具两奇异端点的J-对称微分算子的J-自伴域
|
摘要
研究了J-对称微分算式τ(y)在两端奇异且亏指数不相等时J-自伴扩张的边条件问题.利用J-对称微分算式生成的最大算子域的构造定理,得到了在(-∞,∞)上J-对称微分算子的J-自伴域边条件的解析描述,并给出了几种特殊亏指数的J-自伴域的完全描述,进一步完善了J-自伴域的边条件理论.
The boundary condition of the J-selfadjoint extensions domain of J-symmetric differential expression is investigated when the two ends are singular and the deficiency indices are not equal.Using the construction theorem on the domain of maximal operator generated by J-symmetric differential expression,the analytic expression of the boundary condition of J-selfadjoint extensions of J-symmetric differential operators in(-∞,∞)is obtained,and the complete characterization of J-selfadjoint extensions domain of several special deficiency indices is given,the boundary condition theory of J-selfadjoint extensions domain is further improved.
The boundary condition of the J-selfadjoint extensions domain of J-symmetric differential expression is investigated when the two ends are singular and the deficiency indices are not equal.Using the construction theorem on the domain of maximal operator generated by J-symmetric differential expression,the analytic expression of the boundary condition of J-selfadjoint extensions of J-symmetric differential operators in(-∞,∞)is obtained,and the complete characterization of J-selfadjoint extensions domain of several special deficiency indices is given,the boundary condition theory of J-selfadjoint extensions domain is further improved.
引文
[1]张志敏,许美珍.两区间四阶对称微分算子自伴扩张域的描述[J].应用数学进展,2017,6(1):78-89.Zhang Zhimin,Xu Meizhen.The-selfadjoint realizations of two-interval forth-order-symmetric operators[J].Advances in Applied Mathematics,2017,6(1):78-89.(in Chinese)
[2]尚在久,李文明.关于对称微分算子的若干问题[J].内蒙古大学学报(自然科学版),1991,22(3):285-294.Shang Zaijiu,Li Wenming.On some problems ofsymmetric differential operators[J].Acta Scientiarum Naturalium Universitatis Intramongolicae,1991,22(3):285-294.(in Chinese)
[3]Glazman I M.An analogue of the extensions of J-symmetric operators[J].Journal of Differential Equations,1957,57:258-274.
[4]Race D.The theory of selfadjoint extensions of symmetric operators[J].Journal of Differential Equations,1985,57:258-274.
[5]Cao Z.On self-adjoint extensions of n-th order differential operators in the limit circle case[J].Acta Mathematica Sinica,1985,28:205-217.
[6]Sun J.On the self-adjoint extensions of symmetric ordinary differential operators with middle deficiency indices[J]. Acta Mathematica Sinica, 1986, 2:152-167.
[7]王文娟,王忠.具有周期复系数的2n阶微分算子的谱[J].应用数学学报,2015,38(4):699-707.Wang Wenjuan,Wang Zhong.The spectrum of differential operators of 2n-th order with periodic complex coefficients[J].Acta Mathematics Applicatae Sinica,2015,38(4):699-707.(in Chinese)
[8]钱志祥.2n阶J-自伴向量微分算子的本质谱.数学的实践与认识[J].2016,4(17):261-268.Qian Zhixiang.The essential spectrum of 2n-th order J-selfadjoint vector differential operator[J].Mathematics in Practice and Theory,2016,4(17):261-268.(in Chinese)
[9]林秋红.具有可积系数的高阶J-自伴微分算子的离散谱的条件[J].数学的实践与认识,2018,48(10):247-254.Lin Qiuhong.Conditions for the discrete spectrum of high-order J-selfadjoint differential operators with integral coefficients[J].Mathematics in Practice and Theory,2018,48(10):247-254.(in Chinese)
[10]林秋红.具有对数函数系数的J-自伴微分算子谱是离散的充分条件[J].数学的实践与认识,2018,48(3):225-232.Lin Qiuhong.Sufficient conditions for the discrete spectrum of J-selfadjoint differential operators of coefficients contains logarithmic[J].Mathematics in Practice and Theory,2018,48(3):225-232.(in Chinese)
[11]钱志祥.具有可积系数的高阶J-自伴微分算子的本质谱[J].兰州理工大学学报,2015,41(3):154-159.Qian Zhixiang.Essential spectrum of high-order Jselfadjoint differential operators with integral coefficient[J].Journal of Lanzhou University of Technology,2015,41(3):154-159.(in Chinese)
[12]尚在久,朱瑞英.上对称微分算子的自伴域[J].内蒙古大学学报(自然科学版),1986,17(1):7-27.Shang Zaijiu,Zhu Ruiying.The domains of the selfadjoint extensions of ordinary symmetric differential operators over[J].Acta Scientiarum Naturalium Universitatis Intramongolicae,1986,17(1):7-27.(in Chinese)
[13]李文明.具两奇异端点的对称微分算子的自伴域[J].内蒙古大学学报(自然科学版),1989,20(3):291-298.Li Wenming.The domains of self-adjoint extensions of ordinary symmetric differential operators with two singular end points[J].Acta Scientiarum Naturalium Universitatis Intramongolicae,1989,20(3):291-298.(in Chinese)
[14]Shang Zaijiu.On selfadjoint extensions of symmetric ordinary differential operators[J].Journal of Differential Equations,1988,72:153-177.
