The Subelliptic Heat Kernels of the Quaternionic Hopf Fibration

The main goal of this work is to study the sub-Laplacian of the unit sphere which is obtained by lifting with respect to the Hopf fibration the Laplacian of the quaternionic projective space. We obtain in particular explicit formulas for its heat kernel and deduce an expression for the Green function of the conformal sub-Laplacian and small-time asymptotics. As a byproduct of our study we also obtain several results related to the sub-Laplacian of a projected Hopf fibration.     

The Subelliptic Heat Kernel on the Anti-de Sitter Space

We study the subelliptic heat kernel of the sub-Laplacian on a 2n+1-dimensional anti-de Sitter space ?²ⁿ⁺¹ which also appears as a model space of a CR Sasakian manifold with constant negative sectional curvature. In particular we obtain an explicit and geometrically meaningful formula for the subelliptic heat kernel. The key idea is to work in a set of coordinates that reflects the symmetry coming from the Hopf fibration \(\mathbb{S}^{1}\to \mathbb{H}^{2n+1}\). A direct application is obtaining small time asymptotics of the subelliptic heat kernel. Also we derive an explicit formula for the sub-Riemannian distance on ?²ⁿ⁺¹.     

Spectral asymptotics for nonsmooth perturbations of the harmonic oscillator

We obtain the spectral asymptotics of a nonsmooth perturbation of the one-dimensional harmonic oscillator. We use the technique of perturbation theory which is based on an asymptotic presentation of the part of the kernel of the nonperturbed operator resolvent in some neighborhood of an eigenvalue.     

The mixed boundary value problem, Krein resolvent formulas and spectral asymptotic estimates

For a second-order symmetric strongly elliptic operator A on a smooth bounded open set in Rn, the mixed problem is defined by a Neumann-type condition on a part Σ₊ of the boundary and a Dirichlet condition on the other part Σ₋. We show a Kre?n resolvent formula, where the difference between its resolvent and the Dirichlet resolvent is expressed in terms of operators acting on Sobolev spaces over Σ₊. This is used to obtain a new Weyl-type spectral asymptotics formula for the resolvent difference (where upper estimates were known before), namely , where C0,+ is proportional to the area of Σ₊, in the case where A is principally equal to the Laplacian.     

The subelliptic heat kernel on the CR sphere

We study the heat kernel of the sub-Laplacian $L$ on the CR sphere $\mathbb{S }^{2n+1}$ . An explicit and geometrically meaningful formula for the heat kernel is obtained. As a by-product we recover in a simple way the Green function of the conformal sub-Laplacian $-L+n^2$ that was obtained by Geller (J Differ Geom 15:417–435, 1980), and also get an explicit formula for the sub-Riemannian distance. The key point is to work in a set of coordinates that reflects the symmetries coming from the fibration $\mathbb{S }^{2n+1} \rightarrow \mathbb{CP }^n$ .     

Operator–valued Fourier multiplier theorems on Besov spaces

We investigate analytical properties of a measure geometric Laplacian which is given as the second derivative w.r.t. two atomless finite Borel measures μ and ν with compact supports supp μ ? supp ν on the real line. This class of operators includes a generalization of the well‐known Sturm‐Liouville operator as well as of the measure geometric Laplacian given by . We obtain for this differential operator under both Dirichlet and Neumann boundary conditions similar properties as known in the classical Lebesgue case for the euclidean Laplacian like Gauß‐Green‐formula, inversion formula, compactness of the resolvent and its kernel representation w.r.t. the corresponding Green function. Finally we prove nuclearity of the resolvent and give two representations of its trace.     

On the resolvent of singular Sturm-Liouville operators with transmission conditions

In this article, we investigate the resolvent operator of singular Sturm-Liouville problem with transmission conditions. We obtain integral representations for the resolvent of this operator in terms of the spectral function. Later, we discuss some properties of the resolvent operator, such as Hilbert-Schmidt kernel property, compactness. Finally, we give a formula in terms of the spectral function for the Weyl-Titchmarsh function of this problem.     

The Sharp Lower Bound of the First Eigenvalue of the Sub-Laplacian on a Quaternionic Contact Manifold

The main technical result of the paper is a Bochner type formula for the sub-Laplacian on a quaternionic contact manifold. With the help of this formula we establish a version of Lichnerowicz’s theorem giving a lower bound of the eigenvalues of the sub-Laplacian under a lower bound on the Sp(n)Sp(1) components of the qc-Ricci curvature. It is shown that in the case of a 3-Sasakian manifold the lower bound is reached iff the quaternionic contact manifold is a round 3-Sasakian sphere. Another goal of the paper is to establish a priori estimates for square integrals of horizontal derivatives of smooth compactly supported functions. As an application, we prove a sharp inequality bounding the horizontal Hessian of a function by its sub-Laplacian on the quaternionic Heisenberg group.     

Stability Properties of Linear Volterra Discrete Systems with Nonlinear Perturbation

We consider a Volterra discrete system with nonlinear perturbation x ( n +1)= A ( n ) x ( n )+ ~ s =0 n B ( n , s ) x ( s )+ g ( n , x ( n ) and obtain necessary and sufficient conditions for stability properties of the zero solution employing the resolvent equation coupled with the variation of parameters formula.     

Large Time Behavior of Solutions to Difference Wave Operators

ABSTRACT

The Cauchy problem for two dimensional difference wave operators is considered with potentials and initial data supported in a bounded region. The large time asymptotic behavior of solutions is obtained. In contrast to the continuous case (when the problem in the Euclidian space is considered, not on the lattice) the resolvent of the corresponding stationary problem has singularities on the continuous spectrum, and they contribute to the asymptotics.     

