
Develops a new calculus for studying operators on principal bundles.
problem Investigates G-equivariant operators on principal bundles over manifolds. method Introduces Borel-Weil calculus to analyze G-equivariant (pseudo)differential operators. result Explicit conditions for rapid mixing in dynamical systems and spectral theory results for sub-elliptic Laplacians.
Improved formulation of spinfoam quantum gravity with cosmological constant, ensuring all amplitudes are finite and providing semiclassical asymptotics.
problem Ensuring the finiteness of spinfoam amplitudes and providing semiclassical asymptotics for quantum gravity.
method Using state-integral model of PSL(2, C) Chern-Simons theory and implementing simplicity constraint. result All spinfoam amplitudes are finite and provide semiclassical asymptotics with oscillatory terms related to the Regge action.
In this paper, we study a refined L2 version of the semiclassical approximation of projectively invariant elliptic operators with invariant Morse type potentials on covering spaces of compact manifolds. We work on the level of spectral projections (and not just their traces) and obtain an information about classes of t…
Paper derives closed-form solutions for CEV model using semiclassical approximation.
problem Analyzing the constant elasticity variance (CEV) option pricing model.
method Utilizes semiclassical (WKB) approximation and Van Vleck-Morette determinant.
result Derives an exponential factor not previously considered in the kernel.
We study two quantization schemes for compact symplectic manifolds with almost complex structures. The first of these is the Spin-c quantization. We prove the analog of Kodaira vanishing for the Spin-c Dirac operator, which shows that the index space of this operator provides an honest (not virtual) vector space semicl…
We consider a compact Riemannian manifold with a Hermitian line bundle whose curvature is non-degenerate. The Laplacian acting on high tensor powers (the semiclassical regime) of the bundle exhibits a cluster of low-energy states. We demonstrate that the orthogonal projectors onto these states are the Fourier component…
Study magnetic Laplacians on hyperbolic surfaces, revealing three regimes of eigenfunction behavior.
problem Investigate semiclassical defect measures of magnetic Laplacians on hyperbolic surfaces.
method Analyze eigenfunctions in low, critical, and high energy regimes using quantum ergodicity and equidistribution.
result Eigenfunctions in different regimes converge to distinct measures: invariant, Liouville, or equidistributed.
Study resolvents of Bochner Laplacians on compact manifolds.
problem Analyzing the resolvents of Bochner Laplacians in the semiclassical limit.
method Introducing Heisenberg semiclassical pseudodifferential operators to study sections of line bundles.
result Resolvents and spectral projections of Bochner Laplacians are studied in the large power limit.
Paper bridges quantum and classical mechanics for open systems.
problem Quantum open systems with bi-Lindblad structure.
method Develops a bridge between bi-Hamiltonian structures and GKSL formalism, introducing contact-compatible Lindblad generators.
result Provides a mathematical mechanism for semiclassical limit of quantum open systems.
Introduces generalized products for pseudodifferential operators on manifolds with corners.
problem Developing a new algebraic structure for pseudodifferential operators.
method Introduces generalized products and shows their implications for pseudodifferential operators.
result Generalized products imply the existence of an algebra of pseudodifferential operators.
In this article we examine the concentration and oscillation effects developed by high-frequency eigenfunctions of the Laplace operator in a compact Riemannian manifold. More precisely, we are interested in the structure of the possible invariant semiclassical measures obtained as limits of Wigner measures correspondin…
Quantizes symplectic manifolds with bounded geometry using Berezin-Toeplitz method.
problem Quantization of symplectic manifolds with bounded geometry.
method Berezin-Toeplitz quantization theory.
result Correct semiclassical limit achieved.
Study quantizes eigenstates of Bochner-Laplacian on symplectic manifolds.
problem Quantizing eigenstates of the Bochner-Laplacian on symplectic manifolds.
method Using eigenstates of the renormalized Bochner Laplacian, we apply Berezin-Toeplitz quantization.
result The quantization has correct semiclassical behavior and a corresponding star-product is constructed.
Quantum states are not entangled if submanifold is a product.
problem Understanding entanglement in quantum states associated with product submanifolds.
method Analyzing quantum states ρN on submanifolds of product Kähler manifolds in the semiclassical limit. result States are not entangled when submanifold is a product.
The paper shows instability in Minkowski spacetime for a quantum system.
problem Linear instability of the semiclassical Einstein-Klein-Gordon system in Minkowski spacetime.
method Formulated a forcing problem for metric and state perturbations, used tensor decomposition and quantum Møller operator.
result Metric perturbations grow exponentially, bounded by a universal scale H, indicating quantum backreaction.
Study eta invariant remainder on contact manifolds, improving previous results.
problem Eta invariant remainder in metric contact manifolds.
method Analyzes remainder term in semiclassical limit, using volumes of recurrence sets of Reeb flow.
result Improves remainder term for Anosov Reeb flows and certain elliptic flows.
