By combining the operator Hermite polynomial method and the technique of integration within an ordered product of operators, for the first time we derive the generating function of even- and odd-Hermite polynomials wh...By combining the operator Hermite polynomial method and the technique of integration within an ordered product of operators, for the first time we derive the generating function of even- and odd-Hermite polynomials which will be useful in constructing new optical field states. We then show that the squeezed state and photon-added squeezed state can be expressed by even- and odd-Hermite polynomials.展开更多
Using the coherent state representation of Wigner operator and the technique of integration within an ordered product (IWOP) of operators, this paper derives the Wigner function for the Hermite polynomial state (HP...Using the coherent state representation of Wigner operator and the technique of integration within an ordered product (IWOP) of operators, this paper derives the Wigner function for the Hermite polynomial state (HPS). The tomogram of the HPS is calculated with the aid of newly introduced intermediate coordinate-momentum representation in quantum optics.展开更多
The nonclassicality of the two-variable Hermite polynomial state is investigated. It is found that the two-variable Hermite polynomial state can be considered as a two-mode photon subtracted squeezed vacuum state. A c...The nonclassicality of the two-variable Hermite polynomial state is investigated. It is found that the two-variable Hermite polynomial state can be considered as a two-mode photon subtracted squeezed vacuum state. A compact expression for the Wigner function is also derived analytically by using the Weyl-ordered operator invariance under similar transformations. Especially, the nonclassicality is discussed in terms of the negativity of the Wigner function. Then violations of Bell's inequality for the two-variable Hermite polynomial state are studied.展开更多
By virtue of the operator-Hermite-polynomial method, we derive some new generating function formulas of the product of two bivariate Hermite polynomials. Their applications in studying quantum optical states are prese...By virtue of the operator-Hermite-polynomial method, we derive some new generating function formulas of the product of two bivariate Hermite polynomials. Their applications in studying quantum optical states are presented.展开更多
We derive some new generating function formulae of the two-variable Hermite polynomials, such as ∞∑n=0tm/m!Hn,2m(x),∞∑n=0sntm/n!m!H2n,2m(x,y),and ∞∑n=0sntm/n!m!H2n+l,2m+k(x,y).We employ the operator Herm...We derive some new generating function formulae of the two-variable Hermite polynomials, such as ∞∑n=0tm/m!Hn,2m(x),∞∑n=0sntm/n!m!H2n,2m(x,y),and ∞∑n=0sntm/n!m!H2n+l,2m+k(x,y).We employ the operator Hermite polynomial method and the technique of integration within an ordered product of operators to solve these problems, which will be useful in constructing new optical field states.展开更多
We examine the energy function with respect to the zeros of exceptional Hermite polynomials. The localization of the eigenvalues of the Hessian is given in the general case.In some special arrangements we have a more ...We examine the energy function with respect to the zeros of exceptional Hermite polynomials. The localization of the eigenvalues of the Hessian is given in the general case.In some special arrangements we have a more precise result on the behavior of the energy function. Finally we investigate the energy function with respect to the regular zeros of the exceptional Hermite polynomials.展开更多
In this article,the author extends the validity of a uniform asymptotic expansion of the Hermite polynomials Hn(√2n+1α)to include all positive values of α. His method makes use of the rational functions introduc...In this article,the author extends the validity of a uniform asymptotic expansion of the Hermite polynomials Hn(√2n+1α)to include all positive values of α. His method makes use of the rational functions introduced by Olde Daalhuis and Temme (SIAM J.Math.Anal.,(1994),25:304-321).A new estimate for the remainder is given.展开更多
For directly normalizing the photon non-Gaussian states (e.g., photon added and subtracted squeezed states), we use the method of integration within an ordered product (IWOP) of operators to derive some new bosoni...For directly normalizing the photon non-Gaussian states (e.g., photon added and subtracted squeezed states), we use the method of integration within an ordered product (IWOP) of operators to derive some new bosonic operator-ordering identities. We also derive some new integration transformation formulas about one- and two-variable Hermite polynomials in complex function space. These operator identities and associative integration formulas provide much convenience for constructing non-Gaussian states in quantum engineering.展开更多
