Numerical ranges of unbounded operators arising in quantum physics

Numerical ranges of composition operators

Composition operators on the Hilbert Hardy space of the unit disk are considered. The shape of their numerical range is determined in the case when the symbol of the composition operator is a monomial or an inner function fixing 0. Several results on the numerical range of composition operators of arbitrary symbol are obtained. It is proved that 1 is an extreme boundary point if and only if 0 i...

On Generalized Numerical Ranges of Quadratic Operators

It is shown that the result of Tso-Wu on the elliptical shape of the numerical range of quadratic operators holds also for the essential numerical range. The latter is described quantitatively, and based on that sufficient conditions are established under which the c-numerical range also is an ellipse. Several examples are considered, including singular integral operators with the Cauchy kernel...

Generalized numerical ranges of permanental compounds arising from quantum systems of bosons

Generalized Numerical Ranges and Quantum Error Correction

For a noisy quantum channel, a quantum error correcting code of dimension k exists if and only if the joint rank-k numerical range associated with the error operators of the channel is non-empty. In this paper, geometric properties of the joint rank k-numerical range are obtained and their implications to quantum computing are discussed. It is shown that for a given k if the dimension of the un...

Orderings of Operators in Quantum Physics

For well-known three orderings (normal ordering, anti-normal ordering, and Weyl ordering) in Quantum Physics, we give a simple proof for the intertwining formulas among them. Moreover, for the creation operator a† and the annihilation operator a of a harmonic oscillator, we show Weyl ordering expression of (a†a)n as different types of polynomials with respect to the number operator N ≡ a†a. Non...