J. Szmigielski - Academia.edu (original) (raw)
Papers by J. Szmigielski
J Inst Math Jussieu, 2005
![Research paper thumbnail of On pairs of difference operators satisfying [D, X]=Id](https://attachments.academia-assets.com/95356554/thumbnails/1.jpg)
](Journal of Mathematical Physics, 1998
Different finite difference replacements for the derivative are analyzed in the context of the He... more Different finite difference replacements for the derivative are analyzed in the context of the Heisenberg commutation relation. The type of the finite difference operator is shown to be tied to whether one can naturally consider D and X to be self-adjoint and skew self-adjoint or whether they have to be viewed as creation and annihilation operators. The first class, generalizing the central difference scheme, is shown to give unitary equivalent representations. For the second case we construct a large class of examples, generalizing previously known difference operator realizations of [D, X] = Id.
Dynamics of Partial Differential Equations, 2009
Contemporary Mathematics, 1990
Cauchy Biorthogonal Polynomials appear in the study of special solutions to the dispersive nonlin... more Cauchy Biorthogonal Polynomials appear in the study of special solutions to the dispersive nonlinear partial differential equation called the Degasperis-Procesi (DP) equation, as well as in certain two-matrix random matrix models. Another context in which such biorthogonal polynomials play a role is the cubic string; a third order ODE boundary value problem -f'''=zg f which is a generalization of the inhomogeneous string problem studied by M.G. Krein. A general class of such boundary value problems going beyond the original cubic string problem associated with the DP equation is discussed under the assumption that the source of inhomogeneity g is a discrete measure. It is shown that by a suitable choice of a generalized Fourier transform associated to these boundary value problems one can establish a Parseval type identity which aligns Cauchy biorthogonal polynomials with certain natural orthogonal systems on L^2_g.
Classical results of Stieltjes are used to obtain explicit formulas for the peakon-antipeakon sol... more Classical results of Stieltjes are used to obtain explicit formulas for the peakon-antipeakon solutions of the Camassa-Holm equation. The closed form solution is expressed in terms of the orthogonal polynomials of the related classical moment problem. It is shown that collisions occur only in peakon-antipeakon pairs, and the details of the collisions are analyzed using results from the moment problem. A sharp result on the steepening of the slope at the time of collision is given. Asymptotic formulas are given, and the scattering shifts are calculated explicitly
An explicit vertex operator algebra construction is given of a class of irreducible modules for t... more An explicit vertex operator algebra construction is given of a class of irreducible modules for toroidal Lie algebras.
The paper investigates the properties of certain biorthogonal polynomials appearing in a specific... more The paper investigates the properties of certain biorthogonal polynomials appearing in a specific simultaneous Hermite-Pade' approximation scheme. Associated to any totally positive kernel and a pair of positive measures on the positive axis we define biorthogonal polynomials and prove that their zeroes are simple and positive. We then specialize the kernel to the Cauchy kernel 1/x+y and show that the ensuing biorthogonal polynomials solve a four-term recurrence relation, have relevant Christoffel-Darboux generalized formulae, and their zeroes are interlaced. In addition, these polynomial solve a combination of Hermite-Pade' approximation problems to a Nikishin system of order 2. The motivation arises from two distant areas; on one side, in the study of the inverse spectral problem for the peakon solution of the Degasperis-Procesi equation; on the other side, from a random matrix model involving two positive definite random Hermitian matrices. Finally, we show how to charact...
We introduce a new class of two(multi)-matrix models of positive Hermitean matrices coupled in a ... more We introduce a new class of two(multi)-matrix models of positive Hermitean matrices coupled in a chain; the coupling is related to the Cauchy kernel and differs from the exponential coupling more commonly used in similar models. The correlation functions are expressed entirely in terms of certain biorthogonal polynomials and solutions of appropriate Riemann–Hilbert problems, thus paving the way to a steepest descent analysis and universality results. The interpretation of the formal expansion of the partition function in terms of multicolored ribbon-graphs is provided and a connection to the O(1) model. steepest descent analysis of the partition function reveals that the model is related to a trigonal curve (three-sheeted covering of the plane) much in the same way as the Hermitean matrix model is related to a hyperelliptic curve.
