An explicitly solvable model for tunnelling of relativistic spinless particles through a sphere is suggested. The model operator is constructed by an operator extensions theory method from the orthogonal sum of the Di...An explicitly solvable model for tunnelling of relativistic spinless particles through a sphere is suggested. The model operator is constructed by an operator extensions theory method from the orthogonal sum of the Dirac operators on a semi- axis and on the sphere. The transmission coefficient is obtained. The dependence of the transmission coefficient on the particle energy has a resonant character. One observes pairs of the Breit-Wigner and the Fano resonances. It correlates with the corresponding results for a non-relativistic particle.展开更多
In this paper we research the lower bound of the eigenvalue of Spinc Dirac operator on the Spinc manifold. By the Weisenbock formula, we get an estimate of it, then following the idea of Th Friedrich [2] and X Zhang [...In this paper we research the lower bound of the eigenvalue of Spinc Dirac operator on the Spinc manifold. By the Weisenbock formula, we get an estimate of it, then following the idea of Th Friedrich [2] and X Zhang [6]. We get a finer estimate of it. As an application, we give a condition when the Seiberg-Witten equation only has 0 solution.展开更多
The classical Ambarzumyan’s theorem states that if the Neumann eigenvalues of the Sturm-Liouville operator-d^(2)/dx^(2)+q with an integrable real-valued potential q on[0,π] are {n^(2):n≥0},then q=0 for almost all x...The classical Ambarzumyan’s theorem states that if the Neumann eigenvalues of the Sturm-Liouville operator-d^(2)/dx^(2)+q with an integrable real-valued potential q on[0,π] are {n^(2):n≥0},then q=0 for almost all x∈[0,π].In this work,the classical Ambarzumyan’s theorem is extended to the Dirac operator on equilateral tree graphs.We prove that if the spectrum of the Dirac operator on graphs coincides with the unperturbed case,then the potential is identically zero.展开更多
Following the previous work,we shall study some inverse problems for the Dirac operator on an equilateral star graph.It is proven that the so-called Weyl function uniquely determines the potentials.Furthermore,we pay ...Following the previous work,we shall study some inverse problems for the Dirac operator on an equilateral star graph.It is proven that the so-called Weyl function uniquely determines the potentials.Furthermore,we pay attention to the inverse problem of recovering the potentials from the spectral data,which consists of the eigenvalues and weight matrices,and present a constructive algorithm.The basic tool in this paper is the method of spectral mappings developed by Yurko.展开更多
The boundary value problem plays a crucial role in the analytical investigation of continuum dynamics. In this paper, an analytical method based on the Dirac operator to solve the nonlinear and non-homogeneous boundar...The boundary value problem plays a crucial role in the analytical investigation of continuum dynamics. In this paper, an analytical method based on the Dirac operator to solve the nonlinear and non-homogeneous boundary value problem of rectangular plates is proposed. The key concept behind this method is to transform the nonlinear or non-homogeneous part on the boundary into a lateral force within the governing function by the Dirac operator, which linearizes and homogenizes the original boundary, allowing one to employ the modal superposition method for obtaining solutions to reconstructive governing equations. Once projected into the modal space, the harmonic balance method(HBM) is utilized to solve coupled ordinary differential equations(ODEs)of truncated systems with nonlinearity. To validate the convergence and accuracy of the proposed Dirac method, the results of typical examples, involving nonlinearly restricted boundaries, moment excitation, and displacement excitation, are compared with those of the differential quadrature element method(DQEM). The results demonstrate that when dealing with nonlinear boundaries, the Dirac method exhibits more excellent accuracy and convergence compared with the DQEM. However, when facing displacement excitation, there exist some discrepancies between the proposed approach and simulations;nevertheless, the proposed method still accurately predicts resonant frequencies while being uniquely capable of handling nonuniform displacement excitations. Overall, this methodology offers a convenient way for addressing nonlinear and non-homogenous plate boundaries.展开更多
