342 research outputs found
Differential Equations for Algebraic Functions
It is classical that univariate algebraic functions satisfy linear
differential equations with polynomial coefficients. Linear recurrences follow
for the coefficients of their power series expansions. We show that the linear
differential equation of minimal order has coefficients whose degree is cubic
in the degree of the function. We also show that there exists a linear
differential equation of order linear in the degree whose coefficients are only
of quadratic degree. Furthermore, we prove the existence of recurrences of
order and degree close to optimal. We study the complexity of computing these
differential equations and recurrences. We deduce a fast algorithm for the
expansion of algebraic series
Quantum inner-product metrics via recurrent solution of Dieudonne equation
A given Hamiltonian matrix H with real spectrum is assumed tridiagonal and
non-Hermitian. Its possible Hermitizations via an amended, ad hoc inner-product
metric are studied. Under certain reasonable assumptions, all of these metrics
are shown obtainable as recurrent solutions of the hidden Hermiticity
constraint called Dieudonne equation. In this framework even the two-parametric
Jacobi-polynomial real- and asymmetric-matrix N-site lattice Hamiltonian is
found tractable non-numerically at all N.Comment: 21 p
Gegenbauer-solvable quantum chain model
In an innovative inverse-problem construction the measured, experimental
energies , , ... of a quantum bound-state system are assumed
fitted by an N-plet of zeros of a classical orthogonal polynomial . We
reconstruct the underlying Hamiltonian (in the most elementary
nearest-neighbor-interaction form) and the underlying Hilbert space
of states (the rich menu of non-equivalent inner products is offered). The
Gegenbauer's ultraspherical polynomials are chosen for
the detailed illustration of technicalities.Comment: 29 pp., 1 fi
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