Collapsing and Dirac-Type Operators

We analyze the limit of the spectrum of a geometric Dirac-type operator under a collapse with bounded diameter and bounded sectional curvature. In the case of a smooth limit space B, we show that the limit of the spectrum is given by the spectrum of a certain first-order differential operator on B, which can be constructed using superconnections. In the case of a general limit space X, we express the limit operator in terms of a transversally elliptic operator on a G-manifold X/ with X = X//G. As an application, we give a characterization of manifolds which do not admit uniform upper bounds, in terms of diameter and sectional curvature, on the k-th eigenvalue of the square of a Dirac-type operator. We also give a formula for the essential spectrum of a Dirac-type operator on a finite-volume manifold with pinched negative sectional curvature.     

Relative influence of two small parameters in non uniformly elliptic homogenization problems

We study three elliptic problems depending on two small parameters (? = homogenization parameter and δ = perturbation parameter which causes non uniform ellipticity). In each case, the homogenized operator corresponding to the second order operator is independent of the way (?, δ) → (0, 0), but the convergence results and the limit solution do depend on the relative size of ? and δ; the relevant parameter is δ?^?2.     

Continuity of the spectrum on closed similarity orbits

We present a useful case when the spectral radius of a norm limit of operator similar to a fixed operatorT still equals that ofT.This work was supported in part by grants from the National Science Foundation DMS-8811084, ECS-9001371, ECS-9122106, by the Air Force Office of Scientific Research AFOSR-90-0024 and AFOSR-90-0053, and by the Army Research Office DAAL03-91-G-0019.     

On the structure of the solution of singularly perturbed initial boundary value problems with an unbounded spectrum of the limit operator

Singularly perturbed initial boundary value problems are studied for some classes of linear systems of ordinary differential equations on the semiaxis with an unbounded spectrum of the limit operator. We give a new version of the proof of the existence of a unique and bounded (as ε→+0) solution for which with the help of the splitting method we construct a uniform asymptotic expansion on the entire semiaxis and describe all singularities (reflecting the structure of the corresponding boundary layers) in closed analytic form, including the critical case in which the points of the spectrum of the limit operator can touch the imaginary axis; this supplements previous results. Translated fromMatematicheskie Zametki, Vol. 65, No. 6, pp. 831–835, June, 1999.     

Extensions,dilations, and spectral problems of singular Hamiltonian systems

In this paper, we construct a space of boundary values for minimal symmetric 1D Hamiltonian operator with defect index (1,1) (in limit‐point case at a(b) and limit‐circle case at b(a)) acting in the Hilbert space In terms of boundary conditions at a and b, all maximal dissipative, accumulative, and self‐adjoint extensions of the symmetric operator are given. Two classes of dissipative operators are studied. They are called “dissipative at a” and “dissipative at b.” For 2 cases, a self‐adjoint dilation of dissipative operator and its incoming and outgoing spectral representations are constructed. These constructions allow us to establish the scattering matrix of dilation and a functional model of the dissipative operator. Further, we define the characteristic function of the dissipative operators in terms of the Weyl‐Titchmarsh function of the corresponding self‐adjoint operator. Finally, we prove theorems on completeness of the system of root vectors of the dissipative operators.     

On the image of the limit q‐Bernstein operator

The limit q‐Bernstein operator B_q emerges naturally as an analogue to the Szász–Mirakyan operator related to the Euler distribution. Alternatively, B_q comes out as a limit for a sequence of q‐Bernstein polynomials in the case 0<q<1. Lately, different properties of the limit q‐Bernstein operator and its iterates have been studied by a number of authors. In particular, it has been shown that B_q is a positive shape‐preserving linear operator on C[0, 1] with ∥B_q∥=1, which possesses the following remarkable property: in general, it improves the analytic properties of a function. In this paper, new results on the properties of the image of B_q are presented. Copyright © 2009 John Wiley & Sons, Ltd.     