For a compact complex spin manifold M with a holomorphic isometric embed- ding into the complex projective space,the authors obtain the extrinsic estimates from above and below for eigenvalues of the Dirac operator,wh...For a compact complex spin manifold M with a holomorphic isometric embed- ding into the complex projective space,the authors obtain the extrinsic estimates from above and below for eigenvalues of the Dirac operator,which depend on the data of an isometric embedding of M.Further,from the inequalities of eigenvalues,the gaps of the eigenvalues and the ratio of the eigenvalues are obtained.展开更多
Assume thatτis a finite dimensional complex Lie superalgebra with a non-degenerate super-symmetric invariant bilinear form,p is a finite dimensional complex super vector space with a nondegenerate super-symmetric bil...Assume thatτis a finite dimensional complex Lie superalgebra with a non-degenerate super-symmetric invariant bilinear form,p is a finite dimensional complex super vector space with a nondegenerate super-symmetric bilinear form,and v:τ→osp(p)is a homomorphism of Lie superalgebras.In this paper,we give a necessary and sufficient condition forτ■p to be a quadratic Lie superalgebra.Then,we define the cubic Dirac operator D(g,τ)on g and give a formula of(D(g,τ))^(2).Finally,we get the Vogan’s conjecture for quadratic Lie superalgebras by D(g,τ).展开更多
We consider the inverse eigenvalue problems for stationary Dirac systems with differentiable selfadjoint matrix potential.The theorem of Ambarzumyan for a Sturm-Liouville problem is extended to Dirac operators,which a...We consider the inverse eigenvalue problems for stationary Dirac systems with differentiable selfadjoint matrix potential.The theorem of Ambarzumyan for a Sturm-Liouville problem is extended to Dirac operators,which are subject to separation boundary conditions or periodic(semi-periodic)boundary conditions,and some analogs of Ambarzumyan's theorem are obtained.The proof is based on the existence and extremal properties of the smallest eigenvalue of corresponding vectorial Sturm-Liouville operators,which are the second power of Dirac operators.展开更多
In this paper, we consider the stochastic Dirac operatoron a polish space (Ω,β, P). The relation between the Lyapunov index, rotation number andthe spectrum of Lis discussed. The existence of the Lyapunov index and ...In this paper, we consider the stochastic Dirac operatoron a polish space (Ω,β, P). The relation between the Lyapunov index, rotation number andthe spectrum of Lis discussed. The existence of the Lyapunov index and rotation number isshown. By using the W-T functions and W-function we prove the theorems for Las in Kotani[1], [2] for Schrodinger operatorB, and in Johnson [5] for Dirac operators on compact space.展开更多
In the present paper, the semi-commutative differential oparators associated with the 1-dimensional Dirac operator are constructed. Using this results, the hierarchy of the mKdV (-) polynomials are expressed in terms ...In the present paper, the semi-commutative differential oparators associated with the 1-dimensional Dirac operator are constructed. Using this results, the hierarchy of the mKdV (-) polynomials are expressed in terms of the KdV polynomials. These formulas give a new interpretation of the classical Darboux transformation and the Miura transformation. Moreover, the recursion operator associated with the hierarchy of the mKdV (-) polynomials is constructed by the algebraic method.展开更多
We review the themes relating to the proposition that“quantization commutes with reduction”([Q,R]=0),from symplectic manifolds to Cauchy-Riemann manifolds.
In this paper, applying the method of , we give the complete description of self adjointness of singular Dirac operators, their deficiency indices are supposed to be (2.2) and (1.1), respectively.
