Kernel estimates for a Schrödinger type operator. (English) Zbl 1369.35013
Summary: In this paper, the principal result obtained is the estimate for the heat kernel associated to the Schrödinger-type operator \((1+| x |^\alpha)\varDelta - | x|^\beta\)
\[ k(t, x, y) \leq Ct^{\frac{\varTheta}{2}} \frac{\varphi (x)\varphi (y)}{1+| x| ^\alpha}, \]
where \(\varphi = (1+| x|^\alpha)^{\frac{2-\varTheta}{4}+\frac{1}{\alpha}\frac{\varTheta-N}{2}}\), \(\varTheta \geq N\) and \(0<t\leq 1\), provided that \(N>2\), \(\alpha >2\) and \(\beta > \alpha -2\). This estimate improves a similar estimate obtained in [A. Canale et al., “Kernel estimates for Schrödinger type operators with unbounded diffusion and potential terms”, Z. Anal. Anwend. (to appear)] with respect to the dependence on the spatial component.
\[ k(t, x, y) \leq Ct^{\frac{\varTheta}{2}} \frac{\varphi (x)\varphi (y)}{1+| x| ^\alpha}, \]
where \(\varphi = (1+| x|^\alpha)^{\frac{2-\varTheta}{4}+\frac{1}{\alpha}\frac{\varTheta-N}{2}}\), \(\varTheta \geq N\) and \(0<t\leq 1\), provided that \(N>2\), \(\alpha >2\) and \(\beta > \alpha -2\). This estimate improves a similar estimate obtained in [A. Canale et al., “Kernel estimates for Schrödinger type operators with unbounded diffusion and potential terms”, Z. Anal. Anwend. (to appear)] with respect to the dependence on the spatial component.
MSC:
| 35J10 | Schrödinger operator, Schrödinger equation |
| 47D07 | Markov semigroups and applications to diffusion processes |
| 35K05 | Heat equation |
| 35K10 | Second-order parabolic equations |
