Properties of ground states of some nondimensional Schrödinger equation and a comparison theorem

The article deals with the Schr ̈odinger equation on the line R having a priori the ground state. Under the assumption that the potential V (x) satisfies the conditions (1) V (x) is nonincreasing on (−∞, a) and not decreasing on (a, ∞), (2) V (x) is piecewise continuous over (−∞, ∞) = R, (3) V (x) is bounded below, the mirror properties and ground state comparison relations are obtained using only elementary analytical computations. In particular, if two potentials V1(x) ≤ V2(x), x ∈ (−∞, 0), are given, then E1 ≤ E2, where E1, E2 are the corresponding ground states.