For a compact spin Riemannian manifold(M,g^(TM))of dimension n such that the associated scalar curvature kT M verifies that k^(TM)≥n(n-1),Llarull’s rigidity theorem says that any area-decreasing smooth map f from M ...For a compact spin Riemannian manifold(M,g^(TM))of dimension n such that the associated scalar curvature kT M verifies that k^(TM)≥n(n-1),Llarull’s rigidity theorem says that any area-decreasing smooth map f from M to the unit sphere Sn of nonzero degree is an isometry.In this paper,we present a new proof of Llarull’s rigidity theorem in odd dimensions via a spectral flow argument.This approach also works for a generalization of Llarull’s theorem when the sphere Sn is replaced by an arbitrary smooth strictly convex closed hypersurface in Rn+1.The results answer two questions by Gromov(2023).展开更多
In[T.Wu et al.,arXiv2310.09775,2023],a general Dabrowski-Sitarz-Zalecki type theorem for odd dimensional manifolds with boundary was proved.In this paper,we give the proof of the another general Dabrowski-Sitarz-Zalec...In[T.Wu et al.,arXiv2310.09775,2023],a general Dabrowski-Sitarz-Zalecki type theorem for odd dimensional manifolds with boundary was proved.In this paper,we give the proof of the another general Dabrowski-Sitarz-Zalecki type theorem for the spectral Einstein functional associated with the Dirac operator on odd dimensional manifolds with boundary.展开更多
Euclidean Clifford analysis is a higher dimensional function theory centred around monogenic functions, i.e., null solutions to a first order vector valued rotation in- variant differential operator called the Dirac ...Euclidean Clifford analysis is a higher dimensional function theory centred around monogenic functions, i.e., null solutions to a first order vector valued rotation in- variant differential operator called the Dirac operator. More recently, Hermitian Clifford analysis has emerged as a new branch, offering yet a refinement of the Euclidean case; it focuses on the simultaneous null solutions, called Hermitian monogenic functions, to two Hermitian Dirac operators and which are invariant under the action of the unitary group. In Euclidean Clifford analysis, the Teodorescu operator is the right inverse of the Dirac operator __0. In this paper, Teodorescu operators for the Hermitian Dirac operators c9~_ and 0_~, are constructed. Moreover, the structure of the Euclidean and Hermitian Teodor- escu operators is revealed by analyzing the more subtle behaviour of their components. Finally, the obtained inversion relations are still refined for the differential operators is- suing from the Euclidean and Hermitian Dirac operators by splitting the Clifford algebra product into its dot and wedge parts. Their relationship with several complex variables theory is discussed.展开更多
The endoscopic transfer factor is expressed as difference of characters for the even and odd parts of the spin modules,or Dirac index of the trivial representation.The lifting of tempered characters in terms of index ...The endoscopic transfer factor is expressed as difference of characters for the even and odd parts of the spin modules,or Dirac index of the trivial representation.The lifting of tempered characters in terms of index of Dirac cohomology is calculated explicitly.展开更多
Let L be the sublaplacian on the quaternion Heisenberg group N and T the Dirac type operator with respect to central variables of N. In this article, we characterize the He-valued joint eigenfunctions of L and T havin...Let L be the sublaplacian on the quaternion Heisenberg group N and T the Dirac type operator with respect to central variables of N. In this article, we characterize the He-valued joint eigenfunctions of L and T having eigenvalues from the quaternionic Heisenberg fan.展开更多
In this paper we study important quantities defined from solutions of first order linear systems of ordinary differential equations. It will be proved that many quantities, such as solutions, eigenvalues of one-dimens...In this paper we study important quantities defined from solutions of first order linear systems of ordinary differential equations. It will be proved that many quantities, such as solutions, eigenvalues of one-dimensional Dirac operators, Lyapunov exponents and rotation numbers, depend on the coefficients in a very strong way. That is, they are not only continuous in coefficients with respect to the usual L^p topologies, but also with respect to the weak topologies of the Lp spaces. The continuity results of this paper are a basis to study these quantities in a quantitative way.展开更多
Associated with the Dirac operator and partial derivatives,this paper establishes some real PaleyWiener type theorems to characterize the Clifford-valued functions whose Clifford Fourier transform(CFT) has compact sup...Associated with the Dirac operator and partial derivatives,this paper establishes some real PaleyWiener type theorems to characterize the Clifford-valued functions whose Clifford Fourier transform(CFT) has compact support. Based on the Riemann-Lebesgue theorem for the CFT,the Boas theorem is provided to describe the CFT of Clifford-valued functions that vanish on a neighborhood of the origin.展开更多
