Improved semiclassical eigenvalue estimates for the Laplacian and the Landau Hamiltonian

Let G/K be a rank one or complex non compact symmetric space of dimension l. We prove that if f e Lp, 1⩽p⩽2, the Riesz means of order z of f with respect to the eigenfunction expansion of the Laplacian converge to falmost everywhere for Re(z)>δ(l, p). The critical index δ(l, p) is the same as in the classical result of Stein in the Euclidean case.

Namely, for 1 ≤ p ≤ ∞, α > (n − 1)/2 the boundedness of the operator S α R : L p → L p is proved, where S α R ƒ are Riesz means of order α of Hermite expansions of a function ƒ (cf. loc.cit.). The a.e. convergence of S α R ƒ(x) to ƒ(x) for ƒ ∈ L p (R p ), p ≥ 2, n ≥ 2 and α > (n − 1)(1/2 − 1/p) and the convergence of S α R ƒ(x) to ƒ(x) at every Lebesgue point x of ƒ if α > (n − 1)/2 are proved. Moreover the a.e. convergence of Riesz means σ α n ƒ(r) of order α > (2n − 1)(1/2 − 1/p)of Laguerre expansions of a function ƒ ∈ L p (R + , r 2n−1 dr), 2 ≤ p ≤ ∞ (the notations are from the author.

Given a frequency \(\lambda = (\lambda _n)\) and \(\ell \ge 0\), we introduce the scale of Banach spaces \(H_{\infty ,\ell }^{\lambda }[{{\,\mathrm{Re}\,}}> 0]\) of holomorphic functions f on the open right half-plane \([{{\,\mathrm{Re}\,}}> 0]\), which satisfy (A) the growth condition \(|f(s)| = O((1 + |s|)^\ell )\), and (B) have a \(\lambda\)-Riesz germ, i.e. on some open subset and for some \(m \ge 0\) the function f coincides with the pointwise limit (as \(x \rightarrow \infty\)) of the so-called \((\lambda ,m)\)-Riesz means $$\begin{aligned} \sum_{\lambda _n < x} a_n e^{-\lambda _n s}\left( 1-\frac{\lambda _n}{x}\right) ^m ,\,x >0 \end{aligned}$$of some \(\lambda\)-Dirichlet series \(\sum a_n e^{-\lambda _n s}\). Reformulated in our terminology, an important result of Riesz shows that in this case the function f for every \(k >\ell\) is the pointwise limit of the \((\lambda ,k)\)-Riesz means of D on \([{{\,\mathrm{Re}\,}}> 0]\). Our main contribution is an extension showing that ’after translation’ every bounded set in \(H_{\infty ,\ell }^{\lambda }[{{\,\mathrm{Re}\,}}> 0]\) is uniformly approximable by all its \((\lambda ,k)\)-Riesz means of order \(k>\ell\). This follows from an appropriate maximal theorem, which turns out to be at the very heart of a seemingly interesting structure theory of the Banach spaces \(H_{\infty ,\ell }^{\lambda }[{{\,\mathrm{Re}\,}}> 0]\). One of the many consequences is that \(H_{\infty ,\ell }^{\lambda }[{{\,\mathrm{Re}\,}}> 0]\) basically consists of those holomorphic functions on \([{{\,\mathrm{Re}\,}}>0]\), which have a \(\lambda\)-Riesz germ and are of finite uniform order \(\ell\) on \([{{\,\mathrm{Re}\,}}>0]\). To establish all this and more, we need to reorganize (and to improve) various aspects and keystones of the classical theory of Riesz summability of general Dirichlet series as invented by Hardy and M. Riesz.

