Related papers: Fuchsian equations of type DN

We define the concept of completely regular ordinary differential operators and give various criteria for operators to belong to this class. We give also criteria for Birkhof regularity of ordinary differential operators in terms of the…

Given a non-zero polynomial $P(x)$, we study Fuchsian differential operators of the form $L=\partial_x^2-u(x)$ such that for all $\lambda\in\mathbb{C}$ the operator $L+\lambda P(x)$ is monodromy free. We prove that all such operators are…

We prove the bispectrality of some class of matrix Schr\"odinger operators with polynomial potentials that satisfy a second-order matrix autonomous differential equation. The physical equation is constructed using the formal theory of the…

The Schrodinger equations which are exactly solvable in terms of associated special functions are directly related to some self-adjoint operators defined in the theory of hypergeometric type equations. The fundamental formulae occurring in…

This paper is devoted to the eigenvalue problem for the quantum Gaudin system. We prove the universal correspondence between eigenvalues of Gaudin Hamiltonians and the so-called G-opers without monodromy in general gl(n) case modulo a…

The work analyzes the theory of Dunkl operator as a moment differential operator. This last operator generalizes the first one whenever the sequence of moments satisfies appropriate classical properties, classically considered in the…

A generalized variant of the Calder\'on problem from electrical impedance tomography with partial data for anisotropic Lipschitz conductivities is considered in an arbitrary space dimension $n \geq 2$. The following two results are shown:…

Let $E/F$ be a quadratic extension of non-Archimedean local fields of characteristic 0. Let $D$ be the unique quaternion division algebra over $F$ and fix an embedding of $E$ to $D$. Then, $\mathrm{GL}_m(D)$ can be regarded as a subgroup of…

We give fundamental solutions of arbitrarily sized matrix Fuchsian linear systems, in the case where the coefficients $B^{(i)}$ of the systems are matrix solutions of the Schlesinger system that are upper triangular, and whose eigenvalues…

Let $(p_n)_n$ be a sequence of orthogonal polynomials with respect to the measure $\mu$. Let $T$ be a linear operator acting in the linear space of polynomials $\PP$ and satisfying that $\dgr(T(p))=\dgr(p)-1$, for all polynomial $p$. We…

Burchnall and Chaundy showed that if two ODOs $P$, $Q$ with analytic coefficients commute there exists a polynomial $f(\lambda ,\mu)$ with complex coefficients such that $f(P,Q)=0$, called the BC-polynomial. This polynomial can be computed…

Generalizing the algebra of motion-invariant differential operators on a symmetric space we study invariant operators on equivariant vector bundles. We show that the eigenequation is equivalent to the corresponding eigenequation with…

An evolution equation, arising in the study of the Dynamical Systems Method (DSM) for solving equations with monotone operators, is studied in this paper. The evolution equation is a continuous analog of the regularized Newton method for…

We revisit miscellaneous linear differential operators mostly associated with lattice Green functions in arbitrary dimensions, but also Calabi-Yau operators and order-seven operators corresponding to exceptional differential Galois groups.…

We study the unitarity of monodromies of rank two Fuchsian systems of SL type with $(n+1)$ regular singularities on the Riemann sphere, namely, we give a sufficient and necessary condition for the monodromy group to be conjugate to a…

In this paper, we generalize a recent result of A. Eremenko and A. Gabrielov on irreducibility of the spectral discriminant for the Schr\"odinger equation with quartic potentials. We consider the eigenvalue problem with a complex-valued…

Discrete Differential Equations (DDEs) are functional equations that relate polynomially a power series $F(t,u)$ in $t$ with polynomial coefficients in a "catalytic" variable $u$ and the specializations, say at $u=1$, of $F(t,u)$ and of…

Let $T$ be a bounded quaternionic normal operator on a right quaternionic Hilbert space $\mathcal{H}$. We show that $T$ can be factorized in a strongly irreducible sense, that is, for any $\delta >0$ there exist a compact operator $K$ with…

It is shown that each linear operator on a separable Hilbert space which generates a finite type I von Neumann algebra has, up to unitary equivalence, a unique representation as a direct integral of inflations of mutually unitary…

We study the regularity properties of a general second order H\"ormander operator with Dini continous coefficients $a_{ij}$. Precisely if $X_0, X_1,\cdots X_m$ are smooth self adjoint vector fields satisfying the H\"ormander condition, we…