The Schr?dinger equation on cylinders and the n-torus

In this paper, we study the solutions to the Schr?dinger equation on some conformally flat cylinders and on the n-torus. First, we apply an appropriate regularization procedure. Using the Clifford algebra calculus with an appropriate Witt basis, the solutions can be expressed as multiperiodic eigensolutions to the regularized parabolic-type Dirac operator. We study their fundamental properties, give representation formulas of all these solutions in terms of multiperiodic generalizations of the elliptic functions in the context of the regularized parabolic-type Dirac operator. Furthermore, we also develop some integral representation formulas. In particular, we set up a Green type integral formula for the solutions to the homogeneous regularized Schr?dinger equation on cylinders and n-tori. Then, we treat the inhomogeneous Schr?dinger equation with prescribed boundary conditions in Lipschitz domains on these manifolds. We present an L _p -decomposition where one of the components is the kernel of the first-order differential operator that factorizes the cylindrical (resp. toroidal) Schr?dinger operator. Finally, we study the behavior of our results in the limit case where the regularization parameter tends to zero.     

Limit behaviour of the solution of an evolution boundary‐value problem involving the p‐Laplacian operator for the case of an equivalued condition on a shrinking surface

In this paper, we discuss the limit behaviour of the solution of an evolution boundary‐value problem involving the p‐Laplacian operator for the case of an equivalued condition on a shrinking surface. Copyright © 2004 John Wiley & Sons, Ltd.     

Semiclassical Spectral Asymptotics for a Two-Dimensional Magnetic Schrödinger Operator II: The Case of Degenerate Wells

We continue our study of a magnetic Schrödinger operator on a two-dimensional compact Riemannian manifold in the case when the minimal value of the module of the magnetic field is strictly positive. We analyze the case when the magnetic field has degenerate magnetic wells. The main result of the paper is an asymptotics of the groundstate energy of the operator in the semiclassical limit. The upper bounds are improved in the case when we have a localization by a miniwell effect of lowest order. These results are applied to prove the existence of an arbitrary large number of spectral gaps in the semiclassical limit in the corresponding periodic setting.     

Convolution Inequalities in Lorentz Spaces

In this paper we study boundedness of the convolution operator in different Lorentz spaces. We obtain the limit case of the Young-O’Neil inequality in the classical Lorentz spaces. We also investigate the convolution operator in the weighted Lorentz spaces.     

A representation theorem for smooth Brownian martingales

We show that, under certain smoothness conditions, a Brownian martingale, when evaluated at a fixed time, can be represented via an exponential formula at a later time. The time-dependent generator of this exponential operator only depends on the second order Malliavin derivative operator evaluated along a ‘frozen path’. The exponential operator can be expanded explicitly to a series representation, which resembles the Dyson series of quantum mechanics. Our continuous-time martingale representation result can be proven independently by two different methods. In the first method, one constructs a time-evolution equation, by passage to the limit of a special case of a backward Taylor expansion of an approximating discrete-time martingale. The exponential formula is a solution of the time-evolution equation, but we emphasize in our article that the time-evolution equation is a separate result of independent interest. In the second method, we use the property of denseness of exponential functions. We provide several applications of the exponential formula, and briefly highlight numerical applications of the backward Taylor expansion.     

On the limit of square-Cesàro means of contractions on Hilbert spaces

By combining tools from functional analysis and algebraic number theory, we investigate the qualitative properties of the square-Cesàro means ${{\frac{1}{N}}\sum_{n=1}^N T^{n^2}}By combining tools from functional analysis and algebraic number theory, we investigate the qualitative properties of the square-Cesàro means \frac1N?_n=1^N Tⁿ²{{\frac{1}{N}}\sum_{n=1}^N T^{n^2}}, where T is a contraction on a Hilbert space. The existence of their limit in the strong operator topology as N → ∞ is known even for general polynomial sequences, but the limit itself has not yet been studied apart from the linear case solved by the von Neumann Ergodic Theorem. The case of the sequence n ² is the next step beyond this linear case, and we show that even though the limit operator is generally not a projection, it still has a relatively simple and interesting structure.     

Symmetries for the Ablowitz–Ladik Hierarchy: Part II. Integrable Discrete Nonlinear Schrödinger Equations and Discrete AKNS Hierarchy

In the paper, we continue to consider symmetries related to the Ablowitz–Ladik hierarchy. We derive symmetries for the integrable discrete nonlinear Schrödinger hierarchy and discrete AKNS hierarchy. The integrable discrete nonlinear Schrödinger hierarchy is in scalar form and its two sets of symmetries are shown to form a Lie algebra. We also present discrete AKNS isospectral flows, non‐isospectral flows and their recursion operator. In continuous limit these flows go to the continuous AKNS flows and the recursion operator goes to the square of the AKNS recursion operator. These discrete AKNS flows form a Lie algebra that plays a key role in constructing symmetries and their algebraic structures for both the integrable discrete nonlinear Schrödinger hierarchy and discrete AKNS hierarchy. Structures of the obtained algebras are different structures from those in continuous cases which usually are centerless Kac–Moody–Virasoro type. These algebra deformations are explained through continuous limit and degree in terms of lattice spacing parameter h.     

