Results 1  10
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28
Grounding in communication
 In
, 1991
"... We give a general analysis of a class of pairs of positive selfadjoint operators A and B for which A + XB has a limit (in strong resolvent sense) as h10 which is an operator A, # A! Recently, Klauder [4] has discussed the following example: Let A be the operator(d2/A2) + x2 on L2(R, dx) and let ..."
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Cited by 862 (19 self)
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We give a general analysis of a class of pairs of positive selfadjoint operators A and B for which A + XB has a limit (in strong resolvent sense) as h10 which is an operator A, # A! Recently, Klauder [4] has discussed the following example: Let A be the operator(d2/A2) + x2 on L2(R, dx) and let B = 1 x 1s. The eigenvectors and eigenvalues of A are, of course, well known to be the Hermite functions, H,(x), n = 0, l,... and E, = 2n + 1. Klauder then considers the eigenvectors of A + XB (A> 0) by manipulations with the ordinary differential equation (we consider the domain questions, which Klauder ignores, below). He finds that the eigenvalues E,(X) and eigenvectors &(A) do not converge to 8, and H, but rather AO) + (en 4 Ho+, J%(X)+ gn+1 I n = 0, 2,..., We wish to discuss in detail the general phenomena which Klauder has uncovered. We freely use the techniques of quadratic forms and strong resolvent convergence; see e.g. [3], [5]. Once one decides to analyze Klauder’s phenomenon in the language of quadratic forms, the phenomenon is quite easy to understand and control. In fact, the theory is implicit in Kato’s book [3, VIII.31.
The Classical Moment Problem as a SelfAdjoint Finite Difference Operator
, 1998
"... This is a comprehensive exposition of the classical moment problem using methods from the theory of finite difference operators. Among the advantages of this approach is that the Nevanlinna functions appear as elements of a transfer matrix and convergence of Pade approximants appears as the strong r ..."
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Cited by 95 (7 self)
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This is a comprehensive exposition of the classical moment problem using methods from the theory of finite difference operators. Among the advantages of this approach is that the Nevanlinna functions appear as elements of a transfer matrix and convergence of Pade approximants appears as the strong resolvent convergence of finite matrix approximations to a Jacobi matrix. As a bonus of this, we obtain new results on the convergence of certain Pade approximants for series of Hamburger.
The mathematical theory of resonances whose widths are exponentially small
 Duke Math. J
, 1980
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Long time dynamics near the symmetry breaking bifurcation for nonlinear Schrödinger/GrossPitaevskii equations, Discrete Contin
 Dyn. Syst
"... Abstract. We consider a class nonlinear Schrödinger / GrossPitaevskii equations (NLS/GP) with a focusing (attractive) nonlinear potential and symmetric double well linear potential. NLS/GP plays a central role in the modeling of nonlinear optical and meanfield quantum manybody phenomena. It is kn ..."
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Cited by 9 (2 self)
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Abstract. We consider a class nonlinear Schrödinger / GrossPitaevskii equations (NLS/GP) with a focusing (attractive) nonlinear potential and symmetric double well linear potential. NLS/GP plays a central role in the modeling of nonlinear optical and meanfield quantum manybody phenomena. It is known that there is a critical L 2 norm (optical power / particle number) at which there is a symmetry breaking bifurcation of the ground state. We study the rich dynamical behavior near the symmetry breaking point. The source of this behavior in the full Hamiltonian PDE is related to the dynamics of a finitedimensional Hamiltonian reduction. We derive this reduction, analyze a part of its phase space and prove a shadowing theorem on the persistence of solutions, with oscillating masstransport between wells, on very long, but finite, time scales within the full NLS/GP. The infinite time dynamics for NLS/GP are expected to depart, from the finite dimensional reduction, due to resonant coupling of discrete and continuum / radiation modes. (1.1)
Eigenvalues of PTsymmetric oscillators with polynomial potentials
 J. Phys. A
"... Abstract. We study the eigenvalue problem −u ′ ′ (z) − [(iz) m + Pm−1(iz)]u(z) = λu(z) with the boundary conditions that u(z) decays to zero as z tends to infinity along the rays argz = − π 2π 2 m+2, where Pm−1(z) = a1zm−1 + a2zm−2 + · · · + am−1z is a polynomial and m ≥ 3. We provide an asymp ..."