Peakons are non-smooth soliton solutions appearing in certain nonlinear partial differential equa... more Peakons are non-smooth soliton solutions appearing in certain nonlinear partial differential equations, most notably the Camassa-Holm equation and the Degasperis-Procesi equation. In the latter case the construction of peakons leads to a new class of biorthogonal polynomials. The present paper is the first in the series of papers aimed to establish a general framework in which to study such polynomials. It is shown that they belong to a class of biorthogonal polynomials with respect to a pairing between two Hilbert spaces with measures dα,dβ on the positive semi-axis R+ coupled through the the Cauchy kernel K(x, y) = 1 x+y. Fundamental properties of these polynomials are proved: their zeros are interlaced, they satisfy four-term recurrence relations and generalized Christoffel-Darboux identities, they admit a characterization in terms of a 3 by 3 matrix Riemann-Hilbert problem. The relevance of these polynomials to a third order boundary value problem (the cubic string) is explained...
Journal of the Institute of Mathematics of Jussieu, 2005
It has long been known that a number of periodic completely integrable systems are associated to ... more It has long been known that a number of periodic completely integrable systems are associated to hyperelliptic curves, for which the Abel map linearizes the flow (at least in part). We show that this is true for a relatively recent such system: the periodic discrete reduction of the shallow water equation derived by Camassa and Holm. The associated spectral problem has the same form and evolves in the same way as the spectral problem for a family of finite-dimensional non-periodic Hamiltonian flows introduced by Calogero and Françoise. We adapt the Weyl function method used earlier by us to solve the peakon problem to give an explicit solution to both the periodic discrete Camassa–Holm system and the (non-periodic) Calogero–Françoise system in terms of theta functions. AMS 2000 Mathematics subject classification: Primary 35Q51; 37J35; 35Q53
Journal of Nonlinear Mathematical Physics, 2001
Explicit formulas are given for the multi-peakon-antipeakon solutions of the Camassa-Holm equatio... more Explicit formulas are given for the multi-peakon-antipeakon solutions of the Camassa-Holm equation, and a detailed analysis is made of both short-term and long-term aspects of the interaction between a single peakon and single anti-peakon.
Advances in Mathematics, 1998
The acoustic scattering operator on the real line is mapped to a Schrödinger operator under the L... more The acoustic scattering operator on the real line is mapped to a Schrödinger operator under the Liouville transformation. The potentials in the image are characterized precisely in terms of their scattering data, and the inverse transformation is obtained as a simple, linear quadrature. An existence theorem for the associated Harry Dym flows is proved, using the scattering method. The scattering problem associated with the Camassa-Holm flows on the real line is solved explicitly for a special case, which is used to reduce a general class of such problems to scattering problems on finite intervals.
The acoustic scattering operator on the real line is mapped to a Schr\"odinger operator unde... more The acoustic scattering operator on the real line is mapped to a Schr\"odinger operator under the Liouville transformation. The potentials in the image are characterized precisely in terms of their scattering data, and the inverse transformation is obtained as a simple, linear quadrature. An existence theorem for the associated Harry Dym flows is proved, using the scattering method. The scattering problem associated with the Camassa-Holm flows on the real line is solved explicitly for a special case, which is used to reduce a general class of such problems to scattering problems on finite intervals.
