Completeness in the sense of Cauchy principal value of the eigenfunction systems of infinite dimensional Hamiltonian operator

Alatancang,; Wu, Deyu

doi:10.1007/s11425-008-0110-3

Cited by 18 publications

(12 citation statements)

References 2 publications

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“…Thus theorem 3.2 in Ref. [10] is the special case of Theorem 2 in this paper. It is worth mentioning that the assumption (iv) of theorem 3.2 in Ref.…”

Section: Proofmentioning

confidence: 61%

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Completeness of Eigenfunction Systems for Off-Diagonal Infinite-Dimensional Hamiltonian Operators

Hou¹,

Alatancang²

2010

Commun. Theor. Phys.

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For the off-diagonal infinite dimensional Hamiltonian operators, which have at most countable eigenvalues, a necessary and sufficient condition of the eigenfunction systems to be complete in the sense of Cauchy principal value is presented by using the spectral symmetry and new orthogonal relationship of the operators. Moreover, the above result is extended to a more general case. At last, the completeness of eigenfunction systems for the operators arising from the isotropic plane magnetoelectroelastic solids is described to illustrate the effectiveness of the criterion. The whole results offer theoretical guarantee for separation of variables in Hamiltonian system for some mechanics equations.

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“…Thus theorem 3.2 in Ref. [10] is the special case of Theorem 2 in this paper. It is worth mentioning that the assumption (iv) of theorem 3.2 in Ref.…”

Section: Proofmentioning

confidence: 61%

“…In Ref. [10], the authors presented a preferable conclusion for the completeness of the eigenfunctions system in the sense of Cauchy principal value for some class of infinitedimensional Hamiltonian operators. However, the prior conditions are strict therein, so they cannot be applied to mechanics problems currently.…”

Section: Introductionmentioning

confidence: 99%

Completeness of Eigenfunction Systems for Off-Diagonal Infinite-Dimensional Hamiltonian Operators

Hou¹,

Alatancang²

2010

Commun. Theor. Phys.

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“…The essence of the symplectic elasticity method is the separation method of variables based on a Hamiltonian system, which needs the investigation of the basis property of the eigenfunction systems of Hamiltonian operator matrices. [8][9][10][11][12][13] Applying the analogy theory of computational structural mechanics and optimal control [14], many elasticity equations can be written as the following separable linear Hamiltonian system: [15][16][17] v = H 𝑣 + 𝑓 ,…”

Section: Introductionmentioning

confidence: 99%

Block basis property of a class of 2 × 2 operator matrices and its application to elasticity

Kuan¹,

Hou²,

Alatancang³

2013

Chinese Phys. B

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A necessary and sufficient condition is obtained for the generalized eigenfunction systems of 2 × 2 operator matrices to be a block Schauder basis of some Hilbert space, which offers a mathematical foundation of solving symplectic elasticity problems by using the method of separation of variables. Moreover, the theoretical result is applied to two plane elasticity problems via the separable Hamiltonian systems.

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“…This method extends the traditional separation of variables. However, the mathematical foundation, i.e., the symplectic eigenfunction expansion theorem, is partly solved [9][10][11] , and there are still many unsolved problems. For example, the completeness of the eigenfunction system of the infinite-dimensional Hamiltonian operator appearing in the two-dimensional (2D) elasticity problems based on the stress formulation has not been proved.…”

Section: Introductionmentioning

confidence: 99%

Eigenfunction expansion method of upper triangular operator matrix and application to two-dimensional elasticity problems based on stress formulation

Eburilitu

Alatancang

2012

Appl. Math. Mech.-Engl. Ed.

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This paper studies the eigenfunction expansion method to solve the twodimensional (2D) elasticity problems based on the stress formulation. The fundamental system of partial differential equations of the 2D problems is rewritten as an upper triangular differential system based on the known results, and then the associated upper triangular operator matrix matrix is obtained. By further research, the two simpler complete orthogonal systems of eigenfunctions in some space are obtained, which belong to the two block operators arising in the operator matrix. Then, a more simple and convenient general solution to the 2D problem is given by the eigenfunction expansion method. Furthermore, the boundary conditions for the 2D problem, which can be solved by this method, are indicated. Finally, the validity of the obtained results is verified by a specific example. 224EBURILITU and ALATANCANG method of separation of variables based on the Hamiltonian system. Moreover, given a mechanics equation, in general, we can look for the various differential systems [12] . The Hamiltonian system is only one class of them. In all differential systems, we believe that the upper triangular differential systems have certain advantages in solving which problems. Now, many researchers are studying the upper triangular operator matrices, which has become a research hot point of the operator theory.In Ref.[13], the fundamental system of partial differential equations of the 2D problems is rewritten as an upper triangular differential system. For the associated operator matrix, the existence of the two the complete orthogonal systems of eigenfunctions in some space are obtained, which belong to the two block operators arising in the operator matrix. The general solution to the 2D problem is given by the eigenfunction expansion method. However, the analytic expression formula of the general solution is very tedious and complicate. Therefore, it is inconvenient to be applied.Through an in-depth study of the 2D problem, in a new coordinate system, this paper presents a more simple and practical general solution. Furthermore, the boundary conditions for the 2D problem, which can be solved by the method, are indicated.

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Completeness in the sense of Cauchy principal value of the eigenfunction systems of infinite dimensional Hamiltonian operator

Cited by 18 publications

References 2 publications

Completeness of Eigenfunction Systems for Off-Diagonal Infinite-Dimensional Hamiltonian Operators

Completeness of Eigenfunction Systems for Off-Diagonal Infinite-Dimensional Hamiltonian Operators

Block basis property of a class of 2 × 2 operator matrices and its application to elasticity

Eigenfunction expansion method of upper triangular operator matrix and application to two-dimensional elasticity problems based on stress formulation

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