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131
Transfer functions of regular linear systems Part III: Inversions And Duality
 Trans. Amer. Math. Soc
, 2000
"... We study four transformations which lead from one wellposed linear system to another: timeinversion, flowinversion, timeflowinversion and duality. Timeinversion means reversing the direction of time, flowinversion means interchanging inputs with outputs, while timeflowinversion means doing ..."
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Cited by 78 (13 self)
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We study four transformations which lead from one wellposed linear system to another: timeinversion, flowinversion, timeflowinversion and duality. Timeinversion means reversing the direction of time, flowinversion means interchanging inputs with outputs, while timeflowinversion means doing both of the inversions mentioned before. A wellposed linear system is timeinvertible if and only if its operator semigroup extends to a group. The system is flowinvertible if and only if its inputoutput map has a bounded inverse on some (hence, on every) finite time interval [0; ] ( > 0). This is true if and only if the transfer function of has a uniformly bounded inverse on some right halfplane. The system is timeflowinvertible if and only if on some (hence, on every) finite time interval [0; ], the combined operator from the initial state and the input function to the final state and the output function is invertible. This is the case, for example, if the system is conservative, since then is unitary. Timeowinversion can sometimes, but not always, be reduced to a combination of time and flowinversion. We derive a surprising necessary and sucient condition for to be timeflowinvertible: its system operator must have a uniformly bounded inverse on some left halfplane.
SolitonLike Asymptotics for a Classical Particle Interacting with a Scalar Wave Field
 COMM. PARTIAL DIERENTIAL EQUATIONS
, 1998
"... We consider a scalar wave field translation invariantly coupled to a single particle. This Hamiltonian system admits solitonlike solutions, where the particle and comoving field travel with constant velocity. We prove that a solution of finite energy converges, in suitable local energy seminorms, t ..."
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Cited by 42 (14 self)
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We consider a scalar wave field translation invariantly coupled to a single particle. This Hamiltonian system admits solitonlike solutions, where the particle and comoving field travel with constant velocity. We prove that a solution of finite energy converges, in suitable local energy seminorms, to some solitonlike solutions in the long time limit t ! \Sigma1.
Resonance Expansions Of Scattered Waves
 Comm. Pure Appl. Math
"... The this paper is to describe expansions of solutions to the wave equation on R^n with a compactly supported perturbation present. We show that under a separation condition on resonances, the solutions can be expanded in terms of resonances close to the real axis, modulo an error rapidly decaying in ..."
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Cited by 30 (10 self)
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The this paper is to describe expansions of solutions to the wave equation on R^n with a compactly supported perturbation present. We show that under a separation condition on resonances, the solutions can be expanded in terms of resonances close to the real axis, modulo an error rapidly decaying in time. To avoid the discussion of particular aspects of potential, gravitational or obstacle scattering, the results are stated using the abstract "black box" formalism of Sjöstrand and the second author [19]. When M is a compact Riemannian manifold, and Delta, its Laplacian, then we have a generalized Fourier expansion of the wave group: sin t p p f(x) = X 2 j 2Spec() e i j t w j (x) ; w j = 2 j w j ; (1.1) and the convergence is absolute in the case of smooth data. The simplest case of a noncompact spectral problem is given by taking R^n with the usual Laplacian outside a compact set  we can for instance "glue" any compact Riemannian manifold to R^n or consider the obstacle problem with the Dirichlet Laplacian on R^n\O. Since resonances or scattering poles constitute a natural replacement of discrete spectral data for problems on exterior domains, we expect a similar expansion involving them in place of eigenvalues  this point of view was emphasized early by LaxPhillips [10] (see [18],[32] and [34] for overviews of recent results). The resonances are de ned as poles of the meromorphic continuation of the resolvent or of the scattering matrix but despite the stationary nature of these de nitions, they are fundamentally a dynamical concept: the real part of a resonance describes the rest energy of a state and the imaginary part its rate of decay. Consequently they should be understood in terms of long time behaviour of solutions to evolutio...
Time dependent resonance theory
, 1995
"... An important class of resonance problems involves the study of perturbations of systems having embedded eigenvalues in their continuous spectrum. Problems with this mathematical structure arise in the study of many physical systems, e.g. the coupling of an atom or molecule to a photonradiation fiel ..."
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Cited by 29 (9 self)
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An important class of resonance problems involves the study of perturbations of systems having embedded eigenvalues in their continuous spectrum. Problems with this mathematical structure arise in the study of many physical systems, e.g. the coupling of an atom or molecule to a photonradiation field, and Auger states of the helium atom, as well as in spectral geometry and number theory. We present a dynamic (timedependent) theory of such quantum resonances. The key hypotheses are (i) a resonance condition which holds generically (nonvanishing of the Fermi golden rule) and (ii) local decay estimates for the unperturbed dynamics with initial data consisting of continuum modes associated with an interval containing the embedded eigenvalue of the unperturbed Hamiltonian. No assumption of dilation analyticity of the potential is made. Our method explicitly demonstrates the flow of energy from the resonant discrete mode to continuum modes due to their coupling. The approach is also applicable to nonautonomous linear problems and to nonlinear problems. We derive the time behavior of the resonant states for intermediate and long times. Examples and applications are presented. Among them is a proof of the instability of an embedded eigenvalue at a threshold energy under suitable hypotheses.
