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Asymmetric harmonic oscillator

The solution of one--dimensional asymmetric quantum harmonic oscillator is presented. The asymmetry can be realized, for example, by using two springs, one spring is glued with the mass, and the second spring is freely connected with the mass in the equilibrium point and it is located inside or outside the first spring which acts on the mass only from the contact point on the right. We study the spectrum of a quantum harmonic oscillator, which has a spring constant $ k_-$ to the left of the equilibrium position and a spring constant $ k_ +$ to the right of the equilibrium position. In the presented case the contact point of the second string is the equilibrium point of the first string. The explicit form of eigenfunctions, the way to calculate the eigenvalues and the properties of the eigenfunctions are discussed.