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18
Perturbation results of critical elliptic equations of CaffarelliKohnNirenberg type
 Journal of Differential Equations
"... Abstract. We find for small ε positive solutions to the equation −div (x  −2a λ ( ) p−1 ..."
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Abstract. We find for small ε positive solutions to the equation −div (x  −2a λ ( ) p−1
New QuasiExactly Solvable Hamiltonians in Two Dimensions
"... Quasiexactly solvable Schrodinger operators have the remarkable property that a part of their spectrum can be computed by algebraic methods. Such operators lie in the enveloping algebra of a finitedimensional Lie algebra of first order differential operators  the "hidden symmetry algebra". In t ..."
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Quasiexactly solvable Schrodinger operators have the remarkable property that a part of their spectrum can be computed by algebraic methods. Such operators lie in the enveloping algebra of a finitedimensional Lie algebra of first order differential operators  the "hidden symmetry algebra". In this paper we develop some general techniques for constructing quasiexactly solvable operators. Our methods are applied to provide a wide variety of new explicit twodimensional examples (on both flat and curved spaces) of quasiexactly solvable Hamiltonians, corresponding to both semisimple and more general classes of Lie algebras. x Supported in Part by DGICYT Grant PS 890011.  Supported in Part by an NSERC Grant. y Supported in Part by NSF Grant DMS 9204192. z On leave from School of Mathematics, University of Minnesota, Minneapolis, Minnesota, U.S.A. 55455. 1 Introduction The spectral problems of nonrelativistic quantum mechanics fall within two general categories. In the fi...
Real Lie algebras of differential operators and quasiexactly solvable potentials
, 1995
"... Abstract. We first establish some general results connecting real and complex Lie algebras of firstorder differential operators. These are applied to completely classify all finitedimensional real Lie algebras of firstorder differential operators in R 2. Furthermore, we find all algebras which ar ..."
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Abstract. We first establish some general results connecting real and complex Lie algebras of firstorder differential operators. These are applied to completely classify all finitedimensional real Lie algebras of firstorder differential operators in R 2. Furthermore, we find all algebras which are quasiexactly solvable, along with the associated finitedimensional modules of analytic functions. The resulting real Lie algebras are used to construct new quasiexactly solvable Schrödinger operators on R 2.
Umbral calculus, difference equations and the discrete Schrödinger equation, Preprint nlin S1/0305047
"... We discuss umbral calculus as a method of systematically discretizing linear differential equations while preserving their point symmetries as well as generalized symmetries. The method is then applied to the Schrödinger equation in order to obtain a realization of nonrelativistic quantum mechanics ..."
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We discuss umbral calculus as a method of systematically discretizing linear differential equations while preserving their point symmetries as well as generalized symmetries. The method is then applied to the Schrödinger equation in order to obtain a realization of nonrelativistic quantum mechanics in discrete space–time. In this approach a quantum system on a lattice has a symmetry algebra isomorphic to that of the continuous case. Moreover, systems that are integrable, superintegrable or exactly solvable preserve these properties in the discrete case. Key words: umbral calculus, difference equations, symmetries, integrability, quantum mechanics, discrete space–time. PACS num. 02.20.–a, 02.30.Ik, 02.30.Ks, 03. 1 1
Zeros of eigenfunctions of some anharmonic oscillators
"... We study complex zeros of eigenfunctions of second order linear differential operators with real even polynomial potentials. For potentials of degree 4, we prove that all zeros of all eigenfunctions belong to the union of the real and imaginary axes. For potentials of degree 6, we classify eigenfunc ..."
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We study complex zeros of eigenfunctions of second order linear differential operators with real even polynomial potentials. For potentials of degree 4, we prove that all zeros of all eigenfunctions belong to the union of the real and imaginary axes. For potentials of degree 6, we classify eigenfunctions with finitely many zeros, and show that in this case too, all zeros are real or pure imaginary.
On asymptotics of polynomial eigenfunctions for exactlysolvable differential operators
"... In this paper we study the class of differential operators T = P k j=1 QjDj with polynomial coefficients Qj in one complex variable satisfying the condition deg Qj ≤ j with equality for at least one j. We show that if deg Qk < k then the root with the largest modulus of the nth degree eigenpolynomia ..."