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Cited by 9 (2 self)
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Abstract. We study the eigenvalue problem −u ′ ′ (z) − [(iz) m + Pm−1(iz)]u(z) = λu(z) with the boundary conditions that u(z) decays to zero as z tends to infinity along the rays argz = − π 2π 2 m+2, where Pm−1(z) = a1zm−1 + a2zm−2 + · · · + am−1z is a polynomial and m ≥ 3. We provide an asymptotic expansion of the eigenvalues λn as n → +∞, and prove that for each real polynomial Pm−1, all but finitely many eigenvalues are real and positive. Preprint. 1.
Analytic continuation of eigenvalues of a quartic oscillator
, 2008
"... We consider the Schrödinger operator on the real line with even quartic potential x 4 +αx 2 and study analytic continuation of eigenvalues, as functions of parameter α. We prove several properties of this analytic continuation conjectured by Bender, Wu, Loeffel and Martin. 1. All eigenvalues are giv ..."
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Cited by 8 (7 self)
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We consider the Schrödinger operator on the real line with even quartic potential x 4 +αx 2 and study analytic continuation of eigenvalues, as functions of parameter α. We prove several properties of this analytic continuation conjectured by Bender, Wu, Loeffel and Martin. 1. All eigenvalues are given by branches of two multivalued analytic functions, one for even eigenfunctions and one for odd ones. 2. The only singularities of these multivalued functions in the complex αplane are algebraic ramification points, and there are only finitely many singularities over each compact subset of the αplane.
SINGULAR PERTURBATION OF POLYNOMIAL POTENTIALS WITH APPLICATIONS TO PTSYMMETRIC FAMILIES
"... Abstract. We discuss eigenvalue problems of the form −w ′ ′ + Pw = λw with complex polynomial potential P(z) = tz d +..., where t is a parameter, with zero boundary conditions at infinity on two rays in the complex plane. In the first part of the paper we give sufficient conditions for continuity o ..."
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Cited by 7 (6 self)
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Abstract. We discuss eigenvalue problems of the form −w ′ ′ + Pw = λw with complex polynomial potential P(z) = tz d +..., where t is a parameter, with zero boundary conditions at infinity on two rays in the complex plane. In the first part of the paper we give sufficient conditions for continuity of the spectrum at t = 0. In the second part we apply these results to the study of topology and geometry of the real spectral loci of PTsymmetric families with P of degree 3 and 4, and prove several related results on the location of zeros of their eigenfunctions. MSC: 34M35, 35J10. Keywords: singular perturbation, onedimensional Schrödinger operators, eigenvalue, spectral determinant,
Fifty years of eigenvalue perturbation theory
 Bull. Amer. Math. Soc
, 1991
"... ABSTRACT. We highlight progress in the study of eigenvalue perturbation theory, especially problems connected to quantum mechanics. Six models are discussed in detail: isoelectronic atoms, autoionizing states, the anharmonic oscillator, double wells, and the Zeeman and Stark effects. Berry's ph ..."
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Cited by 7 (1 self)
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ABSTRACT. We highlight progress in the study of eigenvalue perturbation theory, especially problems connected to quantum mechanics. Six models are discussed in detail: isoelectronic atoms, autoionizing states, the anharmonic oscillator, double wells, and the Zeeman and Stark effects. Berry's phase is also discussed 1.
Trapping and Cascading of Eigenvalues In the Large Coupling Limit
 COMMUNICATIONS IN MATHEMATICAL PHYSICS
, 1988
"... We consider eigenvalues E λ of the Hamiltonian H λ = — Δ+ V+ λW, W compactly supported, in the /> oo limit. For W ^ 0 we find monotonic convergence of E λ to the eigenvalues of a limiting operator H ^ (associated with an exterior Dirichlet problem), and we estimate the rate of convergence for ..."
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Cited by 6 (2 self)
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We consider eigenvalues E λ of the Hamiltonian H λ = — Δ+ V+ λW, W compactly supported, in the /> oo limit. For W ^ 0 we find monotonic convergence of E λ to the eigenvalues of a limiting operator H ^ (associated with an exterior Dirichlet problem), and we estimate the rate of convergence for 1dimensional systems. In 1dimensional systems with W^0 9 or with W changing sign, we do not find convergence. Instead, we find a cascade phenomenon, in which, as Λ,>oo, each eigenvalue E λ stays near a Dirichlet eigenvalue for a long interval (of length 0(^/1)) of the scaling range, quickly drops to the next lower Dirichlet eigenvalue, stays there for a long interval, drops again, and so on. As a result, for most large values of λ the discrete spectrum of H λ is close to that of H^, but when λ reaches a transition region, the entire spectrum quickly shifts down by one. We also explore the behavior of several explicit models, as λ+ oo.
Schrödinger operators with magnetic fields I: General interactions
 DUKE MATH. J
, 1978
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