We discuss semiclassical asymptotics for the eigenvalues of the Witten Laplacian for compact manifolds with boundary in the presence of a general Riemannian metric. To this end, we modify and use the variational method suggested by Kordyukov, Mathai and Shubin (2005), with a more extended use of quadratic forms instead…
The semiclassical approximation for the partition function in Chern-Simons gauge theory is derived using the invariant integration method. Volume and scale factors which were undetermined and had to be fixed by hand in previous derivations are automatically taken account of in this framework. Agreement with Witten's ex…
We give a pragmatic/pedagogical discussion of using Euclidean path integral in asset pricing. We then illustrate the path integral approach on short-rate models. By understanding the change of path integral measure in the Vasicek/Hull-White model, we can apply the same techniques to "less-tractable" models such as the …
Study on Higgs bundles and hyperpolygon spaces using Hitchin metrics.
problem Investigating the Hitchin metric on moduli spaces of Higgs bundles.
method Using Hitchin hyperkähler metric and parabolic Deligne-Hitchin moduli space.
result Rescaled Hitchin metric converges to hyperpolygon space's hyperkähler metric in the semiclassical limit.
Study on hyperbolic manifolds finds measures of Laplace eigenfunctions restricted to cosphere bundles.
problem Analyzing semiclassical measures on hyperbolic manifolds.
method Adapting Dyatlov and Jin's argument to higher dimensions and using Ratner theory.
result Semiclassical measures' support contains the cosphere bundle of a compact totally geodesic submanifold.
Researchers prove Weyl laws for Schrödinger operators on noncompact manifolds.
problem Proving Weyl laws for Schrödinger operators on noncompact manifolds.
method Heat kernel asymptotics, Karamata-Hardy-Littlewood Tauberian theorem, and semiclassical analysis.
result Established both classical and semiclassical Weyl laws for Schrödinger operators on noncompact manifolds.
New method speeds up CEV option pricing for small maturities.
problem High computational times for CEV option pricing, especially for small maturities.
method Semiclassical approximation of Feynman's path integral.
result The new method is efficient and accurate compared to standard CEV solution.
The Fokker-Planck equation with diffusion coefficient quadratic in space variable, linear drift coefficient, and nonlocal nonlinearity term is considered in the framework of a model of analysis of asset returns at financial markets. For special cases of such a Fokker-Planck equation we describe a construction of exact …
The abstract semiclassicalises quantum group principal bundles to Poisson geometry.
problem Semiclassicalising quantum group principal bundles to Poisson geometry.
method The theory is developed for Poisson manifolds with Poisson-compatible contravariant connections, and for Poisson-Lie groups with bicovariant Poisson-compatible contravariant connections.
result The construction of the Poisson level of the q-Hopf fibration and the spin connection on a principal bundle. We give a new proof of Witten asymptotic conjecture for Seifert manifolds with non vanishing Euler class and one exceptional fiber. Our method is based on semiclassical analysis on a two dimensional phase space torus. We prove that the Witten-Reshetikhin-Turaev invariant of a Seifert manifold is the scalar product of t…
I consider the semiclassical approximation of the graded Chern-Simons field theories describing certain systems of topological A type branes in the large radius limit of Calabi-Yau compactifications. I show that the semiclassical partition function can be expressed in terms of a certain (differential) numerical invaria…
Quantizes the standard hyperkähler space R^(4n) without a point.
problem Quantizing the standard hyperkähler space R^(4n) without a point.
method Constructs a quantization replacing the family of Berezin-Toeplitz quantizations.
result Provides semiclassical asymptotics for the constructed quantization.
The paper proves regularity of states on manifolds with unstable dynamics.
problem Propagation of regularity in dynamical systems with unstable manifolds.
method Leafwise semiclassical pseudodifferential calculus adapted to foliated spaces.
result Pollicott-Ruelle resonant states are smooth over entire manifolds if smooth on unstable leaves.
The paper studies asymptotic expansions of operators related to Bochner-Schrödinger on Riemannian manifolds.
problem Asymptotic expansions of operators related to Bochner-Schrödinger on Riemannian manifolds.
method Analyzes the Bochner-Schrödinger operator Hp and its function φ(Hp) in L2(X,Lp⊗E), providing an asymptotic expansion of its smooth Schwartz kernel. result The trace of the operator φ(Hp) admits a complete asymptotic expansion in powers of p−1/2 as po∞. We review recent probabilistic results on covariant Schrödinger operators on vector bundles over (possibly locally infinite) weighted graphs, and explain applications like semiclassical limits. We also clarify the relationship between these results and their formal analogues on smooth (possibly noncompact) Riemannian m…