On the Equivalence of Matrix Differential Operators to Schrödinger Form

On the Equivalence of Matrix Differential Operators to Schrödinger Form

We prove a generalization to the case of s × s matrix linear differential operators of the classical theorem of E. Cotton giving necessary and sufficient conditions for equivalence of eigenvalue problems for scalar linear differential operators. The conditions for equivalence to a matrix Schrödinger operator are derived and formulated geometrically in terms of vanishing conditions on the curvat...

The spectral properties of differential operators with matrix coefficients on elliptic systems with boundary conditions

Let $$(Lv)(t)=sum^{n} _{i,j=1} (-1)^{j} d_{j} left( s^{2alpha}(t) b_{ij}(t) mu(t) d_{i}v(t)right),$$ be a non-selfadjoint differential operator on the Hilbert space $L_{2}(Omega)$ with Dirichlet-type boundary conditions. In continuing of papers [10-12], let the conditions made on the operator $ L$ be sufficiently more general than [11] and [12] as defined in Section $1$. In this paper, we estim...

On Commuting Matrix Differential Operators

If the differential expressions P and L are polynomials (over C) of another differential expression they will obviously commute. To have a P which does not arise in this way but satisfies [P,L] = 0 is rare. Yet the question of when it happens has received a lot of attention since Lax presented his description of the KdV hierarchy by Lax pairs (P,L). In this paper the question is answered in the...

Some equivalence classes of operators on B(H)

Quasi exactly solvable matrix Schrödinger operators

Two families of quasi exactly solvable 2 × 2 matrix Schrödinger operators are constructed. The first one is based on a polynomial matrix potential and depends on three parameters. The second is a one-parameter generalisation of the scalar Lamé equation. The relationship between these operators and QES Hamiltonians already considered in the literature is pointed out.