A solution of the Schrödinger equation for the ground state of a particle in a potential field is analyzed. Since the wavefunctions of the ground state are nodeless, potentials of various kinds can be unambiguously determined. It turns out that the ground state corresponds to zero energy for a wide class of model potentials. Moreover, the zero level can be a single one at the boundary of the continuous spectrum. Crater-like potentials monotonically dependent on coordinates in one-, two-, and three-dimensional cases are studied. Instanton-type potentials with two local minima are of interest in the one-dimensional case. For the Coulomb potential, the energy of the ground state is stable with respect to both long- and short-range screening of this potential. Two-soliton solutions of the nonlinear Schrödinger equation are found. It is demonstrated that the proposed version of the inverse scattering transform is efficient in the analysis of solutions of differential equations.

分析了薛定ding方程在势场中粒子基态的解。由于基态的波函数是无节点的，因此可以明确确定各种电位。事实证明，对于一大类模型电势，基态对应于零能量。此外，零电平在连续光谱的边界处可以是一个。研究了在一维，二维和三维情况下单调依赖于陨石坑的类坑势。在一维情况下，具有两个局部极小值的瞬时子型势很受关注。对于库仑电势，相对于该电势的长距离和短程屏蔽，基态的能量都是稳定的。发现了非线性薛定ding方程的两孤子解。