Related papers: Bispectral Darboux Transformations: The Generalize…
Twisted real structures are well-motivated as a way to implement the conformal transformation of a Dirac operator for a real spectral triple without needing to twist the noncommutative 1-forms. We study the coupling of spectral triples with…
Found are conditions on a scalar type spectral operator $A$ in a complex Banach space necessary and sufficient for all weak solutions of the evolution equation \begin{equation*} y'(t)=Ay(t),\ t\ge 0, \end{equation*} to be strongly Gevrey…
A. de Sole, V. G. Kac, and M. Wakimoto (arXiv:1004.5387) have recently introduced a new family of compatible Hamiltonian operators of the form $H^{(N,0)}=D^2\circ((1/u)\circ D)^{2n}\circ D$, where $N=2n+3$, $n=0,1,2,...$, $u$ is the…
In this paper, we generalize finite depth wavelet scattering transforms, which we formulate as $\Lb^q(\mathbb{R}^n)$ norms of a cascade of continuous wavelet transforms (or dyadic wavelet transforms) and contractive nonlinearities. We then…
In this article, the Darboux transformation for the non-isospectral AKNS hierarchy is constructed. We show that the Darboux transformation for the non-isospectral AKNS hierarchy is not an auto-B\"acklund transformation, because the integral…
Normality of the Dirac operator is shown to be necessary for chiral properties. From the global chiral Ward identity, which in the continuum limit gives the index theorem, a sum rule results which constrains the spectrum. The…
Painleve analysis and the singular manifold method are the tools used in this paper to perform a complete study of an equation in 2+1 dimensions. This procedure has allowed us to obtain the Lax pair, Darboux transformation and tau functions…
For a large class of integral operators or second order differential operators, their isospectral (or cospectral) operators are constructed explicitly in terms of $h$-transform (duality). This provides us a simple way to extend the known…
We study Darboux transformations associated with the focusing nonlinear Schr\"odinger equation (NLS_-) and their effect on spectral properties of the underlying Lax operator. The latter is a formally J-self-adjoint (but non-self-adjoint)…
The aim of this paper is to extend the notion of the spectral order for finite families of pairwise commuting bounded and unbounded selfadjoint operators in Hilbert space. It is shown that the multidimensional spectral order $\preccurlyeq$…
In this paper a higher order non-linear differential equation is given and it becomes a higher order Airy equation (in our terminology) under the Cole-Hopf transformation. For the even case a solution is explicitly constructed, which is a…
Let $H$ be a separable complex Hilbert space. Denote by ${\mathcal G}_{\infty}(H)$ the Grassmannian consisting of closed linear subspaces with infinite dimension and codimension. This Grassmannian is partially ordered by the inclusion…
A general classification of linear differential and finite-difference operators possessing a finite-dimensional invariant subspace with a polynomial basis (the generalized Bochner problem) is given. The main result is that any operator with…
In this paper we consider shift operators, self-adjoint, unitary and normal operators on the standard module over a unital C*-algebra A. We define various generalized spectra in A of these operators, give description of such spectra of…
Generalized spectra of differential operators can be related to spectra of preconditioned discretized operators. Obtaining (estimates of) the eigenvalues of the preconditioned discretized operators may lead to better estimating of the…
We investigate one of the simplest multispecies generalization of the asymmetric simple exclusion process on a ring. This process has a rich combinatorial spectral structure and a matrix product form for the stationary state. In the totally…
We construct Darboux transformations for the super-symmetric KP hierarchies of Manin--Radul and Jacobian types. We also consider the binary Darboux transformation for the hierarchies. The iterations of both type of Darboux transformations…
The Darboux transformations for the two dimensional elliptic affine Toda equations corresponding to all seven infinite series of affine Kac-Moody algebras, including $A_l^{(1)}$, $A_{2l}^{(2)}$, $A_{2l-1}^{(2)}$, $B_l^{(1)}$, $C_l^{(1)}$,…
\hspace{.2in}We consider the Darboux type transformations for the spectral problems of supersymmetric KdV systems. The supersymmetric analogies of Darboux and Darboux-Levi transformations are established for the spectral problems of…
We consider random arrays indexed by the leaves of an infinitary rooted tree of finite depth, with the distribution invariant under the rearrangements that preserve the tree structure. We call such arrays hierarchically exchangeable and…