[2]尚在久,李文明.关于对称微分算子的若干问题[J].内蒙古大学学报(自然科学版),1991,22(3):285-294.Shang Zaijiu,Li Wenming.On some problems ofsymmetric differential operators[J].Acta Scientiarum Naturalium Universitatis Intramongolicae,1991,22(3):285-294.(in Chinese)
[3]Glazman I M.An analogue of the extensions of J-symmetric operators[J].Journal of Differential Equations,1957,57:258-274.
[4]Race D.The theory of selfadjoint extensions of symmetric operators[J].Journal of Differential Equations,1985,57:258-274.
[5]Cao Z.On self-adjoint extensions of n-th order differential operators in the limit circle case[J].Acta Mathematica Sinica,1985,28:205-217.
[6]Sun J.On the self-adjoint extensions of symmetric ordinary differential operators with middle deficiency indices[J]. Acta Mathematica Sinica, 1986, 2:152-167.
[7]王文娟,王忠.具有周期复系数的2n阶微分算子的谱[J].应用数学学报,2015,38(4):699-707.Wang Wenjuan,Wang Zhong.The spectrum of differential operators of 2n-th order with periodic complex coefficients[J].Acta Mathematics Applicatae Sinica,2015,38(4):699-707.(in Chinese)
[8]钱志祥.2n阶J-自伴向量微分算子的本质谱.数学的实践与认识[J].2016,4(17):261-268.Qian Zhixiang.The essential spectrum of 2n-th order J-selfadjoint vector differential operator[J].Mathematics in Practice and Theory,2016,4(17):261-268.(in Chinese)
[9]林秋红.具有可积系数的高阶J-自伴微分算子的离散谱的条件[J].数学的实践与认识,2018,48(10):247-254.Lin Qiuhong.Conditions for the discrete spectrum of high-order J-selfadjoint differential operators with integral coefficients[J].Mathematics in Practice and Theory,2018,48(10):247-254.(in Chinese)
[10]林秋红.具有对数函数系数的J-自伴微分算子谱是离散的充分条件[J].数学的实践与认识,2018,48(3):225-232.Lin Qiuhong.Sufficient conditions for the discrete spectrum of J-selfadjoint differential operators of coefficients contains logarithmic[J].Mathematics in Practice and Theory,2018,48(3):225-232.(in Chinese)
[11]钱志祥.具有可积系数的高阶J-自伴微分算子的本质谱[J].兰州理工大学学报,2015,41(3):154-159.Qian Zhixiang.Essential spectrum of high-order Jselfadjoint differential operators with integral coefficient[J].Journal of Lanzhou University of Technology,2015,41(3):154-159.(in Chinese)
[12]尚在久,朱瑞英.上对称微分算子的自伴域[J].内蒙古大学学报(自然科学版),1986,17(1):7-27.Shang Zaijiu,Zhu Ruiying.The domains of the selfadjoint extensions of ordinary symmetric differential operators over[J].Acta Scientiarum Naturalium Universitatis Intramongolicae,1986,17(1):7-27.(in Chinese)
[13]李文明.具两奇异端点的对称微分算子的自伴域[J].内蒙古大学学报(自然科学版),1989,20(3):291-298.Li Wenming.The domains of self-adjoint extensions of ordinary symmetric differential operators with two singular end points[J].Acta Scientiarum Naturalium Universitatis Intramongolicae,1989,20(3):291-298.(in Chinese)
[14]Shang Zaijiu.On selfadjoint extensions of symmetric ordinary differential operators[J].Journal of Differential Equations,1988,72:153-177.