By virtue of the entangled state representation we concisely derive some new operator identities with regard to the two-variable Hermite polynomial (TVHP). By them and the technique of integration within an ordered ...By virtue of the entangled state representation we concisely derive some new operator identities with regard to the two-variable Hermite polynomial (TVHP). By them and the technique of integration within an ordered product (IWOP) of operators we further derive new generating function formulas of the TVHP. They are useful in quantum optical theoretical calculations. It is seen from this work that by combining the IWOP technique and quantum mechanical representations one can derive some new integration formulas even without really performing the integration.展开更多
Based on the operator Hermite polynomials method(OHPM), we study Stirling numbers in the context of quantum mechanics, i.e., we present operator realization of generating function formulas of Stirling numbers with s...Based on the operator Hermite polynomials method(OHPM), we study Stirling numbers in the context of quantum mechanics, i.e., we present operator realization of generating function formulas of Stirling numbers with some applications.As a by-product, we derive a summation formula involving both Stirling number and Hermite polynomials.展开更多
The bilinear generating function for products of two Laguerre 2D polynomials with different arguments is calculated. It corresponds to the formula of Mehler for the generating function of products of two Hermite polyn...The bilinear generating function for products of two Laguerre 2D polynomials with different arguments is calculated. It corresponds to the formula of Mehler for the generating function of products of two Hermite polynomials. Furthermore, the generating function for mixed products of Laguerre 2D and Hermite 2D polynomials and for products of two Hermite 2D polynomials is calculated. A set of infinite sums over products of two Laguerre 2D polynomials as intermediate step to the generating function for products of Laguerre 2D polynomials is evaluated but these sums possess also proper importance for calculations with Laguerre polynomials. With the technique of operator disentanglement some operator identities are derived in an appendix. They allow calculating convolutions of Gaussian functions combined with polynomials in one- and two-dimensional case and are applied to evaluate the discussed generating functions.展开更多
In expansions of arbitrary functions in Bessel functions or Spherical Bessel functions, a dual partner set of polynomials play a role. For the Bessel functions, these are the Chebyshev polynomials of first kind and fo...In expansions of arbitrary functions in Bessel functions or Spherical Bessel functions, a dual partner set of polynomials play a role. For the Bessel functions, these are the Chebyshev polynomials of first kind and for the Spherical Bessel functions the Legendre polynomials. These two sets of functions appear in many formulas of the expansion and in the completeness and (bi)-orthogonality relations. The analogy to expansions of functions in Taylor series and in moment series and to expansions in Hermite functions is elaborated. Besides other special expansion, we find the expansion of Bessel functions in Spherical Bessel functions and their inversion and of Chebyshev polynomials of first kind in Legendre polynomials and their inversion. For the operators which generate the Spherical Bessel functions from a basic Spherical Bessel function, the normally ordered (or disentangled) form is found.展开更多
In this paper we introduce the class of Hermite's matrix polynomials which appear as finite series solutions of second order matrix differential equations Y'-xAY'+BY=0.An explicit expression for the Hermit...In this paper we introduce the class of Hermite's matrix polynomials which appear as finite series solutions of second order matrix differential equations Y'-xAY'+BY=0.An explicit expression for the Hermite matrix polynomials,the orthogonality property and a Rodrigues' formula are given.展开更多
In this paper some novel integrals associated with the product of classical Hermite's polynomials ∫-∞+∞(x2)mexp(-x2){Hr(x)}2dx,∫0∞exp(-x2)H2k(x)H2s+1(x)dx,∫0∞exp(-x2)H2k(x)H2s(x)dx and ∫0...In this paper some novel integrals associated with the product of classical Hermite's polynomials ∫-∞+∞(x2)mexp(-x2){Hr(x)}2dx,∫0∞exp(-x2)H2k(x)H2s+1(x)dx,∫0∞exp(-x2)H2k(x)H2s(x)dx and ∫0∞exp(-x2)H2k+1(x)H2s+1(x)dx, are evaluated using hypergeometric approach and Laplace transform method, which is a different approach from the approaches given by the other authors in the field of spe- cial functions. Also the results may be of significant nature, and may yield numerous other interesting integrals involving the product of classical Hermite's polynomials by suitable simplifications of arbitrary parameters.展开更多
The primary purpose of this paper is to present the Volterra integral equa- tion of the two-variable Hermite matrix polynomials. Moreover, a new representation of these matrix polynomials are established here.
