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Abstract |
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Verified numerical computations for eigenvalues of non-commutative harmonic oscillatorsK. NagatouIn this topic, we study the eigenvalues of the self-adjoint operator \[ Q_{(\alpha, \beta )}\equiv I_{(\alpha, \beta )}(-\frac{\partial_x^2}{2}+\frac{x^2}{2}) +J(x\partial_x +\frac{1}{2}),~x\in {\rm\bf R}, \] where \[ I_{(\alpha, \beta )}\equiv \pmatrix{\alpha & 0 \cr 0 & \beta },~ J\equiv \pmatrix{0 & -1 \cr 1 & 0 } \in {\rm Mat}_2({\rm\bf R}), \] and $\alpha$ and $\beta $ are positive real values satisfying $\alpha \beta >1.$ This operator is concerned with the Hamiltonian of one dimensional harmonic oscillator \[ H=-\frac{\partial^2}{\partial x^2}+x^2, \] but its spectrum is unknown except for the case that $\alpha =\beta $. We have already proposed a numerical enclosure method for elliptic eigenvalue problems which is based on Nakao's verification method for nonlinear elliptic equations. In the present case, using the complete orthonormal system in $L^2({\rm\bf R})$ consisting of the eigenfunctions of one dimensional harmonic oscillators, we will apply the method to the coupling type eigenvalue problems in the unbounded domain. We numerically construct a set containing {\it eigenpairs}which satisfies the hypothesis of Banach's fixed point theorem in a certain Sobolev space by using an approximation and constructive error estimates. We then prove the local uniqueness {\it separately} of eigenvalues and eigenfunctions. This local uniqueness assures the simplicity of the eigenvalue. |
General EnquiriesDr. Malcolm Brown. |
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Page last updated
22/04/99 |