The author studies the properties and applications of quasi-Killing spinors and quasi-twistor spinors and obtains some vanishing theorems. In particular, the author classifies all the types of quasi-twistor spinors on...The author studies the properties and applications of quasi-Killing spinors and quasi-twistor spinors and obtains some vanishing theorems. In particular, the author classifies all the types of quasi-twistor spinors on closed Riemannian spin manifolds. As a consequence, it is known that on a locally decomposable closed spin manifold with nonzero Ricci curvature, the space of twistor spinors is trivial. Some integrability condition for twistor spinors is also obtained.展开更多
基金Project partially financially supported by the Funds from the Government of the Russian Federation(Grant No.074-U01)the Funds from the Ministry of Education and Science of the Russian Federation(GOSZADANIE 2014/190)(Grant Nos.14.Z50.31.0031 and 1.754.2014/K)the President Foundation of the Russian Federation(Grant No.MK-5001.2015.1)
文摘An explicitly solvable model for tunnelling of relativistic spinless particles through a sphere is suggested. The model operator is constructed by an operator extensions theory method from the orthogonal sum of the Dirac operators on a semi- axis and on the sphere. The transmission coefficient is obtained. The dependence of the transmission coefficient on the particle energy has a resonant character. One observes pairs of the Breit-Wigner and the Fano resonances. It correlates with the corresponding results for a non-relativistic particle.
基金Supported in part by Mathematics Tianyuan Fund(10226002)
文摘In this paper we research the lower bound of the eigenvalue of Spinc Dirac operator on the Spinc manifold. By the Weisenbock formula, we get an estimate of it, then following the idea of Th Friedrich [2] and X Zhang [6]. We get a finer estimate of it. As an application, we give a condition when the Seiberg-Witten equation only has 0 solution.
基金supported by the National Natural Science Foundation of China(No.11871031)the Natural Science Foundation of the Jiangsu Province of China(No.BK 20201303).
文摘The classical Ambarzumyan’s theorem states that if the Neumann eigenvalues of the Sturm-Liouville operator-d^(2)/dx^(2)+q with an integrable real-valued potential q on[0,π] are {n^(2):n≥0},then q=0 for almost all x∈[0,π].In this work,the classical Ambarzumyan’s theorem is extended to the Dirac operator on equilateral tree graphs.We prove that if the spectrum of the Dirac operator on graphs coincides with the unperturbed case,then the potential is identically zero.
基金Supported by NSFC(Grant No.11871031)the Natural Science Foundation of the Jiangsu Province of China(Grant No.BK20201303)。
文摘Following the previous work,we shall study some inverse problems for the Dirac operator on an equilateral star graph.It is proven that the so-called Weyl function uniquely determines the potentials.Furthermore,we pay attention to the inverse problem of recovering the potentials from the spectral data,which consists of the eigenvalues and weight matrices,and present a constructive algorithm.The basic tool in this paper is the method of spectral mappings developed by Yurko.
基金Project supported by the National Natural Science Foundation of China (No. 12002195)the National Science Fund for Distinguished Young Scholars (No. 12025204)the Program of Shanghai Municipal Education Commission (No. 2019-01-07-00-09-E00018)。
文摘The boundary value problem plays a crucial role in the analytical investigation of continuum dynamics. In this paper, an analytical method based on the Dirac operator to solve the nonlinear and non-homogeneous boundary value problem of rectangular plates is proposed. The key concept behind this method is to transform the nonlinear or non-homogeneous part on the boundary into a lateral force within the governing function by the Dirac operator, which linearizes and homogenizes the original boundary, allowing one to employ the modal superposition method for obtaining solutions to reconstructive governing equations. Once projected into the modal space, the harmonic balance method(HBM) is utilized to solve coupled ordinary differential equations(ODEs)of truncated systems with nonlinearity. To validate the convergence and accuracy of the proposed Dirac method, the results of typical examples, involving nonlinearly restricted boundaries, moment excitation, and displacement excitation, are compared with those of the differential quadrature element method(DQEM). The results demonstrate that when dealing with nonlinear boundaries, the Dirac method exhibits more excellent accuracy and convergence compared with the DQEM. However, when facing displacement excitation, there exist some discrepancies between the proposed approach and simulations;nevertheless, the proposed method still accurately predicts resonant frequencies while being uniquely capable of handling nonuniform displacement excitations. Overall, this methodology offers a convenient way for addressing nonlinear and non-homogenous plate boundaries.
