Let $G$ be a Lie group with Lie algebra $\mathfrak g$ and let $π$ be a unitary representation of $G$ realized on a reproducing kernel Hilbert space. We use Berezin quantization to study spectral measures associated with operators $-idπ(X)$ for $X\in {\mathfrak g}$. As an application, we show how results about contractions of Lie group representations give rise to results on convergence of sequences of spectral measures. We give some examples including contractions of $SU(1,1)$ and $SU(2)$ to the Heisenberg group.
We discuss the Berezin transform, a Markov operator associated to positive operator valued measures (POVMs), in a number of contexts including the Berezin-Toeplitz quantization, Donaldson's dynamical system on the space of Hermitian products on a complex vector space, representations of finite groups, and quantum noise. In particular, we calculate the spectral gap for quantization in terms of the fundamental tone of the phase space. Our results confirm a prediction of Donaldson for the spectrum of the Q-operator on Kahler manifolds with constant scalar curvature. Furthermore, viewing POVMs as …
We introduce a minimalistic notion of semiclassical quantization and use it to prove that the convex hull of the semiclassical spectrum of a quantum system given by a collection of commuting operators converges to the convex hull of the spectrum of the associated classical system. This gives a quick alternative solution to the isospectrality problem for quantum toric systems. If the operators are uniformly bounded, the convergence is uniform. Analogous results hold for non-commuting operators.
We consider Berezin-Toeplitz operators on compact Kahler manifolds whose symbols are characteristic functions. When the support of the characteristic function has a smooth boundary, we prove a two-term Weyl law, the second term being proportional to the Riemannian volume of the boundary. As a consequence, we deduce the area law for the entanglement entropy of integer quantum Hall states. Another application is for the determinantal processes with correlation kernel the Bergman kernels of a positive line bundle : we prove that the number of points in a smooth domain is asymptotically normal.
Let F be a real-valued convex function on the complex plane, and let Ps(b) be a pseudodifferential operator with symbol b. Under certain natural assumptions about properties of pseudodifferential operators, we prove that the sum of F(z) over all eigenvalues z of the operator Ps(b) counted with their multiplicities does not exceed the real part of the trace of Ps(F(b)) modulo a term of the same order as the error term in the Garding inequality.
We consider Berezin-Toeplitz operators whose multipliers are compactly supported densities carried by a submanifold of ${\mathbb C}^N$ . We compute asymptotically the moments of their spectral measures, and we prove Szegö limit theorems in cases when the submanifold is isotropic or co-isotropic, from which Weyl estimates follow. We also obtain asymptotics of the Schatten norms of such operators. Rescaled versions of these operators can be thought of as quantum mechanical mixed states, and our results give the semi-classical limit of their entropy.
We prove, assuming that the Bohr-Sommerfeld rules hold, that the joint spectrum near a focus-focus critical value of a quantum integrable system determines the classical Lagrangian foliation around the full focus-focus leaf. The result applies, for instance, to h-pseudodifferential operators, and to Berezin-Toeplitz operators on prequantizable compact symplectic manifolds.
In the past decade there has been a flurry of activity at the intersection of spectral theory and symplectic geometry. In this paper we review recent results on semiclassical spectral theory for commuting Berezin-Toeplitz and h-pseudodifferential operators. The paper emphasizes the interplay between spectral theory of operators (quantum theory) and symplectic geometry of Hamiltonians (classical theory), with an eye towards recent developments on the geometry of finite dimensional integrable systems.
We provide asymptotic formulas for the Bergman projector and Berezin-Toeplitz operators on a compact K{ä}hler manifold. These objects depend on an integer N and we study, in the limit N $\rightarrow$ +$\infty$, situations in which one can control them up to an error O(e^{-cN}) for some c > 0. We develop a calculus of Toeplitz operators with real-analytic symbols, which applies to K{ä}hler man-ifolds with real-analytic metrics. In particular, we prove that the Bergman kernel is controlled up to O(e^{-cN}) on any real-analytic K{ä}hler manifold as N $\rightarrow$ +$\infty$. We also prove that To…
Using an abstract notion of semiclassical quantization for self-adjoint operators, we prove that the joint spectrum of a collection of commuting semiclassical self-adjoint operators converges to the classical spectrum given by the joint image of the principal symbols, in the semiclassical limit. This includes Berezin-Toeplitz quantization and certain cases of $\hbar$-pseudodifferential quantization, for instance when the symbols are uniformly bounded, and extends a result by L. Polterovich and the authors. In the last part of the paper we review the recent solution to the inverse problem for q…
In this survey article we consider the operator pair $(H,H_0)$ where $H_0$ is the shifted 3D Schrödinger operator with constant magnetic field, $H : = H_0 + V$, and $V$ is a short-range electric potential of a fixed sign. We describe the asymptotic behavior of the Krein spectral shift function (SSF) $ξ(E; H,H_0)$ as the energy $E$ approaches the Landau levels $2bq$, $q \in {\mathbb Z}_+$, which play the role of thresholds in the spectrum of $H_0$. The main asymptotic term of $ξ(E; H,H_0)$ as $E \to 2bq$ with a fixed $q \in {\mathbb Z}_+$ is written in the terms of appropriate compact Berezin-T…
In this article we consider asymptotics for the spectral function of Schrödinger operators on the real line. Let $P:L^2(\mathbb{R})\to L^2(\mathbb{R})$ have the form $$ P:=-\tfrac{d^2}{dx^2}+W, $$ where $W$ is a self-adjoint first order differential operator with certain modified almost periodic structure. We show that the kernel of the spectral projector, $\mathbb{1}_{(-\infty,λ^2]}(P)$ has a full asymptotic expansion in powers of $λ$. In particular, our class of potentials $W$ is stable under perturbation by formally self-adjoint first order differential operators with smooth, compactly supp…
Let $X$ be the unit circle bundle of a positive line bundle on a Hodge manifold. We study the local scaling asymptotics of the smoothed spectral projectors associated to a first order elliptic Töplitz operator $T$ on $X$, possibly in the presence of Hamiltonian symmetries. The resulting expansion is then used to give a local derivation of an equivariant Weyl law. It is not required that $T$ be invariant under the structure circle action, that is, $T$ needn't be a Berezin-Töplitz operator.
