The reduced semiclassical description method

We develop a new version of the semiclassical analysis of a system of bound states in centrally symmetrical potentials. The set of potentials is in a 1∶1 correspondence with a certain set of pairs of functions of the orbital momentum. The first of these functions determines the usual WKB quantization condition and groups the potentials into equivalence classes. Its Mellin transform demonstrates similar behavior for the typical potentials, which allows describing the equivalence class with a small number of parameters. We can chose these parameters as the asymptotically exact estimates of the number of states. We obtain an equation that allows classifying states in a self-consistent atomic potential without knowing the explicit form of the potential. The second of these functions distinguishes the potentials within an equivalence class and also gives the first correction to the quantization condition. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 120, No. 1, pp. 99–115, July, 1999.     

An integral semiclassical method for calculating the spectra for centrally symmetric potentials

We develop a new approach to semiclassical quantization conditions that does not involve asymptotic expansions and is based on exact general properties of wave equations and their spectra. For centrally symmetric potentials, the quantization conditions depend only on a collection of integrals involving powers of the classical momentum. The energy levels calculated using these conditions are in good agreement with numerical data. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 124, No. 3, pp. 463–480, September, 2000.     

Exact solutions of nonlocal nonlinear field equations in cosmology

We consider a method for seeking exact solutions of the equation of a nonlocal scalar field in a nonflat metric. In the Friedmann-Robertson-Walker metric, the proposed method can be used in the case of an arbitrary potential except linear and quadratic potentials, and it allows obtaining solutions in quadratures depending on two arbitrary parameters. We find exact solutions for an arbitrary cubic potential, which consideration is motivated by string field theory, and also for exponential, logarithmic, and power potentials. We show that the k-essence field can be added to the model to obtain exact solutions satisfying all the Einstein equations.     

Instanton Effects and Quantum Spectral Curves

We study a spectral problem associated to the quantization of a spectral curve arising in local mirror symmetry. The perturbative WKB quantization condition is determined by the quantum periods, or equivalently by the refined topological string in the Nekrasov–Shatashvili (NS) limit. We show that the information encoded in the quantum periods is radically insufficient to determine the spectrum: there is an infinite series of instanton corrections, which are non-perturbative in \({\hbar}\), and lead to an exact WKB quantization condition. Moreover, we conjecture the precise form of the instanton corrections: they are determined by the standard or unrefined topological string free energy, and we test our conjecture successfully against numerical calculations of the spectrum. This suggests that the non-perturbative sector of the NS refined topological string contains information about the standard topological string. As an application of the WKB quantization condition, we explain some recent observations relating membrane instanton corrections in ABJM theory to the refined topological string.     

Quantum Field Theory on Affine Bundles

We develop a general framework for the quantization of bosonic and fermionic field theories on affine bundles over arbitrary globally hyperbolic spacetimes. All concepts and results are formulated using the language of category theory, which allows us to prove that these models satisfy the principle of general local covariance. Our analysis is a preparatory step towards a full-fledged quantization scheme for the Maxwell field, which emphasises the affine bundle structure of the bundle of principal U(1)-connections. As a by-product, our construction provides a new class of exactly tractable locally covariant quantum field theories, which are a mild generalization of the linear ones. We also show the existence of a functorial assignment of linear quantum field theories to affine ones. The identification of suitable algebra homomorphisms enables us to induce whole families of physical states (satisfying the microlocal spectrum condition) for affine quantum field theories by pulling back quasi-free Hadamard states of the underlying linear theories.     

Asymptotics of the quantization errors for in-homogeneous self-similar measures supported on self-similar sets

We study the quantization for in-homogeneous self-similar measures μ supported on self-similar sets.Assuming the open set condition for the corresponding iterated function system, we prove the existence of the quantization dimension for μ of order r ∈(0, ∞) and determine its exact value ξ_r. Furthermore, we show that,the ξ_r-dimensional lower quantization coefficient for μ is always positive and the upper one can be infinite. A sufficient condition is given to ensure the finiteness of the upper quantization coefficient.     