Peakons are non-smooth soliton solutions appearing in certain nonlinear partial differential equa... more Peakons are non-smooth soliton solutions appearing in certain nonlinear partial differential equations, most notably the Camassa-Holm equation and the Degasperis-Procesi equation. In the latter case the construction of peakons leads to a new class of biorthogonal polynomials. The present paper is the first in the series of papers aimed to establish a general framework in which to study such polynomials. It is shown that they belong to a class of biorthogonal polynomials with respect to a pairing between two Hilbert spaces with measures dα, dβ on the positive semi-axis R+ coupled through the the Cauchy kernel K(x, y) = 1 x+y . Fundamental properties of these polynomials are proved: their zeros are interlaced, they satisfy four-term recurrence relations and generalized Christoffel-Darboux identities, they admit a characterization in terms of a 3 by 3 matrix Riemann-Hilbert problem. The relevance of these polynomials to a third order boundary value problem (the cubic string) is explain...
arXiv: Exactly Solvable and Integrable Systems, 2007
The contents of the paper is now covered in two separate papers arXiv:0904.2188 and arXiv:0904.26... more The contents of the paper is now covered in two separate papers arXiv:0904.2188 and arXiv:0904.2602. Please refer to those. Note that you can still access the original version arXiv:0711.4082v1.
Recent developments in infinite-dimensional Lie …, 2002
ABSTRACT. An explicit vertex operator algebra construction is given of a class of irreducible mod... more ABSTRACT. An explicit vertex operator algebra construction is given of a class of irreducible modules for toroidal Lie algebras. 1. lntroduction This paper is about representations of toroidal Lie algebras and vertex operator algebras which are naturally associated to them. ...
Advances in Mathematics
In this work we present new results on the convergence of diagonal sequences of certain mixed typ... more In this work we present new results on the convergence of diagonal sequences of certain mixed type Hermite-Padé approximants of a Nikishin system. The study is motivated by a mixed Hermite-Padé approximation scheme used in the construction of solutions of a Degasperis-Procesi peakon problem and germane to the analysis of the inverse spectral problem for the discrete cubic string.
The contents of the paper is now covered in two separate papers arXiv:0904.2188 and arXiv:0904.26... more The contents of the paper is now covered in two separate papers arXiv:0904.2188 and arXiv:0904.2602. Please refer to those. Note that you can still access the original version arXiv:0711.4082v1.
J Inst Math Jussieu, 2005
Journal of Mathematical Physics, 1998
Different finite difference replacements for the derivative are analyzed in the context of the He... more Different finite difference replacements for the derivative are analyzed in the context of the Heisenberg commutation relation. The type of the finite difference operator is shown to be tied to whether one can naturally consider D and X to be self-adjoint and skew self-adjoint or whether they have to be viewed as creation and annihilation operators. The first class, generalizing the central difference scheme, is shown to give unitary equivalent representations. For the second case we construct a large class of examples, generalizing previously known difference operator realizations of [D, X] = Id.
Dynamics of Partial Differential Equations, 2009
Contemporary Mathematics, 1990
Cauchy Biorthogonal Polynomials appear in the study of special solutions to the dispersive nonlin... more Cauchy Biorthogonal Polynomials appear in the study of special solutions to the dispersive nonlinear partial differential equation called the Degasperis-Procesi (DP) equation, as well as in certain two-matrix random matrix models. Another context in which such biorthogonal polynomials play a role is the cubic string; a third order ODE boundary value problem -f'''=zg f which is a generalization of the inhomogeneous string problem studied by M.G. Krein. A general class of such boundary value problems going beyond the original cubic string problem associated with the DP equation is discussed under the assumption that the source of inhomogeneity g is a discrete measure. It is shown that by a suitable choice of a generalized Fourier transform associated to these boundary value problems one can establish a Parseval type identity which aligns Cauchy biorthogonal polynomials with certain natural orthogonal systems on L^2_g.
