The spectrum of Hill's equation

McKean, H. P.; Moerbeke, Pierre van

doi:10.1007/bf01425567

Cited by 510 publications

(293 citation statements)

References 16 publications

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“…The potential therefore belongs to the class of exactly solvable potentials. It is strongly reminiscent of the formula for finite zones potentials [20,21,22,23]. It can probably also be obtained by an infinite sequence of Darboux transformations [24].…”

Section: Propositionmentioning

confidence: 87%

Continuum Limit of the Volterra Model, Separation of Variables and Non-Standard Realizations of the Virasoro Poisson Bracket

Babelon

2006

Commun. Math. Phys.

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The classical Volterra model, equipped with the Faddeev-Takhtadjan Poisson bracket provides a lattice version of the Virasoro algebra. The Volterra model being integrable, we can express the dynamical variables in terms of the so called separated variables. Taking the continuum limit of these formulae, we obtain the Virasoro generators written as determinants of infinite matrices, the elements of which are constructed with a set of points lying on an infinite genus Riemann surface. The coordinates of these points are separated variables for an infinite set of Poisson commuting quantities including L 0 . The scaling limit of the eigenvector can also be calculated explicitly, so that the associated Schroedinger equation is in fact exactly solvable.

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Section: Propositionmentioning

confidence: 87%

Continuum Limit of the Volterra Model, Separation of Variables and Non-Standard Realizations of the Virasoro Poisson Bracket

Babelon

2006

Commun. Math. Phys.

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“…In this work, we shall solve only a small part of it. For d = 1, the inverse problem has been solved thoroughly [MO75,MM75,MT76,Kri77,MO80,Kor08]. We shall specifically concentrate on the case d = 2, for fixed λ.…”

Section: What Is Done In This Workmentioning

confidence: 99%

On complex Fermi curves of two-dimensional, periodic Schrödinger operators

Klauer

2014

Journal of Applied Analysis

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In dimensions d ≥ 2, the complex Bloch varieties and the associated Fermi curves of periodic Schrödinger operators with quasi-periodic boundary conditions are defined as complex analytic varieties. The Schrödinger potentials are taken from the Lebesgue space L d/2 in the case d > 2, and from the Lorentz-Fourier space F ∞,1 in the case d = 2. Then, an asymptotic analysis of the Fermi curves in the case d = 2 is performed. The decomposition of a Fermi curve into a compact part, an asymptotically free part, and thin handles, is recovered as expected. Furthermore, it is shown that the set of potentials whose associated Fermi curve has finite geometric genus is a dense subset of F ∞,1 . Moreover, the Fourier transforms of the potentials are locally isomorphic to perturbed Fourier transforms induced by the handles. Finally, an asymptotic family of parameters describing the sizes of the handles is introduced. These parameters are good candidates for describing the space of all Fermi curves. Zusammenfassung

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“…One can show that when q is an algebro-geometric potential, S B has only finitely many gaps; see [1,6,24]. So we suppose that…”

Section: Realising Linear Systems For Elliptic Potentialsmentioning

confidence: 99%

On tau functions for orthogonal polynomials and matrix models

Blower¹

2011

J. Phys. A: Math. Theor.

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Abstract. Let v be a real polynomial of even degree, and let ρ be the equilibrium probability measure for v with support S; so that, v(x) ≥ 2 log |x−y| ρ(dy)+C v for some constant C v with equality on S. Then S is the union of finitely many bounded intervals with endpoints δ j , and ρ is given by an algebraic weight w(x) on S. The system of orthogonal polynomials for w gives rise to the Magnus-Schlesinger differential equations. This paper identifies the τ function of this system with the Hankel determinant det[of ρ. The solutions of the Magnus-Schlesinger equations are realised by a linear system, which is used to compute the tau function in terms of a Gelfand-Levitan equation. The tau function is associated with a potential q and a scattering problem for the Schrödinger operator with potential q. For some algebro-geometric q, the paper solves the scattering problem in terms of linear systems. The theory extends naturally to elliptic curves and resolves the case where S has exactly two intervals.MSC (2000) classification: 60B20 (37K15)

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The spectrum of Hill's equation

Cited by 510 publications

References 16 publications

Continuum Limit of the Volterra Model, Separation of Variables and Non-Standard Realizations of the Virasoro Poisson Bracket

Continuum Limit of the Volterra Model, Separation of Variables and Non-Standard Realizations of the Virasoro Poisson Bracket

On complex Fermi curves of two-dimensional, periodic Schrödinger operators

On tau functions for orthogonal polynomials and matrix models

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