Exact controllability to trajectories for entropy solutions to scalar conservation laws in several space dimensions

We describe a new method that allows us to obtain a result of exact controllability to trajectories of multidimensional conservation laws in the context of entropy solutions and under a mere non-degeneracy assumption on the flux and a natural geometric condition.     

A Gauss–Newton Approach for Solving Constrained Optimization Problems Using Differentiable Exact Penalties

We propose a Gauss–Newton-type method for nonlinear constrained optimization using the exact penalty introduced recently by André and Silva for variational inequalities. We extend their penalty function to both equality and inequality constraints using a weak regularity assumption, and as a result, we obtain a continuously differentiable exact penalty function and a new reformulation of the KKT conditions as a system of equations. Such reformulation allows the use of a semismooth Newton method, so that local superlinear convergence rate can be proved under an assumption weaker than the usual strong second-order sufficient condition and without requiring strict complementarity. Besides, we note that the exact penalty function can be used to globalize the method. We conclude with some numerical experiments using the collection of test problems CUTE.     

Quantum generalized Toda system

We construct a ??spectral curve?? for the generalized Toda system, which allows efficiently finding its quantization. In turn, the quantization is realized using the technique of the quantum characteristic polynomial for the Gaudin system and an appropriate Alder-Kostant-Symes reduction. We also discuss some relations of this result to the recent consideration of the Drinfeld Zastava space, the monopole space, and corresponding symmetries of the Borel Yangian.     

The upper and lower quantization coefficient for Markov‐type measures

We study the asymptotic quantization error for Markov‐type measures μ on a class of ratio‐specified graph directed fractals E . Assuming a separation condition for E , we show that the quantization dimension for μ of order r exists and determine its exact value in terms of spectral radius of a related matrix. We prove that the ‐dimensional lower quantization coefficient for μ is always positive. Moreover, we establish a necessary and sufficient condition for the ‐dimensional upper quantization coefficient for μ to be finite.     

Functional Quantization and Small Ball Probabilities for Gaussian Processes

Quantization consists in studying the L ^r -error induced by the approximation of a random vector X by a vector (quantized version) taking a finite number n of values. We investigate this problem for Gaussian random vectors in an infinite dimensional Banach space and in particular, for Gaussian processes. A precise link proved by Fehringer⁽⁴⁾ and Dereich et al. ⁽³⁾ relates lower and upper bounds for small ball probabilities with upper and lower bounds for the quantization error, respectively. We establish a complete relationship by showing that the same holds for the direction from the quantization error to small ball probabilities. This allows us to compute the exact rate of convergence to zero of the minimal L ^r -quantization error from logarithmic small ball asymptotics and vice versa.     

Stability of quantization dimension and quantization for homogeneous Cantor measures

We effect a stabilization formalism for dimensions of measures and discuss the stability of upper and lower quantization dimension. For instance, we show for a Borel probability measure with compact support that its stabilized upper quantization dimension coincides with its packing dimension and that the upper quantization dimension is finitely stable but not countably stable. Also, under suitable conditions explicit dimension formulae for the quantization dimension of homogeneous Cantor measures are provided. This allows us to construct examples showing that the lower quantization dimension is not even finitely stable. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Topological Strings from Quantum Mechanics

We propose a general correspondence which associates a non-perturbative quantum-mechanical operator to a toric Calabi–Yau manifold, and we conjecture an explicit formula for its spectral determinant in terms of an M-theoretic version of the topological string free energy. As a consequence, we derive an exact quantization condition for the operator spectrum, in terms of the vanishing of a generalized theta function. The perturbative part of this quantization condition is given by the Nekrasov–Shatashvili limit of the refined topological string, but there are non-perturbative corrections determined by the conventional topological string. We analyze in detail the cases of local \({{\mathbb{P}}^2}\), local \({{\mathbb{P}}^1 \times {\mathbb{P}}^1}\) and local \({{\mathbb{F}}_1}\). In all these cases, the predictions for the spectrum agree with the existing numerical results. We also show explicitly that our conjectured spectral determinant leads to the correct spectral traces of the corresponding operators. Physically, our results provide a non-perturbative formulation of topological strings on toric Calabi–Yau manifolds, in which the genus expansion emerges as a ’t Hooft limit of the spectral traces. Since the spectral determinant is an entire function on moduli space, it leads to a background-independent formulation of the theory. Mathematically, our results lead to precise, surprising conjectures relating the spectral theory of functional difference operators to enumerative geometry.     