Classical results of Stieltjes are used to obtain explicit formulas for the peakon-antipeakon sol... more Classical results of Stieltjes are used to obtain explicit formulas for the peakon-antipeakon solutions of the Camassa-Holm equation. The closed form solution is expressed in terms of the orthogonal polynomials of the related classical moment problem. It is shown that collisions occur only in peakon-antipeakon pairs, and the details of the collisions are analyzed using results from the moment problem. A sharp result on the steepening of the slope at the time of collision is given. Asymptotic formulas are given, and the scattering shifts are calculated explicitly
An explicit vertex operator algebra construction is given of a class of irreducible modules for t... more An explicit vertex operator algebra construction is given of a class of irreducible modules for toroidal Lie algebras.
The paper investigates the properties of certain biorthogonal polynomials appearing in a specific... more The paper investigates the properties of certain biorthogonal polynomials appearing in a specific simultaneous Hermite-Pade' approximation scheme. Associated to any totally positive kernel and a pair of positive measures on the positive axis we define biorthogonal polynomials and prove that their zeroes are simple and positive. We then specialize the kernel to the Cauchy kernel 1/x+y and show that the ensuing biorthogonal polynomials solve a four-term recurrence relation, have relevant Christoffel-Darboux generalized formulae, and their zeroes are interlaced. In addition, these polynomial solve a combination of Hermite-Pade' approximation problems to a Nikishin system of order 2. The motivation arises from two distant areas; on one side, in the study of the inverse spectral problem for the peakon solution of the Degasperis-Procesi equation; on the other side, from a random matrix model involving two positive definite random Hermitian matrices. Finally, we show how to charact...
We introduce a new class of two(multi)-matrix models of positive Hermitean matrices coupled in a ... more We introduce a new class of two(multi)-matrix models of positive Hermitean matrices coupled in a chain; the coupling is related to the Cauchy kernel and differs from the exponential coupling more commonly used in similar models. The correlation functions are expressed entirely in terms of certain biorthogonal polynomials and solutions of appropriate Riemann–Hilbert problems, thus paving the way to a steepest descent analysis and universality results. The interpretation of the formal expansion of the partition function in terms of multicolored ribbon-graphs is provided and a connection to the O(1) model. steepest descent analysis of the partition function reveals that the model is related to a trigonal curve (three-sheeted covering of the plane) much in the same way as the Hermitean matrix model is related to a hyperelliptic curve.
Peakons are non-smooth soliton solutions appearing in certain nonlinear partial differential equa... more Peakons are non-smooth soliton solutions appearing in certain nonlinear partial differential equations, most notably the Camassa-Holm equation and the Degasperis-Procesi equation. In the latter case the construction of peakons leads to a new class of biorthogonal polynomials. The present paper is the first in the series of papers aimed to establish a general framework in which to study such polynomials. It is shown that they belong to a class of biorthogonal polynomials with respect to a pairing between two Hilbert spaces with measures dα,dβ on the positive semi-axis R+ coupled through the the Cauchy kernel K(x, y) = 1 x+y. Fundamental properties of these polynomials are proved: their zeros are interlaced, they satisfy four-term recurrence relations and generalized Christoffel-Darboux identities, they admit a characterization in terms of a 3 by 3 matrix Riemann-Hilbert problem. The relevance of these polynomials to a third order boundary value problem (the cubic string) is explained...
Journal of the Institute of Mathematics of Jussieu, 2005
It has long been known that a number of periodic completely integrable systems are associated to ... more It has long been known that a number of periodic completely integrable systems are associated to hyperelliptic curves, for which the Abel map linearizes the flow (at least in part). We show that this is true for a relatively recent such system: the periodic discrete reduction of the shallow water equation derived by Camassa and Holm. The associated spectral problem has the same form and evolves in the same way as the spectral problem for a family of finite-dimensional non-periodic Hamiltonian flows introduced by Calogero and Françoise. We adapt the Weyl function method used earlier by us to solve the peakon problem to give an explicit solution to both the periodic discrete Camassa–Holm system and the (non-periodic) Calogero–Françoise system in terms of theta functions. AMS 2000 Mathematics subject classification: Primary 35Q51; 37J35; 35Q53
Journal of Nonlinear Mathematical Physics, 2001
Explicit formulas are given for the multi-peakon-antipeakon solutions of the Camassa-Holm equatio... more Explicit formulas are given for the multi-peakon-antipeakon solutions of the Camassa-Holm equation, and a detailed analysis is made of both short-term and long-term aspects of the interaction between a single peakon and single anti-peakon.