Estimates for powers of sub-Laplacian on the non-isotropic Heisenberg group

Assume that is the sub-Laplacian on the nonisotropic Heisenberg group H _n ;Z _j ,Z _j for j = 1, 2, …,n and T are the basis of the Lie algebra h _n .We apply the Laguerre calculus to obtain the explicit kernel for the fundamental solution of the powers of L _α and the heat kernel exp{−sL _α }.Estimates for this kernel in various function spaces can be deduced easily.     

Asymptotics of the spectrum of a second-order nonselfadjoint degenerate elliptic differential operator on the interval

Some properties are studied of a degenerate elliptic operator P defined on the interval (0, 1); namely, the resolvent of P is estimated. The completeness is investigated of the system of vector functions of P, and the summability is studied by the Abel method with parentheses of the Fourier series of elements in the corresponding Hilbert spaces with respect to systems of the root vector functions of P. An asymtotic formula is obtained for the distribution of the eigenvalues of P that distinguishes the principal term of the asymptotics.     

Spectral Asymptotics for Schrödinger Operators with Periodic Point Interactions

Spectrum of the second-order differential operator with periodic point interactions in L₂(R) is investigated. Classes of unitary equivalent operators of this type are described. Spectral asymptotics for the whole family of periodic operators are calculated. It is proven that the first several terms in the asymptotics determine the class of equivalent operators uniquely. It is proven that the spectrum of the operators with anomalous spectral asymptotics (when the ratio between the lengths of the bands and gaps tends to zero at infinity) can be approximated by standard periodic “weighted” operators with step-wise density functions. It is shown that this sequence of periodic weighted operators converges in the norm resolvent sense to the formal (generalized) resolvent of the periodic “Schrödinger operator” with certain energy-dependent boundary conditions. The operator acting in an extended Hilbert space such that its resolvent restricted to L₂(R) coincides with the formal resolvent is constructed explicitly.     

Wiener path integrals and the fundamental solution for the Heisenberg Laplacian

In this paper, we present an explicit calculation of the heat kernel, fundamental solution and Schwartz kernel of the resolvent for the Heisenberg Laplacian using Wiener path integrals and their realizations via the Trotter product formula. This also gives another derivation of mehler’s formula.     

Wiener measure for Heisenberg group

We build Wiener measure for the path space on the Heisenberg group by using of the heat kernel corresponding to the sub-Laplacian and give the definition of the Wiener integral. Then we give the Feynman-Kac formula.     

Large time estimates for heat kernels in nilpotent Lie groups

The aim of this paper is to obtain some estimate for large time for the Heat kernel corresponding to a sub-Laplacian with drift term on a nilpotent Lie group. We also obtain a uniform Harnack inequality for a “bounded” family of sub-Laplacians with drift in the first commutator of the Lie algebra of the nilpotent group.     

A remark on Schatten-von Neumann properties of resolvent differences of generalized Robin Laplacians on bounded domains

In this note we investigate the asymptotic behavior of the s-numbers of the resolvent difference of two generalized self-adjoint, maximal dissipative or maximal accumulative Robin Laplacians on a bounded domain Ω with smooth boundary ∂Ω. For this we apply the recently introduced abstract notion of quasi boundary triples and Weyl functions from extension theory of symmetric operators together with Krein type resolvent formulae and well-known eigenvalue asymptotics of the Laplace-Beltrami operator on ∂Ω. It is shown that the resolvent difference of two generalized Robin Laplacians belongs to the Schatten-von Neumann class of any order p for which

     

Spectral transform for the sub-Laplacian on the Heisenberg group

We continue our analysis of nilpotent groups related to quantum mechanical systems whose Hamiltonians have polynomial interactions. For the spinless particle in a constant external magnetic field, the associated nilpotent group is the Heisenberg group. We solve the heat equation for the Heisenberg group by diagonalizing the sub-Laplacian. The unitary map to the Hilbert space in which the sub-Laplacian is a multiplication operator with positive spectrum is given. The spectral multiplicity is shown to be related to the irreducible representations of SU(2). A Lax pair, generated from the Heisenberg sub-Laplacian, is used to find operators unitarily equivalent to the sub-Laplacian, but not arising from the SL(2,R) automorphisms of the Heisenberg group. Department of Mathematics, supported in part by NSF. Department of Physics and Astronomy, supported in part by DOE.     

Trace of heat kernel,spectral zeta function and isospectral problem for sub-laplacians

In this article, we first study the trace for the heat kernel for the sub-Laplacian operator on the unit sphere in ℂ ⁿ⁺¹. Then we survey some results on the spectral zeta function which is induced by the trace of the heat kernel. In the second part of the paper, we discuss an isospectral problem in the CR setting.     

On an integro-differential equation with almost difference kernel on the half-line

We study the solvability of a class of integro-differential equations with almost difference kernel on the positive half-line. Using a special three-factor decomposition of the original integro-differential operator, we obtain sufficient conditions for the solvability of this equation in the class of tempered absolutely continuous functions. Under additional conditions on the kernel of the corresponding homogeneous equation with some value of the parameter occurring in it, we prove the existence of a nontrivial absolutely continuous solution, which, depending on the sign of the first moment of the kernel, is either a bounded function or has the asymptotics O(x), x → ∞.