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.

scholarly journals Mass and Extremals Associated with the Hardy–Schrödinger Operator on Hyperbolic Space

2018 ◽  
Vol 18 (4) ◽  
pp. 671-689
Author(s):  
Hardy Chan ◽  
Nassif Ghoussoub ◽  
Saikat Mazumdar ◽  
Shaya Shakerian ◽  
Luiz Fernando de Oliveira Faria
Keyword(s):  
Hyperbolic Space ◽  
Schrödinger Operator ◽  
Ground State ◽  
Existence Of Solutions ◽  
Ground State Solution ◽  
Positive Definite ◽  
Schrodinger Operator ◽  
Hardy Potential ◽  
Critical Equation ◽  
Euclidean Setting

AbstractWe consider the Hardy–Schrödinger operator {L_{\gamma}:=-\Delta_{\mathbb{B}^{n}}-\gamma{V_{2}}} on the Poincaré ball model of the hyperbolic space {\mathbb{B}^{n}} ({n\geq 3}). Here {V_{2}} is a radially symmetric potential, which behaves like the Hardy potential around its singularity at 0, i.e., {V_{2}(r)\sim\frac{1}{r^{2}}}. As in the Euclidean setting, {L_{\gamma}} is positive definite whenever {\gamma<\frac{(n-2)^{2}}{4}}, in which case we exhibit explicit solutions for the critical equation {L_{\gamma}u=V_{2^{*}(s)}u^{2^{*}(s)-1}} in {\mathbb{B}^{n},} where {0\leq s<2}, {2^{*}(s)=\frac{2(n-s)}{n-2}}, and {V_{2^{*}(s)}} is a weight that behaves like {\frac{1}{r^{s}}} around 0. In dimensions {n\geq 5}, the equation {L_{\gamma}u-\lambda u=V_{2^{*}(s)}u^{2^{*}(s)-1}} in a domain Ω of {\mathbb{B}^{n}} away from the boundary but containing 0 has a ground state solution, whenever {0<\gamma\leq\frac{n(n-4)}{4}}, and {\lambda>\frac{n-2}{n-4}(\frac{n(n-4)}{4}-\gamma)}. On the other hand, in dimensions 3 and 4, the existence of solutions depends on whether the domain has a positive “hyperbolic mass” a notion that we introduce and analyze therein.

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>

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).

scholarly journals Ground-state positivity, negativity, and compactness for a Schrödinger operator in RN

2007 ◽  
Vol 245 (1) ◽  
pp. 213-248 ◽  
Author(s):  
Bénédicte Alziary ◽  
Jacqueline Fleckinger-Pellé ◽  
Peter Takáč
Keyword(s):  
Schrödinger Operator ◽  
Ground State ◽  
Schrodinger Operator

scholarly journals SCHRÖDINGER SECANT LOWER BOUNDS TO SEMIRELATIVISTIC EIGENVALUES

2007 ◽  
Vol 22 (10) ◽  
pp. 1899-1904 ◽  
Author(s):  
RICHARD L. HALL ◽  
WOLFGANG LUCHA
Keyword(s):  
Lower Bounds ◽  
Schrödinger Operator ◽  
Ground State ◽  
Schrodinger Operator ◽  
Bounded Below

It is shown that the ground-state eigenvalue of a semirelativistic Hamiltonian of the form [Formula: see text] is bounded below by the Schrödinger operator m + β p2 + V, for suitable β>0. An example is discussed.

Sign in / Sign up

Export Citation Format

Share Document