Spectral Projection Operators Research Articles

Bounds on the Hermite spectral projection operator

We study Lp–Lq bounds on the spectral projection operator Πλ associated to the Hermite operator H=|x|2−Δ in Rd. We are mainly concerned with a localized operator χEΠλχE for a subset E⊂Rd and undertake the task of characterizing the sharp Lp–Lq bounds. We obtain sharp bounds in extended ranges of p,q. First, we provide a complete characterization of the sharp Lp–Lq bounds when E is away from λSd−1. Secondly, we obtain the sharp bounds as the set E gets close to λSd−1. Thirdly, we extend the range of p,q for which the operator Πλ is uniformly bounded from Lp(Rd) to Lq(Rd).

Almost everywhere convergence of Bochner-Riesz means for the Hermite operators

Let H=−Δ+|x|2 be the Hermite operator in Rn. In this paper we study almost everywhere convergence of the Bochner-Riesz means associated with H which is defined by SRλ(H)f(x)=∑k=0∞(1−2k+nR2)+λPkf(x). Here Pkf is the k-th Hermite spectral projection operator. For 2≤p<∞, we prove thatlimR→∞⁡SRλ(H)f=fa.e. for all f∈Lp(Rn) provided that λ>λ(p)/2 and λ(p)=max⁡{n(1/2−1/p)−1/2,0}. Conversely, we also show the convergence generally fails if λ<λ(p)/2 in the sense that there is an f∈Lp(Rn) for 2n/(n−1)≤p such that the convergence fails. This is in surprising contrast with a.e. convergence of the classical Bochner-Riesz means for the Laplacian. For n≥2 and p≥2 our result tells that the critical summability index for a.e. convergence for SRλ(H) is as small as only the half of the critical index for a.e. convergence of the classical Bochner-Riesz means. When n=1, we show a.e. convergence holds for f∈Lp(R) with p≥2 whenever λ>0. Compared with the classical result due to Askey and Wainger who showed the optimal Lp convergence for SRλ(H) on R we only need smaller summability index for a.e. convergence.

Beyond the spectral theorem: Spectrally decomposing arbitrary functions of nondiagonalizable operators

Nonlinearities in finite dimensions can be linearized by projecting them into infinite dimensions. Unfortunately, the familiar linear operator techniques that one would then hope to use often fail since the operators cannot be diagonalized. The curse of nondiagonalizability also plays an important role even in finite-dimensional linear operators, leading to analytical impediments that occur across many scientific domains. We show how to circumvent it via two tracks. First, using the well-known holomorphic functional calculus, we develop new practical results about spectral projection operators and the relationship between left and right generalized eigenvectors. Second, we generalize the holomorphic calculus to a meromorphic functional calculus that can decompose arbitrary functions of nondiagonalizable linear operators in terms of their eigenvalues and projection operators. This simultaneously simplifies and generalizes functional calculus so that it is readily applicable to analyzing complex physical systems. Together, these results extend the spectral theorem of normal operators to a much wider class, including circumstances in which poles and zeros of the function coincide with the operator spectrum. By allowing the direct manipulation of individual eigenspaces of nonnormal and nondiagonalizable operators, the new theory avoids spurious divergences. As such, it yields novel insights and closed-form expressions across several areas of physics in which nondiagonalizable dynamics arise, including memoryful stochastic processes, open nonunitary quantum systems, and far-from-equilibrium thermodynamics. The technical contributions include the first full treatment of arbitrary powers of an operator, highlighting the special role of the zero eigenvalue. Furthermore, we show that the Drazin inverse, previously only defined axiomatically, can be derived as the negative-one power of singular operators within the meromorphic functional calculus and we give a new general method to construct it. We provide new formulae for constructing spectral projection operators and delineate the relations among projection operators, eigenvectors, and left and right generalized eigenvectors. By way of illustrating its application, we explore several, rather distinct examples. First, we analyze stochastic transition operators in discrete and continuous time. Second, we show that nondiagonalizability can be a robust feature of a stochastic process, induced even by simple counting. As a result, we directly derive distributions of the time-dependent Poisson process and point out that nondiagonalizability is intrinsic to it and the broad class of hidden semi-Markov processes. Third, we show that the Drazin inverse arises naturally in stochastic thermodynamics and that applying the meromorphic functional calculus provides closed-form solutions for the dynamics of key thermodynamic observables. Finally, we draw connections to the Ruelle–Frobenius–Perron and Koopman operators for chaotic dynamical systems and propose how to extract eigenvalues from a time-series.

