Separability is an important problem in theory of quantum entanglement. By using the Bloch representation of quantum states in terms of the Heisenberg-Weyl observable basis, we present a new separability criterion for...Separability is an important problem in theory of quantum entanglement. By using the Bloch representation of quantum states in terms of the Heisenberg-Weyl observable basis, we present a new separability criterion for bipartite quantum systems. It is shown that this criterion can be better than the previous ones in detecting entanglement. The results are generalized to multipartite quantum states.展开更多
In analogy to the role of Lommel polynomials ?in relation to Bessel functions Jv(z) the theory of Associated Hermite polynomials in the scaled form ?with parmeter v to Parabolic Cylinder functions Dv(z) is developed. ...In analogy to the role of Lommel polynomials ?in relation to Bessel functions Jv(z) the theory of Associated Hermite polynomials in the scaled form ?with parmeter v to Parabolic Cylinder functions Dv(z) is developed. The group-theoretical background with the 3-parameter group of motions M(2) in the plane for Bessel functions and of the Heisenberg-Weyl group W(2) for Parabolic Cylinder functions is discussed and compared with formulae, in particular, for the lowering and raising operators and the eigenvalue equations. Recurrence relations for the Associated Hermite polynomials and for their derivative and the differential equation for them are derived in detail. Explicit expressions for the Associated Hermite polynomials with involved Jacobi polynomials at argument zero are given and by means of them the Parabolic Cylinder functions are represented by two such basic functions.展开更多
In this paper we study the mean square of the error term in the Weyl's law of an irrational (2l + 1)-dimensional Heisenberg manifold. An asymptotic formula is established.
We use the Weyl correspondence approach to investigate the spectral operators of the Laguerre and the Kautz systems. We show that the basic functions in the Laguerre and the Kautz systems are the eigenvectors of modif...We use the Weyl correspondence approach to investigate the spectral operators of the Laguerre and the Kautz systems. We show that the basic functions in the Laguerre and the Kautz systems are the eigenvectors of modified Sturm-Liouville operators. The results are further generalized to multiple parameters of one complex variable in both the unit disc and the upper half-plane contexts.展开更多
基金Project supported by the National Natural Science Foundation of China(Grant Nos.11501153,11661031,and 11675113)the National Natural Science Foundation of Hainan Province,China(Grant No.20161006)
文摘Separability is an important problem in theory of quantum entanglement. By using the Bloch representation of quantum states in terms of the Heisenberg-Weyl observable basis, we present a new separability criterion for bipartite quantum systems. It is shown that this criterion can be better than the previous ones in detecting entanglement. The results are generalized to multipartite quantum states.
文摘In analogy to the role of Lommel polynomials ?in relation to Bessel functions Jv(z) the theory of Associated Hermite polynomials in the scaled form ?with parmeter v to Parabolic Cylinder functions Dv(z) is developed. The group-theoretical background with the 3-parameter group of motions M(2) in the plane for Bessel functions and of the Heisenberg-Weyl group W(2) for Parabolic Cylinder functions is discussed and compared with formulae, in particular, for the lowering and raising operators and the eigenvalue equations. Recurrence relations for the Associated Hermite polynomials and for their derivative and the differential equation for them are derived in detail. Explicit expressions for the Associated Hermite polynomials with involved Jacobi polynomials at argument zero are given and by means of them the Parabolic Cylinder functions are represented by two such basic functions.
基金supported by National Natural Science Foundation of China (Grant No. 10771127)
文摘In this paper we study the mean square of the error term in the Weyl's law of an irrational (2l + 1)-dimensional Heisenberg manifold. An asymptotic formula is established.
基金supported by National Natural Science Foundation of China(Grant Nos.11571083,11971178 and 11701597)the starting grant of South China Agricultural Universitythe Science and Technology Development Fund,Macao SAR(Grant Nos.154/2017/A3,079/2016/A2 and FDCT 0123/2018/A3)
文摘We use the Weyl correspondence approach to investigate the spectral operators of the Laguerre and the Kautz systems. We show that the basic functions in the Laguerre and the Kautz systems are the eigenvectors of modified Sturm-Liouville operators. The results are further generalized to multiple parameters of one complex variable in both the unit disc and the upper half-plane contexts.
