The properties of the 8-wave for a quasl-free partide with position-dependent mass （PDM） have been discussed in details. Differed from the system with constant mass in which the localization of the s-wave for the fr...The properties of the 8-wave for a quasl-free partide with position-dependent mass （PDM） have been discussed in details. Differed from the system with constant mass in which the localization of the s-wave for the free quantum particle around the origin only occurs in two dimensions, the quasi-free particle with PDM can experience attractive forces in D dimensions except D = 1 when its mass function satisfies some conditions. The effective mass of a particle varying with its position can induce effective interaction, which may be attractive in some cases. The analytical expressions of the eigenfunctions and the corresponding probability densities for the 8-waves of the two- and three-dimensional systems with a special PDM are given, and the existences of localization around the origin for these systems are shown.展开更多
By means of a method of analytic number theory the following theorem is proved. Letp be a quasi-homogeneous linear partial differential operator with degreem,m > 0, w.r.t a dilation $\left\{ {\delta _\tau } \right\...By means of a method of analytic number theory the following theorem is proved. Letp be a quasi-homogeneous linear partial differential operator with degreem,m > 0, w.r.t a dilation $\left\{ {\delta _\tau } \right\}{\text{ }}_{\tau< 0} $ given by ( a1, …, an). Assume that either a1, …, an are positive rational numbers or $m{\text{ = }}\sum\limits_{j = 1}^n {\alpha _j \alpha _j } $ for some $\alpha {\text{ = }}\left( {\alpha _1 ,{\text{ }} \ldots {\text{ }},\alpha _n } \right) \in l _ + ^n $ Then the dimension of the space of polynomial solutions of the equationp[u] = 0 on ?n must be infinite展开更多
基金The project supported by National Natural Science Foundation of China for Distinguished Young Scientists under Grant No. 10125521, the Doctoral Fund of Ministry of Education of China under Grant No. 20010284036, the State Key Basic Research Development Program under Grant No. G2000077400, the Knowledge Innovation Project of the Chinese Academy of Sciences under Grant No. KJCX2-SW-N02, and National Natural Science Foundation of China under Grant No. 60371013
文摘The properties of the 8-wave for a quasl-free partide with position-dependent mass （PDM） have been discussed in details. Differed from the system with constant mass in which the localization of the s-wave for the free quantum particle around the origin only occurs in two dimensions, the quasi-free particle with PDM can experience attractive forces in D dimensions except D = 1 when its mass function satisfies some conditions. The effective mass of a particle varying with its position can induce effective interaction, which may be attractive in some cases. The analytical expressions of the eigenfunctions and the corresponding probability densities for the 8-waves of the two- and three-dimensional systems with a special PDM are given, and the existences of localization around the origin for these systems are shown.
基金the National Natural Science Foundation of China (Grnat No. 19971068) .
文摘By means of a method of analytic number theory the following theorem is proved. Letp be a quasi-homogeneous linear partial differential operator with degreem,m > 0, w.r.t a dilation $\left\{ {\delta _\tau } \right\}{\text{ }}_{\tau< 0} $ given by ( a1, …, an). Assume that either a1, …, an are positive rational numbers or $m{\text{ = }}\sum\limits_{j = 1}^n {\alpha _j \alpha _j } $ for some $\alpha {\text{ = }}\left( {\alpha _1 ,{\text{ }} \ldots {\text{ }},\alpha _n } \right) \in l _ + ^n $ Then the dimension of the space of polynomial solutions of the equationp[u] = 0 on ?n must be infinite
