J. J. Duistermaat

Differential equations in the spectral parameter

We determine all the potentialsV(x) for the Schrödinger equation (−∂x2+V(x))∅=k2∅ such that some family of eigenfunctions ∅ satisfies a differential equation in the spectral parameterk of the formB(k, ∂k)ø=Θ(x)ø. For each suchV(x) we determine the algebra of all possible operatorsB and the corresponding functions Θ(x)

The quantum mechanical spherical pendulum

On decrit le comportement asymptotique du spectre du pendule quantique spherique quand la constante de Planck tend vers zero

Spectra of compact locally symmetric manifolds of negative curvature

Let S be a Riemannian symmetric space of noncompact type, and let G be the group of motions of S. Then the algebra L-~ of G-invariant differential operators on S is commutative, and its spectrum A(S) can be canonically identified with ~/w where ~ is a complex vector space with dimension equal to the rank of S, and to is a finite subgroup of G L ( ~ ) generated by reflexions. Let P be a discrete subgroup of G that acts freely on S and let X = E \ S . Then the members of 5~ may be regarded as differential operators on X. Let us now assume that X is compact and define the spectrum A of X as the set of those elements of A(S) for which one can find a nonzero eigenfunction defined on X. In this paper we study the relationship of A to the geometry of X and determine the asymptotic growth of A as a subset of A(S). In subsequent papers we plan to study the asymptotic behaviour of the eigenfunctions and to examine the problem of obtaining improvements on the error estimates. It is well-known that G, which is transitive on S, is a connected real semisimple Lie group with trivial center, and that the stabilizers in G of the points of S are the maximal compact subgroups of G. So we can take S = G/K, X =F\G/K, where K is a fixed maximal compact subgroup of G, and F is a discrete subgroup of G containing no elliptic elements (= elements conjugate to an element of K) other than e, such that F\G is compact. Let G = K A N be an Iwasawa decomposit ion of G; let o be the Lie algebra of A; and let to be the Weyl group of (G, A). If we take ,~to be the dual of the complexification a c of a, then A ( S ) ~ / w canonically. In what follows we shall commit an abuse of notation and identify A(S) with ,~, but with the proviso that points of ~ in the same w-orbit represent the same element of A(S).

Discrete Integrable Systems

10.1007/978-1-4419-9126-3 Copyright owner: Springer Science+Buisness Media, LLC, 2010 Data set: Springer Source Springer Monographs in Mathematics The rich subject matter in this book brings in mathematics from different domains, especially from the theory of elliptic surfaces and dynamics.The material comes from the authorâ€TMs insights and understanding of a birational transformation of the plane derived from a discrete sine-Gordon equation, posing the question of determining the behavior of the discrete dynamical system defined by the iterates of the map. The aim of this book is to give a complete treatment not only of the basic facts about QRT maps, but also the background theory on which these maps are based. Readers with a good knowledge of algebraic geometry will be interested in Kodairaâ€TMs theory of elliptic surfaces and the collection of nontrivial applications presented here. While prerequisites for some readers will demand their knowledge of quite a bit of algebraicand complex analytic geometry, different categories of readers... more Identifiers series ISSN : 1439-7382 ISBN 978-1-4419-7116-6 e-ISBN 978-1-4419-9126-3 DOI Authors Additional information Publisher Springer New York book Read online Download Add to read later Add to collection Add to followed Share Export to bibliography J.J. Duistermaat Utrecht University, Department of Mathematics, Utrecht, Netherlands Terms of service Accessibility options Report an error / abuse

Constant terms in powers of a Laurent polynomial

Abstract We classify the complex Laurent polynomials with the property that their powers have no constant term. The result confirms a conjecture of Mathieu for the case of tori. (A different case would imply Kellers Jacobian Conjecture.)

Reduced phase space and toric variety coordinatizations of Delzant spaces

In this note we describe the natural coordinatizations of a Delzant space defined as a reduced phase space (symplectic geometry view-point) and give explicit formulas for the coordinate transformations. For each fixed point of the torus action on the Delzant polytope, we have a maximal coordinatization of an open cell in the Delzant space which contains the fixed point. This cell is equal to the domain of definition of one of the natural coordinatizations of the Delzant space as a toric variety (complex algebraic geometry view-point), and we give an explicit formula for the toric variety coordinates in terms of the reduced phase space coordinates. We use considerations in the maximal coordinate neighborhoods to give simple proofs of some of the basic facts about the Delzant space, as a reduced phase space, and as a toric variety. These can be viewed as a first application of the coordinatizations, and serve to make the presentation more self-contained.

A characterization of the Ligon‐Schaaf regularization map

First, we give an explicit description of all the mappings from the phace space of the Kepler problem to the phase space of the geodesics on the sphere, which transform the constants of motion of the Kepler problem to the angular momentum. Second, among these we describe those mappings which in addition send Kepler solutions to parametrized geodesics. Third, we describe those mappings which in addition are canonical transformations of the respective phase space. Finally we prove that among these the Ligon-Schaaf map is the unique one which maps the collison orbits to the geodesics which pass through the north pole. In this way we also give a new proof that the Ligon-Schaaf map has all the properties described above.

Energy and entropy as real morphisms for addition and order

In to-days mathematical language such functions are called morphisms with respect to addition and order from (S, +, <) into (R, +, <). Here (R, +, <) denotes the system R of real numbers, provided with its usual addition and order. The physical concepts of energy and entropy also enjoy the properties of being additive and monotone with respect to certain appropriate structures of addition and order. If there exists a unique measurement of energy and entropy in which only these properties are used, then we can use the proposition that energy and entropy are real morphisms for the corresponding addition and order as an alternative definition of these concepts. The physical interpretation of such morphisms is entirely determined by the physical interpretation of the system S and the structures of addition and order in S. Structures of addition and order play an important part in many physical experiments; therefore it should be expected that their morphisms also play an essential part in the corresponding theories. From a mathematical point of view the definition of real-valued

The Quantum Spherical Pendulum

The configuration space is the 2-dimensional sphere