\(L^ 1\)-properties of intrinsic Schrödinger semigroups.

*(English)*Zbl 0613.47039Authors’ abstract: ”We investigate the \(L^ 1\)-properties of the intrinsic Markov semigroup associated with a Schrödinger operator on \({\mathbb{R}}^ N\) which possesses a positive ground state. We discover cases for which this semigroup is norm analytic for positive times, and others for which the semigroup is norm discontinuous in the strongest possible sense. In the case of the harmonic oscillator we show that the generator of the intrinsic semigroup has totally different spectrum depending on whether one works in \(L^ 1({\mathbb{R}},e^{-x^ 2}dx)\) or \(L^ 2({\mathbb{R}},e^{-x^ 2}dx)\). In more general cases we show that the equality of the \(L^ 1\) and \(L^ 2\) spectrum is closely related to whether the Schrödinger semigroup is intrinsically ultracontractive.”

We shall elaborate on the framework in order to state the authors’ striking result on the quantum mechanical harmonic oscillation. The elliptic operator defined on functions on \(\Omega\subset {\mathbb{R}}\) by \[ L=-\sum^{n}_{i,j=1}\partial /\partial x_ i(A_{ij}(x)\partial /\partial x_ j)+V(x) \] is associated with (under suitable hypotheses) a semibounded self-adjoint operator H on \(L^ 2(\Omega,dx)\). Let \(\phi\) be a ground state for H in the sense that \(\phi\) is a strictly positive \(C^{\infty}\) function on \(\Omega\) which is an eigenvalue of \(H:H_{\phi}= E_{\phi}\). Call \(\tilde H=M_{\phi}^{-1}(H- E)M_{\phi}\) the associated ”intrinsic operator”; here \(M_{\phi}\) is the unitary operator from \(L^ 2(\Omega,\phi(x)^ 2dx)\) to \(L^ 2(\Omega,dx)\) of multiplication by \(\phi\). Let \(-\tilde H_ p\) be the realization of \(\tilde H\) which generates a Markov semigroup on \(L^ p(\Omega,\phi(x)^ 2dx)\), \(1\leq p<\infty\). While the spectrum of \(\tilde H_ p\) is typically independent of p for \(1<p<\infty\), this need not hold for \(p=1\). Consider the oscillator: \(H=2^{-1}(d^ 2/dx^ 2+x^ 2)\) on \(L^ 2({\mathbb{R}})\). The authors show that the spectrum of \(\tilde H_ 1\) is \(\{\) \(z\in {\mathbb{C}}:Re z\geq 0\}\). Moreover, each z with Re z\(>0\) is an eigenvalue of \(\tilde H_ 1\) of multiplicity two. Further, for \(0<s<t\), \(\eta\exp(-s\tilde H_ 1)- \exp(-t\tilde H_ 1)\eta =2\).

We shall elaborate on the framework in order to state the authors’ striking result on the quantum mechanical harmonic oscillation. The elliptic operator defined on functions on \(\Omega\subset {\mathbb{R}}\) by \[ L=-\sum^{n}_{i,j=1}\partial /\partial x_ i(A_{ij}(x)\partial /\partial x_ j)+V(x) \] is associated with (under suitable hypotheses) a semibounded self-adjoint operator H on \(L^ 2(\Omega,dx)\). Let \(\phi\) be a ground state for H in the sense that \(\phi\) is a strictly positive \(C^{\infty}\) function on \(\Omega\) which is an eigenvalue of \(H:H_{\phi}= E_{\phi}\). Call \(\tilde H=M_{\phi}^{-1}(H- E)M_{\phi}\) the associated ”intrinsic operator”; here \(M_{\phi}\) is the unitary operator from \(L^ 2(\Omega,\phi(x)^ 2dx)\) to \(L^ 2(\Omega,dx)\) of multiplication by \(\phi\). Let \(-\tilde H_ p\) be the realization of \(\tilde H\) which generates a Markov semigroup on \(L^ p(\Omega,\phi(x)^ 2dx)\), \(1\leq p<\infty\). While the spectrum of \(\tilde H_ p\) is typically independent of p for \(1<p<\infty\), this need not hold for \(p=1\). Consider the oscillator: \(H=2^{-1}(d^ 2/dx^ 2+x^ 2)\) on \(L^ 2({\mathbb{R}})\). The authors show that the spectrum of \(\tilde H_ 1\) is \(\{\) \(z\in {\mathbb{C}}:Re z\geq 0\}\). Moreover, each z with Re z\(>0\) is an eigenvalue of \(\tilde H_ 1\) of multiplicity two. Further, for \(0<s<t\), \(\eta\exp(-s\tilde H_ 1)- \exp(-t\tilde H_ 1)\eta =2\).

Reviewer: J.A.Goldstein

##### MSC:

47D03 | Groups and semigroups of linear operators |

47D07 | Markov semigroups and applications to diffusion processes |

47B15 | Hermitian and normal operators (spectral measures, functional calculus, etc.) |

81Q10 | Selfadjoint operator theory in quantum theory, including spectral analysis |

##### Keywords:

intrinsic Markov semigroup associated with a Schrödinger operator; positive ground state; norm analytic; harmonic oscillator; intrinsically ultracontractive
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\textit{E. B. Davies} and \textit{B. Simon}, J. Funct. Anal. 65, 126--146 (1986; Zbl 0613.47039)

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##### References:

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