Quadratic Symbols Research Articles

General bilinear forms in the Jacobi symbol over hyperbolic regions

We study averages involving the Jacobi quadratic symbol (frac{n}{m}) in regions where the product mn is bounded by a large parameter. We show that these averages exhibit cancellation whenever the summation is restricted to square-free integers bounded away from the axes.

First moment of central values of quadratic Hecke L-functions in the Gaussian field

We evaluate the smoothed first moment of central values of a family of quadratic Hecke [Formula: see text]-functions in the Gaussian field using the method of double Dirichlet series. The asymptotic formula we obtain has an error term of size [Formula: see text] under the generalized Riemann hypothesis. The same approach also allows us to obtain asymptotic formulas for all [Formula: see text], [Formula: see text] for a smoothed double character sum involving [Formula: see text], where [Formula: see text] denotes the quadratic symbol in the Gaussian field.

Pointwise convergence over fractals for dispersive equations with homogeneous symbol

We study the fractal pointwise convergence for the equation iħ∂tu+P(D)u=0, where the symbol P is real, homogeneous and non-singular. We prove that for initial data f∈Hs(Rn) with s>(n−α+1)/2 the solution u converges to fHα-a.e., where Hα is the α-dimensional Hausdorff measure. We improve upon this result depending on the dispersive strength of P. On the other hand, for a family of polynomials P and given α, we exploit a Talbot-like effect to construct initial data whose solutions u diverge in sets of Hausdorff dimension α. To compute the dimension of the sets of divergence, we adopt the Mass Transference Principle from Diophantine approximation. We also construct counterexamples for quadratic symbols like the saddle to show that our positive results are sometimes best possible.

Spectra of Weighted Composition Operators with Quadratic Symbols

Abstract Previously, spectra of certain weighted composition operators W ѱ, φ on H 2 were determined under one of two hypotheses: either φ converges under iteration to the Denjoy-Wolff point uniformly on all of 𝔻 rather than simply on compact subsets, or φ is “essentially linear fractional.” We show that if φ is a quadratic self-map of 𝔻 of parabolic type, then the spectrum of Wѱ, φ can be found when these maps exhibit both of the aforementioned properties, and we determine which symbols do so.

From semigroups to subelliptic estimates for quadratic operators

Using an approach based on the techniques of FBI transforms, we give a new simple proof of the global subelliptic estimates for non-self-adjoint nonelliptic quadratic differential operators, under a natural averaging condition on the Weyl symbols of the operators, established by the second author in Subelliptic estimates for quadratic differential operators (Amer. J. Math. 133 (2011), no. 1, 39–89). The loss of the derivatives in the subelliptic estimates depends directly on algebraic properties of the Hamilton maps of the quadratic symbols. Using the FBI point of view, we also give accurate smoothing estimates of Gelfand–Shilov type for the associated heat semigroup in the limit of small times.

Null-controllability of hypoelliptic quadratic differential equations

We study the null-controllability of parabolic equations associated to a general class of hypoelliptic quadratic differential operators. Quadratic differential operators are operators defined in the Weyl quantization by complex-valued quadratic symbols. We consider in this work the class of accretive quadratic operators with zero singular spaces. These possibly degenerate non-selfadjoint differential operators are known to be hypoelliptic and to generate contraction semigroups which are smoothing in specific Gelfand-Shilov spaces for any positive time. Thanks to this regularizing effect, we prove by adapting the Lebeau-Robbiano method that parabolic equations associated to these operators are null-controllable in any positive time from control regions, for which null-controllability is classically known to hold in the case of the heat equation on the whole space. Some applications of this result are then given to the study of parabolic equations associated to hypoelliptic Ornstein-Uhlenbeck operators acting on weighted $L^2$ spaces with respect to invariant measures. By using the same strategy, we also establish the null-controllability in any positive time from the same control regions for parabolic equations associated to any hypoelliptic Ornstein-Uhlenbeck operator acting on the flat $L^2$ space extending in particular the known results for the heat equation or the Kolmogorov equation on the whole space.

