scholarly journals Decay estimates for Schrödinger heat semigroup with inverse square potential in Lorentz spaces II

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Kazuhiro Ishige ◽  
Yujiro Tateishi
Keyword(s):  
Laplace Operator ◽  
Schrödinger Operator ◽  
Lorentz Space ◽  
Lorentz Spaces ◽  
Operator Norm ◽  
Heat Semigroup ◽  
Decay Estimates ◽  
Schrodinger Operator ◽  
View Point ◽  
The Laplace Operator

<p style='text-indent:20px;'>Let <inline-formula><tex-math id="M1">\begin{document}$ H: = -\Delta+V $\end{document}</tex-math></inline-formula> be a nonnegative Schrödinger operator on <inline-formula><tex-math id="M2">\begin{document}$ L^2({\bf R}^N) $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M3">\begin{document}$ N\ge 2 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M4">\begin{document}$ V $\end{document}</tex-math></inline-formula> is a radially symmetric inverse square potential. Let <inline-formula><tex-math id="M5">\begin{document}$ \|\nabla^\alpha e^{-tH}\|_{(L^{p, \sigma}\to L^{q, \theta})} $\end{document}</tex-math></inline-formula> be the operator norm of <inline-formula><tex-math id="M6">\begin{document}$ \nabla^\alpha e^{-tH} $\end{document}</tex-math></inline-formula> from the Lorentz space <inline-formula><tex-math id="M7">\begin{document}$ L^{p, \sigma}({\bf R}^N) $\end{document}</tex-math></inline-formula> to <inline-formula><tex-math id="M8">\begin{document}$ L^{q, \theta}({\bf R}^N) $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M9">\begin{document}$ \alpha\in\{0, 1, 2, \dots\} $\end{document}</tex-math></inline-formula>. We establish both of upper and lower decay estimates of <inline-formula><tex-math id="M10">\begin{document}$ \|\nabla^\alpha e^{-tH}\|_{(L^{p, \sigma}\to L^{q, \theta})} $\end{document}</tex-math></inline-formula> and study sharp decay estimates of <inline-formula><tex-math id="M11">\begin{document}$ \|\nabla^\alpha e^{-tH}\|_{(L^{p, \sigma}\to L^{q, \theta})} $\end{document}</tex-math></inline-formula>. Furthermore, we characterize the Laplace operator <inline-formula><tex-math id="M12">\begin{document}$ -\Delta $\end{document}</tex-math></inline-formula> from the view point of the decay of <inline-formula><tex-math id="M13">\begin{document}$ \|\nabla^\alpha e^{-tH}\|_{(L^{p, \sigma}\to L^{q, \theta})} $\end{document}</tex-math></inline-formula>.</p>

scholarly journals Ground State for the Schrödinger Operator with the Weighted Hardy Potential

2011 ◽  
Vol 2011 ◽  
pp. 1-26
Author(s):  
J. Chabrowski ◽  
K. Tintarev
Keyword(s):  
Laplace Operator ◽  
Schrödinger Operator ◽  
Ground State ◽  
Ground States ◽  
Schrodinger Operator ◽  
Hardy Potential ◽  
Higher Integrability ◽  
Hardy Type ◽  
The Laplace Operator ◽  
Type Potential

We establish the existence of ground states on for the Laplace operator involving the Hardy-type potential. This gives rise to the existence of the principal eigenfunctions for the Laplace operator involving weighted Hardy potentials. We also obtain a higher integrability property for the principal eigenfunction. This is used to examine the behaviour of the principal eigenfunction around 0.

On the trace of the difference of Schrödinger heat semigroups

1991 ◽  
Vol 119 (1-2) ◽  
pp. 169-175 ◽  
Author(s):  
M. van den Berg
Keyword(s):  
Laplace Operator ◽  
Lower Bounds ◽  
Schrödinger Operator ◽  
Compact Support ◽  
Upper And Lower Bounds ◽  
Schrodinger Operator ◽  
Heat Semigroups ◽  
The Difference ◽  
The Laplace Operator ◽  
Borel Measurable

SynopsisWe obtain upper and lower bounds for tr (e−th−etΔ), where H = −Δ + V is a Schrödinger operator on L2 (ℝm), and ℝ is the Laplace operator for ℝm. The bounds are obtained for a class of negative valued Borel measurable potentials with compact support and in L∞(ℝm).

Morrey Spaces Related to Schrödinger Operators with Certain Nonnegative Potentials and Littlewood–Paley–Stein Functions on the Heisenberg groups

2020 ◽  
Vol 57 (4) ◽  
pp. 465-507
Author(s):  
Hua Wang
Keyword(s):  
Schrödinger Operator ◽  
Morrey Spaces ◽  
Heat Semigroup ◽  
Schrodinger Operator ◽  
Heisenberg Groups ◽  
Area Integral ◽  
Weak Type Estimates ◽  
Homogeneous Dimension ◽  
Lusin Area Integral ◽  
Reverse Hölder

Let be a Schrödinger operator on the Heisenberg group , where is the sublaplacian on and the nonnegative potential V belongs to the reverse Hölder class with . Here is the homogeneous dimension of . Assume that is the heat semigroup generated by. The Lusin area integral and the Littlewood–Paley–Stein function associated with the Schrödinger operator are defined, respectively, bywhereandWhere is a parameter. In this article, the author shows that there is a relationship between and the operator and for any , the following inequality holds true:Based on this inequality and known results for the Lusin area integral , the author establishes the strong-type and weak-type estimates for the Littlewood–Paley–Stein function on . In this article, the author also introduces a class of Morrey spaces associated with the Schrödinger operator on . By using some pointwise estimates of the kernels related to the nonnegative potential V, the author establishes the boundedness properties of the operator acting on the Morrey spaces for an appropriate choice of . It can be shown that the same conclusions hold for the operator on generalized Morrey spaces as well.

scholarly journals On the two-dimensional Schrödinger operator with an attractive potential of the Bessel-Macdonald type

2020 ◽  
pp. 168385
Author(s):  
Wellisson B. De Lima ◽  
Oswaldo M. Del Cima ◽  
Daniel H.T. Franco ◽  
Bruno C. Neves
Keyword(s):  
Schrödinger Operator ◽  
Attractive Potential ◽  
Two Dimensional ◽  
Schrodinger Operator

scholarly journals Evolution of the first eigenvalue of the Laplace operator and the p-Laplace operator under a forced mean curvature flow

2020 ◽  
Vol 18 (1) ◽  
pp. 1518-1530
Author(s):  
Xuesen Qi ◽  
Ximin Liu
Keyword(s):  
Laplace Operator ◽  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Curvature Flow ◽  
Second Fundamental Form ◽  
Coefficient Function ◽  
First Eigenvalue ◽  
Initial Hypersurface ◽  
The Mean ◽  
The Laplace Operator

Abstract In this paper, we discuss the monotonicity of the first nonzero eigenvalue of the Laplace operator and the p-Laplace operator under a forced mean curvature flow (MCF). By imposing conditions associated with the mean curvature of the initial hypersurface and the coefficient function of the forcing term of a forced MCF, and some special pinching conditions on the second fundamental form of the initial hypersurface, we prove that the first nonzero closed eigenvalues of the Laplace operator and the p-Laplace operator are monotonic under the forced MCF, respectively, which partially generalize Mao and Zhao’s work. Moreover, we give an example to specify applications of conclusions obtained above.

Sign in / Sign up

Export Citation Format

Share Document