The paper connects Schrödinger equations to geodesics on a 2-surface.
problem Understanding the relationship between Schrödinger equations and geodesics.
method Analyzes the geodesic equation of a specific metric on a 2-surface.
result Explicit solutions for the metric and geodesics in terms of the Baker--Akhiezer function for finite-gap potentials.
Symplectic method solves infinite-dimensional Schrödinger equations.
problem Solving Schrödinger equations on infinite-dimensional Hilbert spaces with unbounded Hamiltonians.
method Analytic vectors, manifolds modelled on normed spaces, symplectic differential geometry, Marsden--Weinstein reduction.
result Mapped t-dependent Schrödinger equations onto projective spaces. In this paper, the author discusses the elliptic type gradient estimate for the solution of the time-dependent Schrödinger equations on noncompact manifolds. As its application, the dimension-free Harnack inequality and the Liouville type theorem for the Schrödinger equation are proved.
Paper shows how scattering maps of Schrödinger equations relate to metrics.
problem Relating scattering maps of time-dependent Schrödinger equations to metrics.
method Analyzes scattering maps for specific classes of metrics and diffeomorphisms.
result Scattering maps differ by a compact operator if and only if metrics are related by diffeomorphism.
In this paper we study the gradient estimate for positive solutions of Schrodinger equations on locally finite graph. Then we derive Harnack's inequality for positive solutions of the Schrodinger equations. We also set up some results about Green functions of the Laplacian equation on locally finite graph. Interesting …
Reinterprets Schrödinger equation using Cartan connection for geometric investigation.
problem Investigating the geometry of the space for Schrödinger equation solutions.
method Constructs Cartan connection from scaling Lie-Bäcklund group on jet space.
result Demonstrates a new geometric approach to Schrödinger equation.
New formulation of Schrödinger connections preserves vector lengths in geometry.
problem Preserving vector lengths in non-Euclidean geometries.
method Coordinate-free formulation, differential geometry, torsion, non-metricity.
result Explicit example of non-static Einstein manifold with torsion.
Optimal Strichartz estimates for Schrödinger on Zoll manifolds.
problem Optimal Strichartz estimates for solutions to the Schrödinger equation on Zoll manifolds.
method Arithmetic properties of the spectrum of the Laplacian and bilinear oscillatory integral estimates.
result Optimal Strichartz estimates for all q≥2 in Lt,xq spaces. In this paper, we study the schrodinger equation and wave equation with the Dirichlet boundary condition on a connected finite graph. The explicit expressions for solutions are given and the energy conservations are derived. Applications to the corresponding nonlinear problems are indicated.
Geometric focusing affects dispersive estimates for Schrödinger and wave equations.
problem Long-time decay rate in dispersive estimates for Schrödinger and wave equations on non-trapping asymptotically conic manifolds and exact metric cones.
method Classifying the long-time decay rate in dispersive estimates for the Schrödinger and wave equations on non-trapping asymptotically conic manifolds and exact metric cones in terms of the intensity of geometric focusing.
result Each multiplicity of conjugate points within distance π on Y = ∂X0 leads to a |t|1/2-loss in the long-time decay order and a half-order shift in the regularity index in the dispersive estimate for the Schrödinger equation.
Study provides guarantees on neural network generalization for solving Schrödinger equations.
problem Probability of centralizer being trivial for random matrices.
method Lower bounds on probability using random matrix theory and machine learning theory.
result Guarantees on transformer-based neural networks' in-context learning ability.
Paper develops methods for estimating gradients of Finslerian Schrödinger equations.
problem Estimating gradients of Finslerian Schrödinger equations.
method Develops new Laplacian comparison theorem and applies it to Finslerian Schrödinger equation.
result Global and local Li-Yau type gradient estimates for positive solutions.
Geometrically proves WKB solutions of Schrödinger equations are resurgent.
problem Understanding resurgent behavior of WKB solutions on Riemann surfaces.
method Purely geometric approach using holomorphic Lie groupoids and spectral curves.
result Formal WKB solutions are Borel summable in almost all directions.
Develops theory linking Schrödinger equations to manifold ends, proving finiteness.
problem Understanding the number of ends in Riemannian manifolds.
method Variant of Li-Tam theory, polynomial growth analysis, Sobolev inequality.
result Finiteness of manifold ends under scaling invariant Sobolev inequality.
Derives gradient estimate for a specific nonlinear parabolic equation on Finsler manifolds.
problem Derives gradient estimate for a nonlinear parabolic equation on Finsler manifolds.
method Leverages a new Laplacian comparison theorem to derive a Li-Yau type gradient estimate.
result Establishes a Li-Yau type gradient estimate for the Finslerian logarithmic Schrödinger equation.