In this paper. a quantitative estimate for Hermite interpolant to function ψ(z)=(z^m-β~m)~l on the ze- ros of (z^n-α~n)~r is obtained Using this estimate. a rather wide exiension of the theorem of Walsh is proved a...In this paper. a quantitative estimate for Hermite interpolant to function ψ(z)=(z^m-β~m)~l on the ze- ros of (z^n-α~n)~r is obtained Using this estimate. a rather wide exiension of the theorem of Walsh is proved and five special cases of it are given.展开更多
In this paper properties of Hermite matrix polynomials and Hermite matrix functions are studied. The concept ot total set with respect to a matrix functional is introduced and the total property of the Hermite matrix ...In this paper properties of Hermite matrix polynomials and Hermite matrix functions are studied. The concept ot total set with respect to a matrix functional is introduced and the total property of the Hermite matrix polynomials is proved. Asymptotic behaviour of Hermite matrix polynomials is studied and the relationship of Hermite matrix functions with certain matrix differential equations is developed. A new expression of the matrix exponential for a wide class of matrices in terms of Hermite matrix polynomials is proposed.展开更多
In this paper we are looking forward to finding the approximate analytical solutions for fractional integro-differential equations by using Sumudu transform method and Hermite spectral collocation method.The fractiona...In this paper we are looking forward to finding the approximate analytical solutions for fractional integro-differential equations by using Sumudu transform method and Hermite spectral collocation method.The fractional derivatives are described in the Caputo sense.The applications related to Sumudu transform method and Hermite spectral collocation method have been developed for differential equations to the extent of access to approximate analytical solutions of fractional integro-differential equations.展开更多
Generalized photon-added coherent state (GPACS) is creation and annihilation operations on the coherent state. obtained by repeatedly acting the combination of Bose It is found that GPACS can be regarded as a Hermit...Generalized photon-added coherent state (GPACS) is creation and annihilation operations on the coherent state. obtained by repeatedly acting the combination of Bose It is found that GPACS can be regarded as a Hermiteexcited coherent state due to its normalization factor related to a Hermite polynomial. In addition, we adopt the Hilbert-Schmidt distance to quantify the non-Gaussian character of GPACS and discuss the decoherence of GPACS in dissipative channel by studying the loss of nonclassicality in reference of the negativity of Wigner function.展开更多
In this study, we establish an approximate method which produces an approximate Hermite polynomial solution to a system of fractional order differential equations with variable coefficients. At collocation points, thi...In this study, we establish an approximate method which produces an approximate Hermite polynomial solution to a system of fractional order differential equations with variable coefficients. At collocation points, this method converts the mentioned system into a matrix equation which corresponds to a system of linear equations with unknown Hermite polynomial coefficients. Construction of the method on the aforementioned type of equations has been presented and tested on some numerical examples. Results related to the effectiveness and reliability of the method have been illustrated.展开更多
基金supported by the National Natural Science Foundation of China(Grant No.11175113)the Fundamental Research Funds for the Central Universities of China(Grant No.WK2060140013)
文摘By combining the operator Hermite polynomial method and the technique of integration within an ordered product of operators, for the first time we derive the generating function of even- and odd-Hermite polynomials which will be useful in constructing new optical field states. We then show that the squeezed state and photon-added squeezed state can be expressed by even- and odd-Hermite polynomials.
基金Project supported by the National Natural Science Foundation of China (Grant No 10574060) and the Natural Science Foundation of Shandong Province of China (Grant No Y2004A09).
文摘Using the coherent state representation of Wigner operator and the technique of integration within an ordered product (IWOP) of operators, this paper derives the Wigner function for the Hermite polynomial state (HPS). The tomogram of the HPS is calculated with the aid of newly introduced intermediate coordinate-momentum representation in quantum optics.