Resolvent estimates for elliptic quadratic differential operators

Sharp resolvent bounds for nonselfadjoint semiclassical elliptic quadratic differential operators are established, in the interior of the range of the associated quadratic symbol.

Exponential return to equilibrium for hypoelliptic quadratic systems

We study the problem of convergence to equilibrium for evolution equations associated to general quadratic operators. Quadratic operators are non-selfadjoint differential operators with complex-valued quadratic symbols. Under appropriate assumptions, a complete description of the spectrum of such operators is given and the exponential return to equilibrium with sharp estimates on the rate of convergence is proven. Some applications to the study of chains of oscillators and to the generalized Langevin equation are given.

Subelliptic estimates for quadratic differential operators

We prove global subelliptic estimates for quadratic differential operators. Quadratic differential operators are operators defined in the Weyl quantization by complex-valued quadratic symbols. In a previous joint work with M. Hitrik, we pointed out the existence of a particular linear subvector space in the phase space intrinsically associated to their Weyl symbols, called singular space, which rules spectral properties of non-elliptic quadratic operators. The purpose of the present paper is to prove that quadratic operators whose singular spaces are reduced to zero, are subelliptic with a loss of "derivatives" depending directly on particular algebraic properties of the Hamilton maps of their Weyl symbols. More generally, when singular spaces are symplectic spaces, we prove that quadratic operators are subelliptic in any direction of the symplectic orthogonal complements of their singular spaces.

Spectra and semigroup smoothing for non-elliptic quadratic operators

We study non-elliptic quadratic differential operators. Quadratic differential operators are non-selfadjoint operators defined in the Weyl quantization by complex-valued quadratic symbols. When the real part of their Weyl symbols is a non-positive quadratic form, we point out the existence of a particular linear subspace in the phase space intrinsically associated to their Weyl symbols, called a singular space, such that when the singular space has a symplectic structure, the associated heat semigroup is smoothing in every direction of its symplectic orthogonal space. When the Weyl symbol of such an operator is elliptic on the singular space, this space is always symplectic and we prove that the spectrum of the operator is discrete and can be described as in the case of global ellipticity. We also describe the large time behavior of contraction semigroups generated by these operators.

On the pseudospectrum of elliptic quadratic differential operators

We study the pseudospectrum of a class of nonselfadjoint differential operators. Our work consists of a microlocal study of the properties that rule the spectral stability or instability phenomena appearing under small perturbations for elliptic quadratic differential operators. The class of elliptic quadratic differential operators stands for the class of operators defined in the Weyl quantization by complex-valued elliptic quadratic symbols. We establish in this article a simple, necessary, and sufficient condition on the Weyl symbol of these operators which ensures the stability of their spectra. When this condition is violated, we prove that strong spectral instabilities occur for the high energies of these operators in some regions that can be far away from their spectra

Contraction semigroups of elliptic quadratic differential operators

We study the contraction semigroups of elliptic quadratic differential operators. Elliptic quadratic differential operators are the non-selfadjoint operators defined in the Weyl quantization by complex-valued elliptic quadratic symbols. We establish in this paper that under the assumption of ellipticity, as soon as the real part of their Weyl symbols is a non-zero non-positive quadratic form, the norm of contraction semigroups generated by these operators decays exponentially in time.

Quadratic Reciprocity in a Finite Group

In this article we define a quadratic symbol for a finite group and prove a law of reciprocity for its value.

Some remarks on Weyl pseudodifferential operators

We study properties of the Weyl symbols of functions of pseudodifferential operators, especially of pseudodifferential operators with quadratic symbols.

Local Solvability Of First Order Differential Operators Near A Critical Point, Operators With Quadratic Symbols And The Heisenberg Group

We study questions of solvability for operators of the form p(x,D)+b, where p(x,ξ) is a real quadratic form and b∊C. As one consequence, we obtain a necessary and sufficient condition for the local solvability of operators of the form L= near the critical point x=0, and prove the existence of tempered fundamental solutions whenever L is locally solvable.Our analysis of these operators is largely based on recent results about the solvabilitiy of left–invariant second order differential operators on the Heisenberg group and a transference principle for the Schrodinger representation.