基金the Science Research Development Fund of Nanjing University of Science and Technology(No.AB96228).
文摘For a compact complex spin manifold M with a holomorphic isometric embed- ding into the complex projective space,the authors obtain the extrinsic estimates from above and below for eigenvalues of the Dirac operator,which depend on the data of an isometric embedding of M.Further,from the inequalities of eigenvalues,the gaps of the eigenvalues and the ratio of the eigenvalues are obtained.
基金Supported by National Natural Science Foundation of China(Grant Nos.11571182 and 11931009)the Talents Foundation of Central South University of Forestry and Technology(Grant No.104-0089)Natural Science Foundation of Tianjin(Grant No.19JCYBJC30600)。
文摘Assume thatτis a finite dimensional complex Lie superalgebra with a non-degenerate super-symmetric invariant bilinear form,p is a finite dimensional complex super vector space with a nondegenerate super-symmetric bilinear form,and v:τ→osp(p)is a homomorphism of Lie superalgebras.In this paper,we give a necessary and sufficient condition forτ■p to be a quadratic Lie superalgebra.Then,we define the cubic Dirac operator D(g,τ)on g and give a formula of(D(g,τ))^(2).Finally,we get the Vogan’s conjecture for quadratic Lie superalgebras by D(g,τ).
基金supported in part by the National Natural Science Foundation of China(11871031)by the Natural Science Foundation of the Jiangsu Province of China(BK 20201303)。
文摘We consider the inverse eigenvalue problems for stationary Dirac systems with differentiable selfadjoint matrix potential.The theorem of Ambarzumyan for a Sturm-Liouville problem is extended to Dirac operators,which are subject to separation boundary conditions or periodic(semi-periodic)boundary conditions,and some analogs of Ambarzumyan's theorem are obtained.The proof is based on the existence and extremal properties of the smallest eigenvalue of corresponding vectorial Sturm-Liouville operators,which are the second power of Dirac operators.
文摘In this paper, we consider the stochastic Dirac operatoron a polish space (Ω,β, P). The relation between the Lyapunov index, rotation number andthe spectrum of Lis discussed. The existence of the Lyapunov index and rotation number isshown. By using the W-T functions and W-function we prove the theorems for Las in Kotani[1], [2] for Schrodinger operatorB, and in Johnson [5] for Dirac operators on compact space.
基金supported by JSPS KAKENHI Grant Number 2354-0255.
文摘In the present paper, the semi-commutative differential oparators associated with the 1-dimensional Dirac operator are constructed. Using this results, the hierarchy of the mKdV (-) polynomials are expressed in terms of the KdV polynomials. These formulas give a new interpretation of the classical Darboux transformation and the Miura transformation. Moreover, the recursion operator associated with the hierarchy of the mKdV (-) polynomials is constructed by the algebraic method.
文摘We review the themes relating to the proposition that“quantization commutes with reduction”([Q,R]=0),from symplectic manifolds to Cauchy-Riemann manifolds.
文摘In this paper, applying the method of , we give the complete description of self adjointness of singular Dirac operators, their deficiency indices are supposed to be (2.2) and (1.1), respectively.
基金supported by Nankai Zhide Foundationsupported by National Natural Science Foundation of China(Grant Nos.12271266 and 11931007)+4 种基金supported by National Natural Science Foundation of China(Grant No.12101361)Nankai Zhide Foundationthe Fundamental Research Funds for the Central Universities(Grant No.100-63233103)the Project of Young Scholars of Shandong Universitythe Fundamental Research Funds of Shandong University(Grant No.2020GN063)。
文摘For a compact spin Riemannian manifold(M,g^(TM))of dimension n such that the associated scalar curvature kT M verifies that k^(TM)≥n(n-1),Llarull’s rigidity theorem says that any area-decreasing smooth map f from M to the unit sphere Sn of nonzero degree is an isometry.In this paper,we present a new proof of Llarull’s rigidity theorem in odd dimensions via a spectral flow argument.This approach also works for a generalization of Llarull’s theorem when the sphere Sn is replaced by an arbitrary smooth strictly convex closed hypersurface in Rn+1.The results answer two questions by Gromov(2023).