Study fourth order Schrödinger equation on Cartan-Hadamard manifolds, proving existence, scattering, and blow-up results.
problem Fourth order Schrödinger equation with mixed dispersion on Cartan-Hadamard manifolds.
method Fourier transform for hyperbolic space, weighted Strichartz estimates for rotationally symmetric manifolds, localized virial argument.
result Existence, scattering, and blow-up results for the equation.
This paper bridges Kahler geometry and quantum mechanics in lognormal statistical models.
problem Evolution of spectral curves in Siegel Jacobi space through Schrodinger equation.
method Kahler geometry induced on lognormal statistical manifold, Dombrowski's construction.
result Time-dependent Schrodinger equation with varying energy.
Adaptive wave model for financial option pricing is proposed, as a high-complexity alternative to the standard Black--Scholes model. The new option-pricing model, representing a controlled Brownian motion, includes two wave-type approaches: nonlinear and quantum, both based on (adaptive form of) the Schrödinger equatio…
Study dispersive estimates for Schrödinger and wave equations on a cone with specific metric.
problem Pointwise decay estimates for Schrödinger and wave equations on a product cone.
method Modified Hadamard parametrix on Y with ε>π to prove dispersive estimates. result Threshold of conjugate radius ε>π for pointwise dispersive estimates. New method preserves unitarity for Schrödinger equation learning, reducing errors and improving time generalization.
problem Learning the evolution operator for time-dependent Schrödinger equation with varying Hamiltonians.
method Linear estimator preserving weak unitarity, with theoretical error bounds and time generalization.
result Achieves up to two orders of magnitude smaller relative errors than existing methods.
Improved Strichartz estimates for Schrödinger equation on negatively curved manifolds.
problem Improving Strichartz estimates for Schrödinger equation on negatively curved compact manifolds.
method Analyzing the Schrödinger equation on negatively curved compact manifolds, obtaining improved Strichartz estimates.
result Improved Strichartz estimates, including no-loss estimates for hyperbolic surfaces.
Using the one dimensional free particle symmetries, the quantum finance symmetries are obtained. Namely, it is shown that Black-Scholes equation is invariant under Schrödinger group. In order to do this, the one dimensional free non-relativistic particle and its symmetries are revisited. To get the Black-Scholes equati…
By using the geometric concept of PDEs with prescribed curvature representations, we show that the 1+2 dimensional Landau-Lifshitz equation is gauge equivalent to a 1+2 dimensional nonlinear Schrödinger-type system. From the nonlinear Schrödinger-type system, we construct blowing up H3(R2)-solutions to the 1+…
New method estimates Schrödinger bridge potentials via empirical risk minimization.
problem Estimating Schrödinger bridge potentials from samples.
method Rewriting Schrödinger system as a fixed-point equation and estimating the potential via empirical risk minimization.
result Uniform concentration of empirical risk around population counterpart under sub-Gaussian assumptions.
We define a class of geometric flows on a complete Kähler manifold to unify some physical and mechanical models such as the motion equations of vortex filament, complex-valued mKdV equations, derivative nonlinear Schrödinger equations etc. Furthermore, we consider the existence for these flows from S1 into a complet…
Improved Strichartz estimates for Schrödinger equation on manifolds with nonpositive curvature.
problem Improving Strichartz estimates for Schrödinger equation on compact manifolds with nonpositive sectional curvature.
method Improved global kernel estimates for microlocalized operators exploiting geometric assumptions.
result No-loss LtpLxq-estimates on intervals of length logλ⋅λ−1 for all admissible pairs (p,q). We propose a new cognitive framework for option price modelling, using quantum neural computation formalism. Briefly, when we apply a classical nonlinear neural-network learning to a linear quantum Schrödinger equation, as a result we get a nonlinear Schrödinger equation (NLS), performing as a quantum stochastic filter…
Paper solves local well-posedness for Schrödinger flow into sphere with natural boundary conditions.
problem Local well-posedness of Schrödinger flow into S2 with natural boundary conditions. method Developed a new approximation scheme to solve the problem.
result Solved the local well-posedness problem for the Schrödinger flow into S2 with natural boundary conditions. We give a sharp upper bound on the vanishing order of solutions to Schrödinger equation, in the case that the potential is of class C1 on a smooth compact manifold.
Novel heat flow estimates on ALE manifolds for Schrödinger operators.
problem Estimating heat flows on ALE manifolds with non-trivial L2-kernel. method Combining Fredholm theory for Dirac type operators and heat kernel advances.
result Established Lp−Lq decay estimates for heat flows. New uncertainty principle for Schrödinger equations on hyperbolic manifolds.
problem Uncertainty principle for Schrödinger equations on hyperbolic manifolds.
method General strategy of Escauriaza-Kenig-Ponce-Vega, new Carleman estimates, logarithmic convexity, new mollifier and weight function.
result Similar rigidity phenomenon as in Euclidean space persists in hyperbolic geometry.