The conjugate heat equation and Ancient solutions of the Ricci flow

We prove Gaussian type bounds for the fundamental solution of the conjugate heat equation evolving under the Ricci flow. As a consequence, for dimension 4 and higher, we show that the backward limit of Type I κ-solutions of the Ricci flow must be a non-flat gradient shrinking Ricci soliton. This extends Perelman?s previous result on backward limits of κ-solutions in dimension 3, in which case the curvature operator is nonnegative (it follows from Hamilton–Ivey curvature pinching estimate). As an application, this also addresses an issue left in Naber (2010) [23], where Naber proves the interesting result that there exists a Type I dilation limit that converges to a gradient shrinking Ricci soliton, but that soliton might be flat. The Gaussian bounds that we obtain on the fundamental solution of the conjugate heat equation under evolving metric might be of independent interest.     

A Generalization of the Regularization Method to the Singularly Perturbed Integro-Differential Equations With Partial Derivatives

We generalize the Lomov’s regularization method to partial integro-differential equations. It turns out that the procedure for regularization and the construction of a regularized asymptotic solution essentially depend on the type of the integral operator. The most difficult is the case, when the upper limit of the integral is not a variable of differentiation. In this paper, we consider its scalar option. For the integral operator with the upper limit coinciding with the variable of differentiation, we investigate the vector case. In both cases, we develop an algorithm for constructing a regularized asymptotic solution and carry out its full substantiation. Based on the analysis of the principal term of the asymptotic solution, we study the limit in solution of the original problem (with the small parameter tending to zero) and solve the so-called initialization problem about allocation of a class of input data, in which the passage to the limit takes place on the whole considered period of time, including the area of boundary layer.     

The Low-Temperature Limit of Transfer Operators in Fixed Dimension

We construct the 0'th order low-temperature WKB-phase for the first eigenfunction of a transfer operator in a domain around a non-degenerate critical point for the potential. The 0'th order low-temperature phase is shown to solve the eikonal equation in the strong-coupling limit and we obtain estimates on the 0'th order phase, which are preserved in the limit. We furthermore use the IMS localization technique to study the two highest eigenvalues of the transfer operator in the case where the potential is allowed to have many non-degenerate global minima.     

The best approximation of Laplace operator by linear bounded operators in the space <Emphasis Type="Italic">L</Emphasis><Subscript><Emphasis Type="Italic">p</Emphasis></Subscript>

We obtain close two-sided estimates for the best approximation of Laplace operator by linear bounded operators on the class of functions for which the square of the Laplace operator belongs to the L _p -space. We estimate the best constant in the corresponding Kolmogorov inequality and the error of the optimal recovery of values of the Laplace operator on functions from this class defined with an error. In a particular case (p = 2) we solve all three problems exactly.     

A $$\mathbb{Z}_{2}$$-orbifold model of the symplectic fermionic vertex operator superalgebra

We give an example of an irrational C ₂-cofinite vertex operator algebra whose central charge is −2d for any positive integer d. This vertex operator algebra is given as the even part of the vertex operator superalgebra generated by d pairs of symplectic fermions, and it is just the realization of the c = −2-triplet algebra given by Kausch in the case d = 1. We also classify irreducible modules for this vertex operator algebra and determine its automorphism group. This research is supported in part by a grant from Japan Society for the Promotion of Science.     

Convergence in Measure and τ-Compactness of τ-Measurable Operators,Affiliated with a Semifinite von Neumann Algebra

Let τ be a faithful normal semifinite trace on a von Neumann algebra. We establish the Leibniz criterion for sign-alternating series of τ-measurable operators and present an analogue of the criterion of series “sandwich” series for τ-measurable operators. We prove a refinement of this criterion for the τ-compact case. In terms of measure convergence topology, the criterion of τ-compactness of an arbitrary τ-measurable operator is established. We also give a sufficient condition of 1) τ-compactness of the commutator of a τ-measurable operator and a projection; 2) convergence of τ-measurable operator and projection commutator sequences to the zero operator in the measure τ.