We give a general analysis of a class of pairs of positive self-adjoint operators A and B for which A + XB has a limit (in strong resolvent sense) as h-10 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 1-s. 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.

. 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 Pad'e 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 Pad'e approximants for series of Hamburger. x1. Introduction The classical moment problem was central to the development of analysis in the period from 1894 (when Stieltjes wrote his famous memoir [38]) until the 1950's (when Krein completed his series on the subject [15, 16, 17]). The notion of measure (Stieltjes integrals), Pad'e approximants, orthogonal polynomials, extensions of positive linear functionals (Riesz-Markov theorem), boundary values of analytic functions, and the Herglotz-Nevanlinna-Riesz representation theorem all have their ro...

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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 multi-valued analytic functions, one for even eigenfunctions and one for odd ones. 2. The only singularities of these multi-valued functions in the complex α-plane are algebraic ramification points, and there are only finitely many singularities over each compact subset of the α-plane.

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.

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 PT-symmetric 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,

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 1-dimensional systems. In 1-dimensional 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.

ABSTRACT. In this paper, we generalize a recent result of A. Eremenko and A. Gabrielov on irreducibility of the spectral discriminant for the Schrödinger equation with quartic potentials. We consider the eigenvalue problem with a complex-valued polynomial potential of arbitrary degree d and show that the spectral determinant of this problem is connected and irreducible. In other words, every eigenvalue can be reached from any other by analytic continuation. We also prove connectedness of the parameter spaces of the potentials that admit eigenfunctions satisfying k> 2 boundary conditions, except for the case d is even and k = d/2. In the latter case, connected components of the parameter space are distinguished by the number of zeros of

Abstract. For tri-diagonal matrices arising in the simplified Jaynes– Cummings model, we give an asymptotics of the eigenvalues, prove a trace formula and show that the Spectral Riemann Surface is irreducible. MSC: 47B36

Littlewood, when he makes use of an algebraic identity, always saves himself the trouble of proving it; he maintains that an identity, if true, can be verified in few lines by anybody obtuse enough to feel the need of verification. Freeman Dyson [7] We study elementary eigenfunctions y = pe h of operators L(y) = y ′ ′ + Py, where p, h and P are polynomials in one variable. For the case when h is an odd cubic polynomial, we investigate the real level crossing points and asymptotics of eigenvalues. This study leads to an interesting identity with elementary integrals.