In this paper we establish the equivalence of solutions between Schrödinger map into S2 or H2 and their associated gauge invariant Schrödinger equations. We also establish the existence of global weak solutions into H2 in two space dimensions. We extend these ideas for maps into com…
In this paper, we introduce a new notion named as Schrödinger soliton. So-called Schrödinger solitons are defined as a class of special solutions to the Schrödinger flow equation from a Riemannian manifold or a Lorentzian manifold M into a Kähler manifold N. If the target manifold N admits a Killing potential, th…
The local induction equation, or the binormal flow on space curves is a well-known model of deformation of space curves as it describes the dynamics of vortex filaments, and the complex curvature is governed by the nonlinear Schrödinger equation. In this paper, we present its discrete analogue, namely, a model of defor…
Deep QMC methods use neural networks to solve quantum chemistry problems.
problem Solving the electronic Schrödinger equation from first principles.
method Quantum Monte Carlo with neural network wavefunctions.
result Highly accurate solutions at reduced computational cost.
Conservation laws, heirarchies, scattering theory and Bäcklund transformations are known to be the building blocks of integrable partial differential equations. We identify these as facets of a theory of Poisson group actions, and apply the theory to the ZS-AKNS nxn heirarchy (which includes the non-linear Schrödinger …
A new portfolio method using quantum mechanics improves risk diversification.
problem Improving risk-based portfolio construction methods for multi-asset portfolios.
method Schrödinger principal component analysis applied to extract common factors from asset fluctuations.
result The proposed method outperforms conventional risk parity and other risk diversification methods.
Conditions for a soliton's dual form to be harmonic or Ricci harmonic are derived.
problem Characterizing solitons and their dual forms.
method Necessary and sufficient conditions for the dual form to be harmonic or Ricci harmonic are derived.
result Conditions for the dual form of a soliton to be harmonic or Ricci harmonic are provided.
We study the question of well-posedness of the Cauchy problem for Schrödinger maps from $\rone \times \rtwo$ to the sphere $\stwo$ or to H2, the hyperbolic space. The idea is to choose an appropriate gauge change so that the derivatives of the map will satisfy a certain nonlinear Schrödinger system of equa…
Clarifies relation for solving control-affine Schrödinger bridge problems.
problem Solving control-affine Schrödinger bridge problems via Hopf-Cole transform.
method Applies Hopf-Cole transform to conditions of optimality, resulting in nonlinear PDEs.
result Generic control-affine Schrödinger bridge requires further algorithmic development.
A hyperkähler 4-metric with a triholomorphic SU(2) action gives rise to a family of confocal quadrics in Euclidean 3-space when cast in the canonical form of a hyperkähler 4-metric metric with a triholomorphic circle action. Moreover, at least in the case of geodesics orthogonal to the U(1) fibres, both the covariant S…
New quantum algorithm simplifies complex financial derivatives pricing.
problem Complex financial derivatives pricing with high dimensionality.
method Quantum-inspired variational algorithms combined with neural-network quantum states.
result Simplified pricing of European options with many correlated assets.
Uniqueness of 1D bi-Schrödinger flow proven from flat torus to compact space.
problem Proving uniqueness of a smooth flow from flat torus to compact space.
method Extrinsic approach using isometric embedding into Euclidean space, modifying classical H2-energy to handle loss of derivatives. result Uniqueness of the generalized bi-Schrödinger flow established.
This note is devoted to optimal spectral estimates for Schrödinger operators on compact connected Riemannian manifolds without boundary. These estimates are based on the use of appropriate interpolation inequalities and on some recent rigidity results for nonlinear elliptic equations on those manifolds.
We study the regularity properties for solutions of a class of Schrödinger equations (Δ+V)u=0 on a stratified space M endowed with an iterated edge metric. The focus is on obtaining optimal Hölder regularity of these solutions assuming fairly minimal conditions on the underlying metric and potential.
The geometric non-linear Schrodinger equation (GNLS) on the complex Grassmannian manifold M is the Hamiltonian equation for the energy functional on C(R,M) with respect to the symplectic form induced from the Kahler form on M. It has a Lax pair that is gauge equivalent to the Lax pair of the matrix non-linear Schroding…
New algorithm improves on existing methods for solving transport problems.
problem Finding a map to transport one distribution to another.
method Iterative Markovian Fitting (IMF) and Diffusion Schrödinger Bridge Matching (DSBM).
result DSBM significantly improves over previous SB numerics and recovers various transport methods.
The study establishes uncertainty principles on harmonic manifolds of rank one.
problem Developing uncertainty principles for harmonic manifolds of rank one.
method Derivation of various uncertainty principles including Heisenberg, Morgen, Schrödinger, and Hömanders principles.
result Generalization of Hausdorff-Young inequality to harmonic manifolds of rank one.