Uniform gradient estimates on manifolds with a boundary and applications

We revisit the problem of obtaining uniform gradient estimates for Dirichlet and Neumann heat semigroups on Riemannian manifolds with boundary. As applications, we obtain isoperimetric inequalities, using Ledoux’s argument, and uniform quantitative gradient estimates, firstly for $$C^2_b$$ functions with boundary conditions and then for the unit spectral projection operators of Dirichlet and Neumann Laplacians.

Endpoint bounds for a class of spectral multipliers on compact manifolds

It is well known that the Stein-Tomas $L^2$ Fourier restriction theorem can be used to derive sharp $L^p$ bounds for radial Fourier multipliers such as the Bochner-Riesz means. In a similar manner, $L^p \to L^2$ estimates for spectral projection operators have been utilized in order to obtain sharp $L^p$ bounds for spectral multipliers of self-adjoint elliptic pseudo-differential operators on compact manifolds. In this paper, we refine an endpoint result for spectral multipliers due to Seeger, providing endpoint bounds in terms of Besov spaces. Our proof is based on the ideas from the recent work by Heo, Nazarov and Seeger, and Lee, Rogers and Seeger on radial Fourier multipliers.

Localized Lp-estimates of eigenfunctions: A note on an article of Hezari and Rivière

We use a straightforward variation on a recent argument of Hezari and Rivière [8] to obtain localized Lp-estimates for all exponents larger than or equal to the critical exponent pc=2(n+1)n−1. We are able to do this directly by just using the Lp-bounds for spectral projection operators from our much earlier work [12]. The localized bounds we obtain here imply, for instance, that, for a density one sequence of eigenvalues on a manifold whose geodesic flow is ergodic, all of the Lp, 2<p≤∞, bounds of the corresponding eigenfunctions are relatively small compared to the general ones in [12], which are saturated on round spheres. The connection with quantum ergodicity was established for exponents 2<p<pc in the recent results of the author [13] and Blair and the author [3]; however, the article of Hezari and Rivière [8] was the first one to make this connection (in the case of negatively curved manifolds) for the critical exponent, pc. As is well known, and we indicate here, bounds for the critical exponent, pc, imply ones for all the other exponents 2<p≤∞. The localized estimates involve L2-norms over small geodesic balls Br of radius r, and we shall go over what happens for these in certain model cases on the sphere and on manifolds of nonpositive curvature. We shall also state a problem of determining when one can improve on the trivial O(r12) estimates for these L2(Br) bounds. If r=λ−1, one can improve on the trivial estimates if one has improved Lpc(M) bounds just by using Hölder's inequality; however, obtaining improved bounds for r≫λ−1 seems to be subtle.

The gradient estimate of a Neumann eigenfunction on a compact manifold with boundary

Let eλ(x) be a Neumann eigenfunction with respect to the positive Laplacian Δ on a compact Riemannian manifold M with boundary such that Δeλ = λ2eλ in the interior of M and the normal derivative of eλ vanishes on the boundary of M. Let χλ be the unit band spectral projection operator associated with the Neumann Laplacian and f be a square integrable function on M. The authors show the following gradient estimate for χλ f as \(\lambda \geqslant 1:{\left\| {\nabla {\chi _\lambda }{\kern 1pt} \left. f \right\|} \right._\infty } \leqslant C\left( {\lambda \left\| {{\chi _\lambda }{{\left. f \right\|}_\infty } + \left. {{\lambda ^{ - 1}}} \right\|} \right.\Delta {\chi _\lambda }{{\left. f \right\|}_\infty }} \right)\), where C is a positive constant depending only on M. As a corollary, the authors obtain the gradient estimate of eλ: For every λ ≥ 1, it holds that \(\left\| {\nabla {{\left. {{e_\lambda }} \right\|}_\infty } \leqslant C\left. \lambda \right\|} \right.{\left. {{e_\lambda }} \right\|_\infty }\).

