## Perturbations of the Wigner-von Neumann potential leaving the embedded eigenvalue fixed.(English)Zbl 1021.81016

The authors investigate the stability of eigenvalues embedded in the essential spectrum for the one-dimensional operator $$H_{Q,p}= {d^2\over dx^2}+{\gamma\over x}\sin(\alpha x)+ V(x)$$ on $$L^p(\mathbb{R})$$ with $$V\in L^1(\mathbb{R})$$ where $$1\leq p< \infty$$, $$\gamma\neq 0$$ and $$\alpha> 0$$. The operator $$H_{Q,p}$$ is defined as the infinitesimal generator of the $$C_0$$-semigroup on $$L^p(\mathbb{R})$$ characterized via the Feynman-Kac formula. There main result is that there are no positive eigenvalues for $$|\gamma|\leq{2\alpha\over p}$$ whereas there exists a positive eigenvalue with value $${\alpha^2\over 4}$$ for $$|\gamma|> {2\alpha\over p}$$. In the latter case they prove that the set $$M(\alpha,\gamma)$$ of functions $$V\in L^1(\mathbb{R})$$ for which the eigenvalues is stable is a smooth manifold of codimension one.
The proof relies on the investigation of the differential equation $$-\psi''+ ({\gamma\over x}\sin(\alpha x)+ V)\psi= k^2\psi$$ related to the operator $$H_{Q,p}$$. In Theorem 2.1 it is shown that the solutions behave asymptotically like $$\sin(kr)$$ or $$\cos(kr)$$ plus corrections vanishing at infinity for $$k\neq{\alpha\over 2}$$. For $$k={\alpha\over 2}$$ the solutions have asymptotics like $$r^{-(\gamma/2\alpha)}(\cos(kr)+ o(1))$$ and $$r^{(\gamma/2\alpha)}(\sin(kr)+ o(1))$$ as $$r\to\infty$$. Their proof of the asymptotic behaviour follows the work of J. Cassell [The asymptotic integration of some oscillatory differential equations, Q. J. Math., Oxf. II. Ser. 33, 281-296 (1982; Zbl 0519.34027)].
Next, the authors show that the solutions $$\psi_{\pm}$$ behaving like $$r^{-(\gamma/2\alpha)}(\cos(kr)+ o(1))$$ for $$r\to\pm\infty$$ of the differential equation for $$k={\alpha\over 2}$$ depend smoothly on $$V\in L^1(\mathbb{R})$$. Thus the Wronskian is a smooth function of $$V$$ and it is shown that $$\{0\}$$ is a regular value of the Wronskian giving that $$M(\alpha,\gamma)$$ is a smooth manifold of codimension one.
Reviewer: Giere

### MSC:

 81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis 34L30 Nonlinear ordinary differential operators 34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.) 34L15 Eigenvalues, estimation of eigenvalues, upper and lower bounds of ordinary differential operators 34C11 Growth and boundedness of solutions to ordinary differential equations

Zbl 0519.34027
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