‘‘Polynomial constants’’ for the quantized NLS equation

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Abstract
The classical nonlinear Schrödinger equation (NLS) is known to have an infinite number of polynomial constants. While recursion relations to compute these are available, no general expressions in terms of the fields have been found. However, general expressions have been obtained in terms of the reflection coefficients. When we turn to the quantum case where the fields become operators with conventional commutation relations, the polynomials with suitable ordering are still constants. The classical expression for the constants in terms of the reflection coefficients strongly suggests what the quantum form should be. This conjecture is proved for the repulsive case. The expression is significantly simpler than the classical one. It is In =(1/2π)∫∞−∞(k)nR*(k)R(k)dk.