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Cited by 6 (1 self)
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In this paper we study the class of differential operators T = P k j=1 QjDj with polynomial coefficients Qj in one complex variable satisfying the condition deg Qj ≤ j with equality for at least one j. We show that if deg Qk < k then the root with the largest modulus of the nth degree eigenpolynomial pn of T tends to infinity when n → ∞, as opposed to the case when deg Qk = k, which we have treated previously in [2]. Moreover we present an explicit conjecture and partial results on the growth of the largest modulus of the roots of pn. Based on this conjecture we deduce the algebraic equation satisfied by the Cauchy transform of the asymptotic root measure of the appropriately scaled eigenpolynomials, for which the union of all roots is conjecturally contained in a compact set. 1
QuasiExact Solvability
 Contemp. Math
, 1994
"... . This paper surveys recent work on quasiexactly solvable Schrodinger operators and Lie algebras of differential operators. 1. Introduction. Lie algebraic and Lie group theoretic methods have played a significant role in the development of quantum mechanics since its inception. In the classical ap ..."
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. This paper surveys recent work on quasiexactly solvable Schrodinger operators and Lie algebras of differential operators. 1. Introduction. Lie algebraic and Lie group theoretic methods have played a significant role in the development of quantum mechanics since its inception. In the classical applications, the Lie group appears as a symmetry group of the Hamiltonian operator, and the associated representation theory provides an algebraic means for computing the spectrum. Of particular importance are the exactly solvable problems, such as the harmonic oscillator or the hydrogen atom, whose point spectrum can be completely determined using purely algebraic methods. The fundamental concept of a "spectrum generating algebra" was introduced by Arima and Iachello, [4], [5], to study nuclear physics, and subsequently, by Iachello, Alhassid, Gursey, Levine, Wu and their collaborators, was also successfully applied to molecular dynamics and spectroscopy, [19], [22], and scattering theory, [...
QuasiExactly Solvable Spin 1/2 Schrödinger Operators
 J. MATH. PHYS
, 1996
"... The algebraic structures underlying quasiexact solvability for spin 1/2 Hamiltonians in one dimension are studied in detail. Necessary and sufficient conditions for a matrix secondorder differential operator preserving a space of wave functions with polynomial components to be equivalent to a Schr ..."
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The algebraic structures underlying quasiexact solvability for spin 1/2 Hamiltonians in one dimension are studied in detail. Necessary and sufficient conditions for a matrix secondorder differential operator preserving a space of wave functions with polynomial components to be equivalent to a Schrodinger operator are found. Systematic simplifications of these conditions are analyzed, and are then applied to the construction of new examples of multiparameter QES spin 1/2 Hamiltonians in one dimension.
Quasiexactlysolvable differential equations, in: CRC Handbook of Lie Group Analysis of Differential Equations
, 1996
"... Abstract. A general classification of linear differential and finitedifference operators possessing a finitedimensional invariant subspace with a polynomial basis is given. The main result is that any operator with the above property must have a representation as a polynomial element of the univer ..."
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Abstract. A general classification of linear differential and finitedifference operators possessing a finitedimensional invariant subspace with a polynomial basis is given. The main result is that any operator with the above property must have a representation as a polynomial element of the universal enveloping algebra of the algebra of differential (difference) operators in finitedimensional representation. In onedimensional case a classification is given by algebras sl2(R) (for differential operators in R) and sl2(R)q (for finitedifference operators in R), osp(2,2) (operators in one real and one Grassmann variable, or equivalently, 2 × 2 matrix operators in R) and gl2(R)K ( for the operators containing the differential operators and the parity operator). A classification of linear operators possessing infinitely many finitedimensional invariant subspaces with a basis in polynomials is presented.
A QuasiExactly Solvable Travel Guide
, 1997
"... nian H is is said to be Lie algebraic if it lies in the universal enveloping algebra of a finitedimensional Lie algebra g, which is spanned by first order differential operators J a = d X i=1 ¸ ai (x) @ @x i + j a (x); a = 1; : : : ; r: (1) In particular, a secondorder differential ope ..."
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nian H is is said to be Lie algebraic if it lies in the universal enveloping algebra of a finitedimensional Lie algebra g, which is spanned by first order differential operators J a = d X i=1 ¸ ai (x) @ @x i + j a (x); a = 1; : : : ; r: (1) In particular, a secondorder differential operator is Lie algebraic if it can be written as a constant coefficient quadratic combination H = X a;b c ab J a J b + X a c a J a + c 0 : (2) Note that if J 2 g, then the commutator [J; H], while still of the same Lie algebraic form (2), is