Advances in Mathematics, 1998
The acoustic scattering operator on the real line is mapped to a Schrödinger operator under the L... more The acoustic scattering operator on the real line is mapped to a Schrödinger operator under the Liouville transformation. The potentials in the image are characterized precisely in terms of their scattering data, and the inverse transformation is obtained as a simple, linear quadrature. An existence theorem for the associated Harry Dym flows is proved, using the scattering method. The scattering problem associated with the Camassa-Holm flows on the real line is solved explicitly for a special case, which is used to reduce a general class of such problems to scattering problems on finite intervals.
The acoustic scattering operator on the real line is mapped to a Schr\"odinger operator unde... more The acoustic scattering operator on the real line is mapped to a Schr\"odinger operator under the Liouville transformation. The potentials in the image are characterized precisely in terms of their scattering data, and the inverse transformation is obtained as a simple, linear quadrature. An existence theorem for the associated Harry Dym flows is proved, using the scattering method. The scattering problem associated with the Camassa-Holm flows on the real line is solved explicitly for a special case, which is used to reduce a general class of such problems to scattering problems on finite intervals.
Peakons are non-smooth soliton solutions appearing in certain nonlinear partial differential equa... more Peakons are non-smooth soliton solutions appearing in certain nonlinear partial differential equations, most notably the Camassa-Holm equation and the Degasperis-Procesi equation. In the latter case the construction of peakons leads to a new class of biorthogonal polynomials. The present paper is the first in the series of papers aimed to establish a general framework in which to study such polynomials. It is shown that they belong to a class of biorthogonal polynomials with respect to a pairing between two Hilbert spaces with measures dα, dβ on the positive semi-axis R+ coupled through the the Cauchy kernel K(x, y) = 1 x+y . Fundamental properties of these polynomials are proved: their zeros are interlaced, they satisfy four-term recurrence relations and generalized Christoffel-Darboux identities, they admit a characterization in terms of a 3 by 3 matrix Riemann-Hilbert problem. The relevance of these polynomials to a third order boundary value problem (the cubic string) is explain...
arXiv: Exactly Solvable and Integrable Systems, 2007
The contents of the paper is now covered in two separate papers arXiv:0904.2188 and arXiv:0904.26... more The contents of the paper is now covered in two separate papers arXiv:0904.2188 and arXiv:0904.2602. Please refer to those. Note that you can still access the original version arXiv:0711.4082v1.
Recent developments in infinite-dimensional Lie …, 2002
ABSTRACT. An explicit vertex operator algebra construction is given of a class of irreducible mod... more ABSTRACT. An explicit vertex operator algebra construction is given of a class of irreducible modules for toroidal Lie algebras. 1. lntroduction This paper is about representations of toroidal Lie algebras and vertex operator algebras which are naturally associated to them. ...
Advances in Mathematics
In this work we present new results on the convergence of diagonal sequences of certain mixed typ... more In this work we present new results on the convergence of diagonal sequences of certain mixed type Hermite-Padé approximants of a Nikishin system. The study is motivated by a mixed Hermite-Padé approximation scheme used in the construction of solutions of a Degasperis-Procesi peakon problem and germane to the analysis of the inverse spectral problem for the discrete cubic string.
The contents of the paper is now covered in two separate papers arXiv:0904.2188 and arXiv:0904.26... more The contents of the paper is now covered in two separate papers arXiv:0904.2188 and arXiv:0904.2602. Please refer to those. Note that you can still access the original version arXiv:0711.4082v1.