基金supported by the National Natural Science Foundation of China (Grant No. 11047133)the Natural Science Foundation of Jiangxi Province of China (Grant No. 2010GQW0027)the Research Foundation of the Education Department of Jiangxi Province of China (Grant No. GJJ11390)
文摘The nonclassicality of the two-variable Hermite polynomial state is investigated. It is found that the two-variable Hermite polynomial state can be considered as a two-mode photon subtracted squeezed vacuum state. A compact expression for the Wigner function is also derived analytically by using the Weyl-ordered operator invariance under similar transformations. Especially, the nonclassicality is discussed in terms of the negativity of the Wigner function. Then violations of Bell's inequality for the two-variable Hermite polynomial state are studied.
基金supported by the National Natural Science Foundation of China(Grant No.11175113)the Fundamental Research Funds for the Central Universities of China(Grant No.WK2060140013)the Natural Science Foundation of Jiangsu Higher Education Institution of China(Grant No.14KJD140001)
文摘By virtue of the operator-Hermite-polynomial method, we derive some new generating function formulas of the product of two bivariate Hermite polynomials. Their applications in studying quantum optical states are presented.
基金Project supported by the National Natural Science Foundation of China(Grnat No.11175113)the Fundamental Research Funds for the Central Universities of China(Grant No.WK2060140013)
文摘We derive some new generating function formulae of the two-variable Hermite polynomials, such as ∞∑n=0tm/m!Hn,2m(x),∞∑n=0sntm/n!m!H2n,2m(x,y),and ∞∑n=0sntm/n!m!H2n+l,2m+k(x,y).We employ the operator Hermite polynomial method and the technique of integration within an ordered product of operators to solve these problems, which will be useful in constructing new optical field states.
基金Supported by Hungarian National Foundation for Scientific Research,Grant No.K-100461
文摘We examine the energy function with respect to the zeros of exceptional Hermite polynomials. The localization of the eigenvalues of the Hessian is given in the general case.In some special arrangements we have a more precise result on the behavior of the energy function. Finally we investigate the energy function with respect to the regular zeros of the exceptional Hermite polynomials.
文摘In this article,the author extends the validity of a uniform asymptotic expansion of the Hermite polynomials Hn(√2n+1α)to include all positive values of α. His method makes use of the rational functions introduced by Olde Daalhuis and Temme (SIAM J.Math.Anal.,(1994),25:304-321).A new estimate for the remainder is given.
基金Project supported by the National Natural Science Foundation of China(Grant No.11175113)
文摘For directly normalizing the photon non-Gaussian states (e.g., photon added and subtracted squeezed states), we use the method of integration within an ordered product (IWOP) of operators to derive some new bosonic operator-ordering identities. We also derive some new integration transformation formulas about one- and two-variable Hermite polynomials in complex function space. These operator identities and associative integration formulas provide much convenience for constructing non-Gaussian states in quantum engineering.
基金supported by the National Natural Science Foundation of China (Grant No. 11174114)the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (Grant No. 12KJD140001)the Research Foundation of Changzhou Institute of Technology of China (Grant No. YN1106)
文摘By virtue of the entangled state representation we concisely derive some new operator identities with regard to the two-variable Hermite polynomial (TVHP). By them and the technique of integration within an ordered product (IWOP) of operators we further derive new generating function formulas of the TVHP. They are useful in quantum optical theoretical calculations. It is seen from this work that by combining the IWOP technique and quantum mechanical representations one can derive some new integration formulas even without really performing the integration.
基金Project supported by the National Natural Science Foundation of China(Grant No.11175113)
文摘Based on the operator Hermite polynomials method(OHPM), we study Stirling numbers in the context of quantum mechanics, i.e., we present operator realization of generating function formulas of Stirling numbers with some applications.As a by-product, we derive a summation formula involving both Stirling number and Hermite polynomials.
文摘The bilinear generating function for products of two Laguerre 2D polynomials with different arguments is calculated. It corresponds to the formula of Mehler for the generating function of products of two Hermite polynomials. Furthermore, the generating function for mixed products of Laguerre 2D and Hermite 2D polynomials and for products of two Hermite 2D polynomials is calculated. A set of infinite sums over products of two Laguerre 2D polynomials as intermediate step to the generating function for products of Laguerre 2D polynomials is evaluated but these sums possess also proper importance for calculations with Laguerre polynomials. With the technique of operator disentanglement some operator identities are derived in an appendix. They allow calculating convolutions of Gaussian functions combined with polynomials in one- and two-dimensional case and are applied to evaluate the discussed generating functions.