Smooth exact penalty functions: a general approach

In the article, we present a new perspective on the method of smooth exact penalty functions that is becoming more and more popular tool for solving constrained optimization problems. In particular, our approach to smooth exact penalty functions allows one to apply previously unused tools (namely, parametric optimization) to the study of these functions. We give a new simple proof of local exactness of smooth penalty functions that significantly generalizes all similar results existing in the literature. We also provide new necessary and sufficient conditions for a smooth penalty function to be globally exact.     

New exact solutions with functional parameters of the Nizhnik—Veselov—Novikov equation with constant asymptotic values at infinity

We use the Zakharov—Manakov δ-dressing method to construct new classes of exact solutions with functional parameters of the hyperbolic and elliptic versions of the Nizhnik—Veselov—Novikov equation with constant asymptotic values at infinity. We show that the constructed solutions contain classes of multisoliton solutions, which at a fixed time are exact potentials of the perturbed telegraph equation (the perturbed string equation) and the two-dimensional stationary Schrödinger equation. We interpret the stationary states of a microparticle in soliton-type potential fields physically in accordance with the constructed exact wave functions for the two-dimensional stationary Schrödinger equation.     

Approximate inference in Bayesian networks using binary probability trees

The present paper introduces a new kind of representation for the potentials in a Bayesian network: Binary Probability Trees. They enable the representation of context-specific independences in more detail than probability trees. This enhanced capability leads to more efficient inference algorithms for some types of Bayesian networks. This paper explains the procedure for building a binary probability tree from a given potential, which is similar to the one employed for building standard probability trees. It also offers a way of pruning a binary tree in order to reduce its size. This allows us to obtain exact or approximate results in inference depending on an input threshold. This paper also provides detailed algorithms for performing the basic operations on potentials (restriction, combination and marginalization) directly to binary trees. Finally, some experiments are described where binary trees are used with the variable elimination algorithm to compare the performance with that obtained for standard probability trees.     

Rate of convergence of Feynman approximations of semigroups generated by the oscillator Hamiltonian

We determine the rate with which finitely multiple approximations in the Feynman formula converge to the exact expression for the equilibrium density operator of a harmonic oscillator in the linear ??-quantization. We obtain an explicit analytic expression for a finitely multiple approximation of the equilibrium density operator and the related Wigner function. We show that in the class of ??-quantizations, the equilibrium Wigner function of a harmonic oscillator is positive definite only in the case of the Weyl quantization.     

WKB method for the Dirac equation with a scalar-vector coupling

We outline a recursive method for obtaining WKB expansions of solutions of the Dirac equation in an external centrally symmetric field with a scalar-vector Lorentz structure of the interaction potentials. We obtain semiclassical formulas for radial functions in the classically allowed and forbidden regions and find conditions for matching them in passing through the turning points. We generalize the Bohr-Sommerfeld quantization rule to the relativistic case where a spin-1/2 particle interacts simultaneously with a scalar and an electrostatic external field. We obtain a general expression in the semiclassical approximation for the width of quasistationary levels, which was earlier known only for barrier-type electrostatic potentials (the Gamow formula). We show that the obtained quantization rule exactly produces the energy spectrum for Coulomb- and oscillatory-type potentials. We use an example of the funnel potential to demonstrate that the proposed version of the WKB method not only extends the possibilities for studying the spectrum of energies and wave functions analytically but also ensures an appropriate accuracy of calculations even for states with n_r 1.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 143, No. 1, pp. 83–111, April, 2005.