Exact solution of two-term time-fractional Thornley's problem by operational method

A two-term time-fractional diffusion equation is studied, subject to a non-local boundary condition. The fractional time derivatives are described in the Caputo sense. We develop a bivariate operational calculus for this problem and apply it to obtain a Duhamel-type representation of the solution. This is a compact representation, containing two convolution products of special solutions and the arbitrary initial and boundary functions. With respect to the space variable a non-classical convolution is used. To find the special solutions, appropriate spectral projection operators are applied. In this way, the solution is constructed in the form of a series expansion on the generalized eigenfunctions of a non-selfadjoint Sturm–Liouville problem and three-parameter Mittag-Leffler functions. Thanks to the uniqueness property of the employed spectral expansion, the uniqueness of the solution is also proven.

Exact solution for the fractional cable equation with nonlocal boundary conditions

AbstractThe fractional cable equation is studied on a bounded space domain. One of the prescribed boundary conditions is of Dirichlet type, the other is of a general form, which includes the case of nonlocal boundary conditions. In real problems nonlocal boundary conditions are prescribed when the data on the boundary can not be measured directly. We apply spectral projection operators to convert the problem to a system of integral equations in any generalized eigenspace. In this way we prove uniqueness of the solution and give an algorithm for constructing the solution in the form of an expansion in terms of the generalized eigenfunctions and three-parameter Mittag-Leffler functions. Explicit representation of the solution is given for the case of double eigenvalues. We consider some examples and as a particular case we recover a recent result. The asymptotic behavior of the solution is also studied.

Spectral Asymptotics Revisited

We discuss two heuristic ideas concerning the spectrum of a Laplacian, and we give theorems and conjectures from the realms of manifolds, graphs and fractals that validate these heuristics. The first heuristic concerns Laplacians that do not have discrete spectra: here we discuss the notion of “spectral mass” closely related to the “integrated density of states”, an average of the diagonal of the kernel of the spectral projection operator, and show that this can serve as a substitute for the eigenvalue counting function. The second heuristic is an “asymptotic splitting law” that describes the proportions of the spectrum that transforms according to the irreducible representations of a finite group that acts as a symmetry group of the Laplacian. For this to be valid we require the existence of a fundamental domain with relatively small boundary. We also give a version in the case that the symmetry group is a compact Lie group. Many of our results are reformulations of known results, and some are merely conjectures, but there is something to be gained by looking at them together with a unified perspective.

Controller Based on Spectral Decomposition for Neutral Delay Systems

A controller design is proposed based on spectral decomposition of operator semigroup for linear neutral delay systems (NDSs).When the norm of the neutral term is less than one,by spectral theory and projection operator,the in- finite dimensional solution space of system is decomposed into the direct sum of finite dimensional unstable generalized eigen- subspace and infinite dimensional stable subspace.Then for the finite dimensional subsystem,the control law of state feedback is obtained.And the asymptotic stability of the colsed loop system with state feedback control is proved by the projection operator.Finally,a numerical example demonstrates the effectiveness of the proposed approach.

Derivatives of the Spectral Function and Sobolev Norms of Eigenfunctions on a Closed Riemannian Manifold

Let e(x, y, λ) be the spectral function and χλ the unit spectral projection operator, with respect to the Laplace–Beltrami operator on a closed Riemannian manifold M. We generalize the one-term asymptotic expansion of e(x, x, λ) by Hormander (Acta Math. 88 (1968), 341–370) to that of ∂ x α∂ y β e(x,y,λ)| x=y for any multiindices α, β in a sufficiently small geodesic normal coordinate chart of M. Moreover, we extend the sharp (L 2,L p) (2 ≤p≤∞) estimates of χλ by Sogge (J. Funct. Anal. 77 (1988), 123–134; London Math. Soc. Lecture Note Ser. 137, Cambridge University Press, Cambridge, 1989; Vol. 1, pp. 416–422) to the sharp (L 2, Sobolev L p) estimates of χλ.