Given a frequency $\lambda = (\lambda_n)$ and $\ell \ge 0$, we introduce the scale of Banach spaces $H_{\infty,\ell}^{\lambda}[Re > 0]$ of holomorphic functions $f$ on the open right half-plane $[Re > 0]$, which satisfy $(A)$ the growth condition $|f(s)| = O((1 + |s|)^\ell)$, and $(B)$ have a Riesz germ, i.e. on some open subset and for some $m \ge 0$ the function $f$ coincides with the pointwise limit (as $x o \infty$) of the so-called $(\lambda,m)$-Riesz means $\sum_{\lambda_n < x} a_n e^{-\lambda_n s}\big( 1-\frac{\lambda_n}{x}\big)^m ,\,x >0$ of some $\lambda$-Dirichlet series $\sum a_n e^{-\lambda_n s}$. Reformulated in our terminology, an important result of M. Riesz shows that in this case the function $f$ for every $k >\ell$ is the pointwise limit of the $(\lambda,k)$-Riesz means of $D$ on $[Re > 0]$. Our main contribution is an extension -- showing that 'after translation' every bounded set in $H_{\infty,\ell}^{\lambda}[Re > 0]$ is uniformly approximable by all its $(\lambda,k)$-Riesz means of order $k>\ell$. This follows from an appropriate maximal theorem, which in fact turns out to be at the very heart of a seemingly interesting structure theory of the Banach spaces $H_{\infty,\ell}^{\lambda}Re > 0]$. One of the many consequences is that $H_{\infty,\ell}^{\lambda}[Re > 0]$ basically consists of those holomorphic functions on $[Re >0]$, which have a Riesz germ and are of finite uniform order $\ell$ on $[Re >0]$. To establish all this and more, we need to reorganize (and to improve) various aspects and keystones of the classical theory of Riesz summability of general Dirichlet series as invented by Hardy and M. Riesz. Joint work with Ingo Schoolmann. 

We study the eigenvalues of the Dirichlet Laplace operator on an arbitrary bounded, open set in \mathbb{R}^d , d \geq 2 . In particular, we derive upper bounds on Riesz means of order \sigma \geq 3/2 , that improve the sharp Berezin inequality by a negative second term. This remainder term depends on geometric properties of the boundary of the set and reflects the correct order of growth in the semi-classical limit. Under certain geometric conditions these results imply new lower bounds on individual eigenvalues, which improve the Li–Yau inequality.

Estimates for Riesz kernels of eigenfunction expansions of elliptic differential operators on compact manifolds

Let {λ2} and {ϕ λ } be the eigenvalues and an orthonormal system of eigenvectors of a second order elliptic differential operator Δ on a compact manifoldM of dimensionN. We prove that the Riesz means of order δ, defined by\(R_\Lambda ^\delta f = \sum\limits_{\lambda< \Lambda } {\left( {1 - \frac{{\lambda ^2 }}{{\Lambda ^2 }}} \right)^\delta \hat f(} \lambda ) \varphi _\lambda \), are uniformly bounded from the Hardy spaceH p (M) into Weak-L p (M), if 0<p<1 and δ=N/p−(N+1)/2.

We prove that for a certain class of n dimensional rank one locally symmetric spaces, if $$f \in L^p$$ , $$1\le p \le 2$$ , then the Riesz means of order z of f converge to f almost everywhere, for $$\mathrm {Re}z> (n-1)(1/p-1/2)$$ .

We show that for δ below certain critical indices there are functions whose Jacobi or Laguerre expansions have almost everywhere divergent Cesaro and Riesz means of order 6.

We consider the eigenvalues of the magnetic Laplacian on a bounded domain Ω of R2 with uniform magnetic field β>0 and magnetic Neumann boundary conditions. We find upper and lower bounds for the ground state energy λ1 and we provide semiclassical estimates in the spirit of Kröger for the first Riesz mean of the eigenvalues. We also discuss upper bounds for the first eigenvalue for non-constant magnetic fields β=β(x) on a simply connected domain in a Riemannian surface.In particular: we prove the upper bound λ1<β for a general plane domain for a constant magnetic field, and the upper bound λ1<maxx∈Ω‾⁡|β(x)| for a variable magnetic field when Ω is simply connected.For smooth domains, we prove a lower bound of λ1 depending only on the intensity of the magnetic field β and the rolling radius of the domain.The estimates on the Riesz mean imply an upper bound for the averages of the first k eigenvalues which is sharp when k→∞ and consists of the semiclassical limit 2πk|Ω| plus an oscillating term.We also construct several examples, showing the importance of the topology: in particular we show that an arbitrarily small tubular neighborhood of a generic simple closed curve has lowest eigenvalue bounded away from zero, contrary to the case of a simply connected domain of small area, for which λ1 is always small.

