## 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

singular support
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