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On the fundamental solution of a perturbed harmonic oscillator. (English) Zbl 0892.35035

The paper is devoted to the investigation of the smoothness of the fundamental solution of the equation \[ {1\over i}{\partial \psi \over \partial t} -{1\over 2}\Delta\psi + {1\over 2}| x| ^2\psi+w(x)\psi= 0 \] in \(\mathbb{R}^n\). If \(w(x)\) is subquadratic perturbation that satisfies some smoothness conditions like \(| \partial ^\alpha_x w(x)| \leq c_\alpha\langle x\rangle^{\nu| \alpha|}\) for all \(\alpha\in\mathbb{Z}^n_+\) with \(| \alpha| \geq 1\), for some \(\nu < 1\), or \(| \partial ^\alpha_x w(x)| = o(1)\) as \(| x| \to \infty\) for all \(\alpha\in\mathbb{Z}^n_+\) with \(| \alpha| \geq 2\), the authors prove \(C^\infty\) smoothness of the fundamental solution \(E(t,x,y)\) for non-resonant \(t\). Some additional estimates of \(E(t,x,y)\) and investigation of the singular support are also done.

MSC:

35B65 Smoothness and regularity of solutions to PDEs
35A08 Fundamental solutions to PDEs
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