基金supported by the National Natural Science Foundation of China (Grant No.11771070).
文摘In[T.Wu et al.,arXiv2310.09775,2023],a general Dabrowski-Sitarz-Zalecki type theorem for odd dimensional manifolds with boundary was proved.In this paper,we give the proof of the another general Dabrowski-Sitarz-Zalecki type theorem for the spectral Einstein functional associated with the Dirac operator on odd dimensional manifolds with boundary.
文摘Euclidean Clifford analysis is a higher dimensional function theory centred around monogenic functions, i.e., null solutions to a first order vector valued rotation in- variant differential operator called the Dirac operator. More recently, Hermitian Clifford analysis has emerged as a new branch, offering yet a refinement of the Euclidean case; it focuses on the simultaneous null solutions, called Hermitian monogenic functions, to two Hermitian Dirac operators and which are invariant under the action of the unitary group. In Euclidean Clifford analysis, the Teodorescu operator is the right inverse of the Dirac operator __0. In this paper, Teodorescu operators for the Hermitian Dirac operators c9~_ and 0_~, are constructed. Moreover, the structure of the Euclidean and Hermitian Teodor- escu operators is revealed by analyzing the more subtle behaviour of their components. Finally, the obtained inversion relations are still refined for the differential operators is- suing from the Euclidean and Hermitian Dirac operators by splitting the Clifford algebra product into its dot and wedge parts. Their relationship with several complex variables theory is discussed.
基金Supported by grants(Grant No.16303218)from Research Grant Council of HKSAR。
文摘The endoscopic transfer factor is expressed as difference of characters for the even and odd parts of the spin modules,or Dirac index of the trivial representation.The lifting of tempered characters in terms of index of Dirac cohomology is calculated explicitly.
基金supported by National Natural Science Foundation of China(Grant Nos.10871003 and 10990012)
文摘Let L be the sublaplacian on the quaternion Heisenberg group N and T the Dirac type operator with respect to central variables of N. In this article, we characterize the He-valued joint eigenfunctions of L and T having eigenvalues from the quaternionic Heisenberg fan.
基金supported by National Basic Research Program of China (Grant No. 2006CB805903)National Natural Science Foundation of China (Grant Nos. 10325102 and 10531010)
文摘In this paper we study important quantities defined from solutions of first order linear systems of ordinary differential equations. It will be proved that many quantities, such as solutions, eigenvalues of one-dimensional Dirac operators, Lyapunov exponents and rotation numbers, depend on the coefficients in a very strong way. That is, they are not only continuous in coefficients with respect to the usual L^p topologies, but also with respect to the weak topologies of the Lp spaces. The continuity results of this paper are a basis to study these quantities in a quantitative way.
基金supported by National Natural Science Foundation of China(Grant No.11371007)
文摘Associated with the Dirac operator and partial derivatives,this paper establishes some real PaleyWiener type theorems to characterize the Clifford-valued functions whose Clifford Fourier transform(CFT) has compact support. Based on the Riemann-Lebesgue theorem for the CFT,the Boas theorem is provided to describe the CFT of Clifford-valued functions that vanish on a neighborhood of the origin.
基金supported by the National Natural Science Foundation of China(Nos.11301202,11571131)
文摘The author studies the properties and applications of quasi-Killing spinors and quasi-twistor spinors and obtains some vanishing theorems. In particular, the author classifies all the types of quasi-twistor spinors on closed Riemannian spin manifolds. As a consequence, it is known that on a locally decomposable closed spin manifold with nonzero Ricci curvature, the space of twistor spinors is trivial. Some integrability condition for twistor spinors is also obtained.