General recursion via coinductive types
 Logical Methods in Computer Science
"... Vol. 1 (2:1) 2005, pp. 1–28 ..."
Scattering theory for systems with different spatial asymptotics to the left and right
 COMMUN.MATH.PHYS. 63
, 1978
"... We discuss the existence and completeness of scattering for onedimensional systems with different spatial asymptotics at ± oo, for example 2 4 V(x) where V(x) = 0 (resp. sin x) if x < 0 (resp. x> 0). We then extend our results to higher dimensional systems periodic, except for a localised impurit ..."
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Cited by 29 (11 self)
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We discuss the existence and completeness of scattering for onedimensional systems with different spatial asymptotics at ± oo, for example 2 4 V(x) where V(x) = 0 (resp. sin x) if x < 0 (resp. x> 0). We then extend our results to higher dimensional systems periodic, except for a localised impurity, in all but one space dimension. A new method, "the twisting trick", is presented for proving the absence of singular continuous spectrum, and some independent applications of this trick are given in an appendix.
Spohn H.: Longtime asymptotics for the coupled MaxwellLorentz equations
 Comm. Part. Diff. Eq
, 2000
"... 1Supported partly by FrenchRussian A.M.Liapunov Center of Moscow State University, and by research grants of RFBR (960100527) and of VolkswagenStiftung. 1 Introduction We consider a single charge coupled to the Maxwell field and subject to prescribed timeindependent external potentials. If q(t) ..."
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Cited by 28 (13 self)
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1Supported partly by FrenchRussian A.M.Liapunov Center of Moscow State University, and by research grants of RFBR (960100527) and of VolkswagenStiftung. 1 Introduction We consider a single charge coupled to the Maxwell field and subject to prescribed timeindependent external potentials. If q(t) 2 IR3 denotes the position of charge at time t, then the coupled MaxwellLorentz equations read
Resonances in physics and geometry
 Notices Amer. Math. Soc
, 1999
"... Resonances are most readily associated with musical instruments or with the Tacoma bridge disaster. The latter is described in many physics and ODE books, and at the Ontario Science Center one can even find a model allowing one to find the destructive resonant frequency. The resonances I would like ..."
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Cited by 27 (10 self)
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Resonances are most readily associated with musical instruments or with the Tacoma bridge disaster. The latter is described in many physics and ODE books, and at the Ontario Science Center one can even find a model allowing one to find the destructive resonant frequency. The resonances I would like to write about are closely related but have their origins in quantum or electromagnetic scattering. To introduce them in a rough way, let us first recall the notion of eigenvalues. Eigenvalues of selfadjoint operators describe, among other things, the energies of bound states, states that exist forever if unperturbed. These do exist in real life; for instance, we can tell the composition of stars from our knowledge of atomic spectra. In most situations, however, states do not exist forever, and a more accurate model is given by a decaying state that oscillates at some rate. The decay might be caused by damping or by a possibility of escape to infinity. To describe these more realistic states, we use resonances. They have a very long tradition in mathematical physics, but they also appear naturally in pure mathematics. The last ten years brought many new ideas and new results into the subject. Old problems concerning the proximity of resonances to the real axis, their relation to quasimodes, and their distribution for scattering by convex bodies have been solved. Upper bounds for counting functions of resonances have become well understood, and the new area of lower bounds has become active. New directions were opened by considering resonances in geometry, where in fact Maciej Zworski is professor of mathematics at the University of California at Berkeley. His email address is
Isometries, shifts, Cuntz algebras and multiresolution wavelet analysis of scale N
, 1996
"... In this paper we show how wavelets originating from multiresolution analysis of scale N give rise to certain representations of the Cuntz algebras ON, and conversely how the wavelets can be recovered from these representations. The representations are given on the Hilbert space L 2 (T) by (Siξ)(z) ..."
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Cited by 26 (13 self)
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In this paper we show how wavelets originating from multiresolution analysis of scale N give rise to certain representations of the Cuntz algebras ON, and conversely how the wavelets can be recovered from these representations. The representations are given on the Hilbert space L 2 (T) by (Siξ)(z) = mi (z)ξ ( z N). We characterize the Wold decomposition of such operators. If the operators come from wavelets they are shifts, and this can be used to realize the representation on a certain Hardy space over L 2 (T). This is used to compare the usual scale2 theory of wavelets with the scaleN theory. Also some other representations of ON of the above form called diagonal representations are characterized and classified up to unitary equivalence by a homological invariant.