In this chapter, we study the one-dimensional Schrödinger operator L(q) with even, PT-symmetric and some other special potentials. In Sect. 3.1, we investigate the operator L(q) with complex-valued even potential \(q(-x)=q(x)\) and prove that it may has at most finite number of ESSEssential spectral singularity. In Sect. 3.2, we investigate the spectrum and spectral singularitiesSpectral singularities of the operator L(q) with a periodic PT-symmetric complex-valued potentialComplex-valued potential  \(\overline{q(-x)}=q(x)\) . A basic mathematical question of PT-symmetric quantum mechanics concerns the reality of the spectrum of the considered non-self-adjoint operatorNon-self-adjoint operator. We prove that the main part of the spectrum of L(q) with a periodic PT-symmetric complex-valued potential is real and contains the large part of \([0,\infty ){.}\) Then, we find necessary and sufficient conditions on the potential for finiteness of the number of the nonreal arcs in the spectrum of L(q). Besides, we find necessary and sufficient conditions for the equality of the spectrum of L(q) to the half line. Moreover, we consider the connections between spectrality of L(q) and the reality of its spectrum for some class of PT-symmetric periodic potentials. Finally, we find explicit conditions on the periodic PT-symmetric complex-valued potentialComplex-valued potential q for which the number of gaps in the real part of the spectrum of the operator L(q) is finite. In Sect. 3.3, we investigate the one-dimensional Schrödinger operatorSchrödinger operator L(q) with complex-valued periodic potentialPeriodic potential q when \(q\in L_{1}[0,1]\) and \(q_{n}=0\) for \(n=0,-1,-2,...\) , where \(q_{n}\) are the Fourier coefficients of q with respect to the system \(\{e^{i2\pi nx}\}{.}\) We prove that the Bloch eigenvaluesBloch eigenvalue are \((2\pi n+t)^{2}\) for \(n\in \mathbb {Z}\) , \(t\in \mathbb {C}\) and find explicit formulas for the Bloch functionsBloch function. Then we consider an inverse problemInverse problem for this operator.

On the Special Potentials

章节
发表日期：

2025年

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Oktay Veliev

摘要

In this chapter, we study the one-dimensional Schrödinger operator L(q) with even, PT-symmetric and some other special potentials. In Sect. 3.1, we investigate the operator L(q) with complex-valued even potential \(q(-x)=q(x)\) and prove that it may has at most finite number of ESSEssential spectral singularity. In Sect. 3.2, we investigate the spectrum and spectral singularitiesSpectral singularities of the operator L(q) with a periodic PT-symmetric complex-valued potentialComplex-valued potential \(\overline{q(-x)}=q(x)\) . A basic mathematical question of PT-symmetric quantum mechanics concerns the reality of the spectrum of the considered non-self-adjoint operatorNon-self-adjoint operator. We prove that the main part of the spectrum of L(q) with a periodic PT-symmetric complex-valued potential is real and contains the large part of \([0,\infty ){.}\) Then, we find necessary and sufficient conditions on the potential for finiteness of the number of the nonreal arcs in the spectrum of L(q). Besides, we find necessary and sufficient conditions for the equality of the spectrum of L(q) to the half line. Moreover, we consider the connections between spectrality of L(q) and the reality of its spectrum for some class of PT-symmetric periodic potentials. Finally, we find explicit conditions on the periodic PT-symmetric complex-valued potentialComplex-valued potential q for which the number of gaps in the real part of the spectrum of the operator L(q) is finite. In Sect. 3.3, we investigate the one-dimensional Schrödinger operatorSchrödinger operator L(q) with complex-valued periodic potentialPeriodic potential q when \(q\in L_{1}[0,1]\) and \(q_{n}=0\) for \(n=0,-1,-2,...\) , where \(q_{n}\) are the Fourier coefficients of q with respect to the system \(\{e^{i2\pi nx}\}{.}\) We prove that the Bloch eigenvaluesBloch eigenvalue are \((2\pi n+t)^{2}\) for \(n\in \mathbb {Z}\) , \(t\in \mathbb {C}\) and find explicit formulas for the Bloch functionsBloch function. Then we consider an inverse problemInverse problem for this operator.