We discuss the spectral properties of the operator ${}_{ℳ}\left(\alpha \right):=-d²/dt²+(1/2t²-\alpha )²$ on the line. We first briefly describe how this operator appears in various problems in the analysis of operators on nilpotent Lie groups, in the spectral properties of a Schrödinger operator with magnetic field and in superconductivity. We then give a new proof that the minimum over α of the groundstate energy is attained at a unique point and also prove that the minimum is non-degenerate. Our study can also be seen as a refinement for a specific...

In this paper, the general ordinary quasi-differential expression $M_p$ of $n$-th order with complex coefficients and its formal adjoint $M_p^{+}$ on any finite number of intervals $I_p=(a_p,b_p)$, $p=1,\cdots ,N$, are considered in the setting of the direct sums of $L_{w_p}^2(a_p,b_p)$-spaces of functions defined on each of the separate intervals, and a number of results concerning the location of the point spectra and the regularity fields of general differential operators generated by such expressions are obtained. Some of these are extensions or generalizations...