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Register a new userExact treatment of operator difference equations with nonconstant noncommutative coefficients
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Added by
Anastasia Jivulescu
Publication date
2012
fields
Physics
and research's language is
English
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In this paper we study in a Hilbert space a homogeneous linear second order difference equation with nonconstant and noncommuting operator coefficients. We build its exact resolutive formula consisting in the explicit non-iterative expression of a generic term of the unknown sequence of vectors of the Hilbert space. Some non-trivial applications are reported with the aim of showing the usefulness and the broad applicability of our result.
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The exact solution of a Cauchy problem related to a linear second-order difference equation with constant noncommutative coefficients is reported.
The detailed construction of the general solution of a second order non-homogenous linear operatordifference equation is presented. The wide applicability of such an equation as well as the usefulness of its resolutive formula is shown by studying some applications belonging to different mathematical contexts.
The singularity structure of a second-order ordinary differential equation with polynomial coefficients often yields the type of solution. If the solution is a special function that is studied in the literature, then the result is more manageable using the properties of that function. It is straightforward to find the regular and irregular singular points of such an equation by a computer algebra system. However, one needs the corresponding indices for a full analysis of the singularity structure. It is shown that the $theta$-operator method can be used as a symbolic computational approach to obtain the indicial equation and the recurrence relation. Consequently, the singularity structure which can be visualized through a Riemann P-symbol leads to the transformations that yield a solution in terms of a special function, if the equation is suitable. Hypergeometric and Heun-type equations are mostly employed in physical applications. Thus only these equations and their confluent types are considered with SageMath routines which are assembled in the open-source package symODE2.
We construct the Baxters operator and the corresponding Baxters equation for a quantum version of the Ablowitz Ladik model. The result is achieved by looking at the quantum analogue of the classical Backlund transformations. For comparison we find the same result by using the well-known Bethe ansatz technique. General results about integrable models governed by the same r-matrix algebra will be given. The Baxters equation comes out to be a q-difference equation involving both the trace and the quantum determinant of the monodromy matrix. The spectrality property of the classical Backlund transformations gives a trace formula representing the classical analogue of the Baxters equation. An explicit q-integral representation of the Baxters operator is discussed.
The method, proposed in the given work, allows the application of well developed standard methods used in quantum mechanics for approximate solution of the systems of ordinary linear differential equations with periodical coefficients.
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