Estimates for spectral projection operators of the sub-laplacian on the heisenberg group

In this paper, we use Laguerre calculus to find theLP spectrum (λ, Μ) of the pair (L, iT). Here\({\cal L} = - \frac{1}{2}\sum\limits_{j = 1}^n {\left( {Z_j \bar Z_j + \bar Z_j Z_j } \right)} \) md T = ∂/∂t with\(\left\{ {Z_1 ,...,Z_n ,\bar Z_1 ,,...,\bar Z_n , T} \right\}\)a basis for the left-invariant vector fields on the Heisenberg group. We find kernels for the spectral projection operators on the ray λ > 0 in the Heisenberg brush and show that they are Calderon-Zygmund-Mikhlin operators. Estimates for these operators in L k p (Hn), HP(Hn), and S k pv (Hn) spaces can therefore be deduced.

Quantum-classical correspondence via Liouville dynamics. II. Correspondence for chaotic Hamiltonian systems

We prove quantum-classical correspondence for bound conservative classically chaotic Hamiltonian systems. In particular, quantum Liouville spectral projection operators and spectral densities, and hence classical dynamics, are shown to approach their classical analogs in the h!0 limit. Correspondence is shown to occur via the elimination of essential singularities. In addition, applications to matrix elements of observables in chaotic systems are discussed. @S1050-2947~96!05212-2# The validity of quantum mechanics as a description of the macroscopic world is contingent upon the reduction of the laws of quantum mechanics to Newton’s laws in the limit where the characteristic actions of a system are large with respect to Planck’s constant @1#. Thus, diagonal and offdiagonal matrix elements must reduce to their classical analogs and quantum dynamics must reproduce the predictions of classical mechanics as h!0. Despite the fundamental importance of quantum-classical correspondence it has only been satisfactorily demonstrated @2‐5# in the very restrictive case of regular systems, i.e., systems that classically possess as many constants of the motion as degrees of freedom. Indeed some authors have suggested that bound quantum systems with a discrete quantum spectrum and a chaotic classical analog may violate the correspondence principle @6#. These doubts about the validity of the correspondence principle for chaotic systems stem from the difficulty of reconciling the quasiperiodic nature of bound state quantum dynamics with the chaotic nature of classical dynamics for the same Hamiltonian. The issue of correspondence for quantum systems whose classical analogs exhibit chaos ~irregular systems! is thus of great interest. Verification of correspondence should be distinguished from the development of semiclassical approximation methods. While semiclassical theories provide a natural starting point for an exploration of the classical limit their existence does not guarantee correspondence. For example, semiclassical theories for regular systems preceded the development of modern quantum mechanics @7#, but an understanding of correspondence for regular systems has only recently been achieved @2,3,5#. By comparison, attempts to develop semiclassical quantization rules for chaotic systems have had some success @8#, whereas the correspondence limit remains largely unexplored @9#. In this paper we demonstrate that the existing semiclassical theories of quantum dynamics for classically chaotic systems are sufficiently well developed to allow us to show that such systems do in fact approach their proper correspondence limits as Planck’s constant approaches zero. This completes the Liouville correspondence program outlined in the preceding paper @5#, and significantly extends the results of our study of quantum maps @10,11#, where we rigorously demonstrated that a nonchaotic quantum map dynamics can completely recover a fully chaotic classical dynamics in the limit h!0. The Liouville picture affords a means of gaining insight into the connections between quantum and classical mechanics @3,12,13#, and is a natural framework for studies of correspondence. As outlined in the preceding paper ~henceforth referred to as paper I !@ 5 #the essential ingredients for Liouville dynamics are eigenstates and eigenvalues of the Liouville operators in both mechanics. In particular, the dynamics is completely characterized by the Liouville eigenfunctions and eigenvalues or the spectral projectors once the class of allowed initial distributions is specified. Here we consider correspondence in chaotic systems from this Liouville perspective.

The calculation of transmission and resonance properties of quantum devices using methods from chemical reactive scattering

Two relatively new methods, the spectral projection method and the stabilization method, of implementing scattering calculations are described, and are here applied to two devices. Both methods use essentially short range spectral projection operators to produce a complete set of solutions of the wave equation that need be valid only inside the interaction region. While the spectral projection method is more generic than the stabilization method which is based on using the more difficult to compute spectral density operator, the latter becomes very efficient when narrow resonances exist. For problems of small size both methods are practical in the sense that they involve only real, symmetric matrices resulting from Hamiltonians represented on L 2basis sets. For more challenging larger systems the spectral projection method lends itself to a very efficient time independent iterative procedure that obtains results simultaneously at all energies. This procedure uses modified Chebyshev recursion relations to essentially expand the operator ( E− H) −1. It requires minimal storage and the resulting series converges rapidly in a manner that is uniform in energy.

