Riv. Mat. Univ. Parma, Vol. 7, No. 2, 2016

Anna Canale[a] and Cristian Tacelli[b]

Kernel estimates for a Schrödinger type operator

Pages: 341-350
Accepted in revised form: 17 October 2016
Mathematics Subject Classification (2010): 47D07, 35J10, 35K05, 35K10.
Keywords: Schrödinger operators, unbounded coefficients, kernel estimates.
[a], [b]: University of Salerno, Dept. of Information Eng., Electrical Eng. and Applied Mathematics Via Giovanni Paolo II, 132 Fisciano (SA), 84084, Italy

Abstract: In this paper the principal result obtained is the estimate for the heat kernel associated to the Schrödinger type operator $$(1+|x|^\alpha)\Delta-|x|^\beta$$ $k(t,x,y)\leq Ct^{-\frac{\theta}{2}}\frac {\varphi(x)\varphi(y)}{1+|x|^\alpha},$ where $$\varphi=(1+|x|^\alpha)^{\frac{2-\theta}{4}+\frac{1}{\alpha}\frac{\theta-N}{2}}$$, $$\theta\geq N$$ and $$0 < t \leq 1$$, provided that $$N > 2$$, $$\alpha > 2$$ and $$\beta > \alpha-2$$. This estimate improves a similar estimate obtained in [3] with respect to the dependence on spatial component.

References

[1] D. Bakry, F. Bolley, I. Gentil and P. Maheux, Weighted Nash inequalities, Rev. Mat. Iberoam. 28 (2012), no.3, 879-906. MR2949623
[2] M. Bertoldi and L. Lorenzi, Analytical methods for Markov semigroups, Chapman & Hall/CRC, Boca Raton, FL 2007. MR2313847
[3] A. Canale, A. Rhandi and C. Tacelli, Kernel estimates for Schrödinger type operators with unbounded diffusion and potential terms, Z. Anal. Anwend., to appear.
[4] A. Canale, A. Rhandi and C. Tacelli, Schrödinger type operators with unbounded diffusion and potential terms, Ann. Sc. Norm. Super. Pisa Cl. Sci. 16 (2016), no.2, 581-601. MR3559611
[5] T. Durante, R. Manzo and C. Tacelli, Kernel estimates for Schrödinger type operators with unbounded coefficients and singular potential terms, Ric. Mat. 65 (2016), no. 1, 289-305. MR3513915
[6] S. Fornaro and L. Lorenzi, Generation results for elliptic operators with unbounded diffusion coefficients in $${L}^p$$- and $${C}_b$$-spaces, Discrete Contin. Dyn. Syst. 18 (2007), 747-772. MR2318266
[7] M. Kunze, L. Lorenzi and A. Rhandi, Kernel estimates for nonautonomous Kolmogorov equations with potential term, in "New prospects in direct, inverse and control problems for evolution equations", Springer INdAM Ser., 10, Springer, Cham 2014, 229-251. MR3362995
[8] M. Kunze, L. Lorenzi and A. Rhandi, Kernel estimates for nonautonomous Kolmogorov equations, Adv. Math. 287 (2016), 600-639. MR3422687
[9] L. Lorenzi and A. Rhandi, On Schrödinger type operators with unbounded coefficients: generation and heat kernel estimates, J. Evol. Equ. 15 (2015), 53-88. MR3315665
[10] G. Metafune, D. Pallara and M. Wacker, Feller semigroups on $$\mathbb{R}^n$$, Semigroup Forum 65 (2002), 159-205. MR1911723
[11] G. Metafune and C. Spina, Elliptic operators with unbounded diffusion coefficients in $${L}^p$$ spaces, Ann. Sc. Norm. Super. Pisa Cl. Sci. 11 (2012), no. 2, 303-340. MR3011993
[12] G. Metafuneì and C. Spina, Heat kernel estimates for some elliptic operators with unbounded diffusion coefficients, Discrete Contin. Dyn. Syst. 32 (2012), 2285-2299. MR2885811
[13] G. Metafune, C. Spina and C. Tacelli, Elliptic operators with unbounded diffusion and drift coefficients in $${L}^p$$ spaces, Adv. Differential Equations 19 (2014), no. 5-6, 473-526. MR3189092
[14] G. Metafune, C. Spina and C. Tacelli, On a class of elliptic operators with unbounded diffusion coefficients, Evol. Equ. Control Theory 3(2014), no. 4, 671-680. MR3274653
[15] E. M. Ouhabaz,Analysis of heat equations on domains, London Math. Soc. Monogr. Ser., 31 , Princeton Univ. Press, Princeton, NJ 2005. MR2124040
[16] F.-Y. Wang, Functional inequalities and spectrum estimates:the infinite measure case, J. Funct. Anal. 194 (2002), 288-310. MR1934605

Home Riv.Mat.Univ.Parma