文摘In expansions of arbitrary functions in Bessel functions or Spherical Bessel functions, a dual partner set of polynomials play a role. For the Bessel functions, these are the Chebyshev polynomials of first kind and for the Spherical Bessel functions the Legendre polynomials. These two sets of functions appear in many formulas of the expansion and in the completeness and (bi)-orthogonality relations. The analogy to expansions of functions in Taylor series and in moment series and to expansions in Hermite functions is elaborated. Besides other special expansion, we find the expansion of Bessel functions in Spherical Bessel functions and their inversion and of Chebyshev polynomials of first kind in Legendre polynomials and their inversion. For the operators which generate the Spherical Bessel functions from a basic Spherical Bessel function, the normally ordered (or disentangled) form is found.
文摘In this paper we introduce the class of Hermite's matrix polynomials which appear as finite series solutions of second order matrix differential equations Y'-xAY'+BY=0.An explicit expression for the Hermite matrix polynomials,the orthogonality property and a Rodrigues' formula are given.
文摘In this paper some novel integrals associated with the product of classical Hermite's polynomials ∫-∞+∞(x2)mexp(-x2){Hr(x)}2dx,∫0∞exp(-x2)H2k(x)H2s+1(x)dx,∫0∞exp(-x2)H2k(x)H2s(x)dx and ∫0∞exp(-x2)H2k+1(x)H2s+1(x)dx, are evaluated using hypergeometric approach and Laplace transform method, which is a different approach from the approaches given by the other authors in the field of spe- cial functions. Also the results may be of significant nature, and may yield numerous other interesting integrals involving the product of classical Hermite's polynomials by suitable simplifications of arbitrary parameters.
文摘The primary purpose of this paper is to present the Volterra integral equa- tion of the two-variable Hermite matrix polynomials. Moreover, a new representation of these matrix polynomials are established here.
基金The Project is supported by National Natural Science Foundation of China.
文摘In this paper. a quantitative estimate for Hermite interpolant to function ψ(z)=(z^m-β~m)~l on the ze- ros of (z^n-α~n)~r is obtained Using this estimate. a rather wide exiension of the theorem of Walsh is proved and five special cases of it are given.
文摘In this paper properties of Hermite matrix polynomials and Hermite matrix functions are studied. The concept ot total set with respect to a matrix functional is introduced and the total property of the Hermite matrix polynomials is proved. Asymptotic behaviour of Hermite matrix polynomials is studied and the relationship of Hermite matrix functions with certain matrix differential equations is developed. A new expression of the matrix exponential for a wide class of matrices in terms of Hermite matrix polynomials is proposed.
文摘In this paper we are looking forward to finding the approximate analytical solutions for fractional integro-differential equations by using Sumudu transform method and Hermite spectral collocation method.The fractional derivatives are described in the Caputo sense.The applications related to Sumudu transform method and Hermite spectral collocation method have been developed for differential equations to the extent of access to approximate analytical solutions of fractional integro-differential equations.
基金supported by the National Natural Science Foundation of China (Grant No.11174114)the Research Foundation of Changzhou Institute of Technology,China (Grant No.YN1007)
文摘Generalized photon-added coherent state (GPACS) is creation and annihilation operations on the coherent state. obtained by repeatedly acting the combination of Bose It is found that GPACS can be regarded as a Hermiteexcited coherent state due to its normalization factor related to a Hermite polynomial. In addition, we adopt the Hilbert-Schmidt distance to quantify the non-Gaussian character of GPACS and discuss the decoherence of GPACS in dissipative channel by studying the loss of nonclassicality in reference of the negativity of Wigner function.
文摘In this study, we establish an approximate method which produces an approximate Hermite polynomial solution to a system of fractional order differential equations with variable coefficients. At collocation points, this method converts the mentioned system into a matrix equation which corresponds to a system of linear equations with unknown Hermite polynomial coefficients. Construction of the method on the aforementioned type of equations has been presented and tested on some numerical examples. Results related to the effectiveness and reliability of the method have been illustrated.