Geometry of quantal adiabatic evolution for systems with a degenerate Hamiltonian

The holonomy group of the connection in the Hermitian vector bundle for quantal adiabatic evolution is studied. The necessary and sufficient condition of “ray-integrability” is given. The curvature tensor is given in terms of the spectral projection operator of the parameter-dependent Hamiltonian.

Concerning the Lp norm of spectral clusters for second-order elliptic operators on compact manifolds

In this paper we study the (Lp, L2) mapping properties of a spectral projection operator for Riemannian manifolds. This operator is a generalization of the harmonic projection operator for spherical harmonics on Sn (see C. D. Sogge, Duke Math. J. 53 1986, 43–65). Among other things, we generalize the L2 restriction theorems of C. Fefferman, E. M. Stein, and P. Tomas (Bull. Amer. Math. Soc. 81 1975, 477–478) for the Fourier transform in Rn to the setting of Riemannian manifolds. We obtain these results for the spectral projection operator as a corollary of a certain “Sobolev inequality” involving Δ + τ2 for large τ. This Sobolev inequality generalizes certain results for Rn of C. Kenig, A. Ruiz, and the author (Duke Math. J. 55 1987, 329–347). The main tools in the proof of the Sobolev inequalities for Riemannian manifolds are the Hadamard parametrix (cf. L. Hörmander, Acta Math. 88 1968, 341–370, and “The Analysis of Linear Partial Differential Equations,” Vol. III, Springer-Verlag, New York 1985) and oscillatory integral theorems of L. Carleson and P. Sjölin (Studia Math. 44 1972, 287–299) and Stein (Ann. Math. Stud. 112 1986, 307–357).

ASYMPTOTICS OF THE SPECTRUM OF PROBLEMS WITH CONSTRAINTS

By means of the method of an approximate spectral projection operator the classical asymptotic formula for the distribution function of eigenvalues with an estimate of the remainder is proved both for problems with an unsolvable constraint such as an incompressibility condition (the Navier-Stokes and Maxwell systems) and those with a solvable constraint (an example is the spectral problem of the theory of electroelasticity). Problems in a bounded Lipschitz domain are considered. We note that an estimate of the remainder for the linearized Navier-Stokes system was obtained earlier only for the case of a domain with boundary of class , while for problems with solvable constraints only the leading term of the asymptotics was known; the asymptotics of the spectrum in the problem of the theory of electroelasticity has not been studied previously.Bibliography: 13 titles.

Groups generated by analytic Schrödinger operators

If the potential in a two-particle system is the boundary value of an analytic function, the physical Hamiltonian H(0) has an analytic continuation H( φ) which is not normal. In case the potential is local and belongs to suitable L p -spaces, there is a bounded operator P(0, φ) projecting onto the continuous subspace of H( φ). This paper shows that P(0, φ) H( φ) e −2 iφ generates a strongly differentiable group. It is proved that P(0, φ) H( φ) is spectral, and details of the spectral projection operators are presented. The reasoning is based on the Paley-Wiener theorem for functions in a strip. It applies to larger systems provided the resolvent of the multiparticle operator H( φ) satisfies certain regularity conditions that come from the theory of smooth operators. There are no smallness conditions on the potential.

Conditions for the Existence of the Generalized Wave Operators

If E0(G) is the spectral projection operator associated with the free Hamiltonian H0, corresponding to a bounded measurable subset G of R, and E1(G) is associated with the total Hamiltonian H = H0 + V, where the operator E1(G)VE0(G) is of trace class, it is proved that the element g = E0(G)f belongs to the domain of the generalized wave operators Ω± if and only if lim lim t→±∞‖(1−E1(G))e−iH0tg‖=0. A stronger version of this result is also proved, from the theory of time-dependent scattering, and is applicable to scattering systems for which families {G1}, {G2} of measurable sets may be found such that E1(G2)VE0(G1) is of trace class.