An analytical theory of the de Haas\char21{}van Alphen effect, under the condition $\ensuremath{\mu}/\ensuremath{\Elzxh}{\ensuremath{\omega}}_{c}\ensuremath{\gg}1$ (where $\ensuremath{\mu}$ is the chemical potential and ${\ensuremath{\omega}}_{c}$ the cyclotron frequency) is investigated in two-dimensional and quasi-two-dimensional metals, taking into account the effects of spin splitting, impurity scattering, finite temperature, and a field-independent reservoir of electrons. The equation for the chemical potential as a function of magnetic field, temperature, and a non-field-quantized reservoir of states is derived. It follows that the semiclassical expression in low-dimensional systems is generally no longer a Fourier-like series. The difficulties in an unequivocal effective-mass determination from the temperature dependence of the oscillation amplitude in low-dimensional metals are pointed out. The influence of the chemical potential oscillations on the shape of the magnetization oscillations is shown analytically: the sawtoothed, inverted sawtoothed, and symmetrical wave forms are found in high-purity two-dimensional metals at very low temperatures from a semiclassical expression.

A new representation for the subdynamics is given such that the resultant coupling matrix becomes identical with the semiclassical expression away from turning points. A perturbation theory is developed which also reduces to semiclassical theory away from the turning points. The subdynamic perturbation expression is therefore capable of making up for the deficiencies of the semiclassical theory at the turning points. A numerical illustration is given.

We consider a semiclassical (large string tension ~ \lambda^1/2) limit of 4-point correlator of two "heavy" vertex operators with large quantum numbers and two "light" operators. It can be written in a factorized form as a product of two 3-point functions, each given by the integrated "light" vertex operator on the classical string solution determined by the "heavy" operators. We check consistency of this factorization in the case of a correlator with two dilatons as "light" operators. We study in detail the example when all 4 operators are chiral primary scalars, two of which carry large charge J of order of string tension. In the large J limit this correlator is nearly extremal. Its semiclassical expression is, indeed, found to be consistent with the general protected form expected for an extremal correlator. We demonstrate explicitly that our semiclassical result matches the large J limit of the known free N=4 SYM correlator for 4 chiral primary operators with charges J,-J,2,-2; we also compare it with an existing supergravity expression. As an example of a 4-point function with two non-BPS "heavy" operators, we consider the case when the latter are representing folded spinning with large AdS spin and two "light" states being chiral primary scalars.

The purpose of this paper is to derive the asymptotic solutions to a class of inhomogeneous integral equations that reduce to algebraic equations when a parameter ε goes to zero (the kernel becoming proportional to a Dirac δ function). This class includes the integral equations obtained from the system of Vlasov and Poisson equations for the Fourier transform in space and the Laplace transform in time of the electrostatic potential, when the equilibrium magnetic field is uniform and the equilibrium plasma density depends on εx, with the co-ordinate z being the direction of the magnetic field. In this case the inhomogeneous term is given by the initial conditions and possibly by sources, and the Laplace-transform variable ω is the eigenvalue parameter.

The reduced linearized equations of ideal magnetohydrodynamics which are highly nonlinear in the eigenvalue parameter, are linearized about a prescribed value of that parameter, enabling the equation to be expressed as a Schrodinger equation with piecewise uniform coefficients. Reflection and transmission coefficients are obtained using standard techniques, and in addition to the possibility of total transmission of an incident wave occurring (together with complex-valued resonance energies), the magnetic field introduces other total transmission energy levels which have no counterpart in the absence of a magnetic field.