In 1966, H. Widom proved an asymptotic formula for the distribution of eigenvalues of the $N\times N$ truncated Hilbert matrix for large values of $N$. In this paper, we extend this formula to Hankel matrices with symbols in the class of piece-wise continuous functions on the unit circle. Furthermore, we show that the distribution of the eigenvalues is independent of the choice of truncation (e.g. square or triangular truncation).
For a class of non-selfadjoint semiclassical operators in dimension one, we get a complete asymptotic description of all eigenvalues near a critical value of the leading symbol of the operator on the boundary of the pseudospectrum.
We develop the spectral and scattering theory for self-adjoint Hankel operators $H$ with piecewise continuous symbols. In this case every jump of the symbol gives rise to a band of the absolutely continuous spectrum of $H$. We construct wave operators relating simple "model" (that is, explicitly diagonalizable) Hankel operators for each jump and the given Hankel operator $H$. We show that the set of all these wave operators is asymptotically complete. This determines the absolutely continuous part of $H$. We also prove that the singular continuous spectrum of $H$ is empty and that its eigenval…
The non-cutoff Kac operator is a kinetic model for the non-cutoff radially symmetric Boltzmann operator. For Maxwellian molecules, the linearization of the non-cutoff Kac operator around a Maxwellian distribution is shown to be a function of the harmonic oscillator, to be diagonal in the Hermite basis and to be essentially a fractional power of the harmonic oscillator. This linearized operator is a pseudodifferential operator, and we provide a complete asymptotic expansion for its symbol in a class enjoying a nice symbolic calculus. Related results for the linearized non-cutoff radially symmet…
Let $L^2(D)$ be the space of measurable square-summable functions on the unit disk. Let $L^2_a(D)$ be the Bergman space, i.e., the (closed) subspace of analytic functions in $L^2(D)$. $P_+$ stays for the orthogonal projection going from $L^2(D)$ to $L^2_a(D)$. For a function $φ\in L^\infty(D)$, the Toeplitz operator $T_φ: L^2_a(D)\to L^2_a(D)$ is defined as $$ T_φf=P_+φf, \quad f\in L^2_a(D). $$ The main result of this article are spectral asymptotics for singular (or eigen-) values of compact Toeplitz operators with logarithmically decaying symbols, that is $$ φ(z)=φ_1(e^{iθ})\, (1+\log(1/(1-…
We construct the propagator of the massless Dirac operator $W$ on a closed Riemannian 3-manifold as the sum of two invariantly defined oscillatory integrals, global in space and in time, with distinguished complex-valued phase functions. The two oscillatory integrals -- the positive and the negative propagators -- correspond to positive and negative eigenvalues of $W$, respectively. This enables us to provide a global invariant definition of the full symbols of the propagators (scalar matrix-functions on the cotangent bundle), a closed formula for the principal symbols and an algorithm for the…
The paper pursues three objectives. Firstly, we provide an expanded version of spectral analysis of self-adjoint Toeplitz operators, initially built by M. Rosenblum in the 1960's. We offer some improvements to Rosenblum's approach: for instance, our proof of the absolute continuity, relying on a weak version of the limiting absorption principle, is more direct. Secondly, we study in detail Toeplitz operators with finite spectral multiplicity. In particular, we introduce generalized eigenfunctions and investigate their properties. Thirdly, we develop a more detailed spectral analysis for piecew…
We consider an elliptic self-adjoint first order differential operator acting on pairs (2-columns) of complex-valued half-densities over a connected compact 3-dimensional manifold without boundary. The principal symbol of our operator is assumed to be trace-free. We study the spectral function which is the sum of squares of Euclidean norms of eigenfunctions evaluated at a given point of the manifold, with summation carried out over all eigenvalues between zero and a positive lambda. We derive an explicit two-term asymptotic formula for the spectral function as lambda tends to plus infinity, ex…
We describe the asymptotics of the spectral norm of finite Toeplitz matrices generated by functions with Fisher-Hartwig singularities as the matrix dimension goes to infinity. In the case of positive generating functions, our result provides the asymptotics of the largest eigenvalue, which is of interest in time series with long-range memory.
Let M be a compact manifold without boundary. Associated to a metric g on M there are various Laplace operators, for example the de Rham Laplacian on forms and the conformal Laplacian on functions. For a general Laplace type operator we consider its spectral zeta function Z(s). For a fixed value of s we calculate the hessian of Z(s) with respect to the metric and show that it is given by a pseudodifferential operator T(s)=U(s)+V(s) where U(s) is polyhomogeneous of degree n-2s and V(s) is polyhomogeneous of degree 2. The operators U(s) and V(s) are meromorphic in s. The symbol expansion of U(s)…
