Compactness of Schrödinger semigroups with unbounded below potentials. (English) Zbl 1156.47043
Let \({\mathcal E}_0\) be a symmetric Dirichlet form with Markov generator \(L_0\) on \(L^2(\mu)\) over a \(\sigma\)-finite measure space \((E,{\mathcal F},\mu)\). It is assumed that \(L_0\) satisfies the following Poincaré-like inequality
\[ \int_Ef^2\,d\mu\leq r{\mathcal E}_0(f,f)+\beta_0(r)\left(\int_E| f| \,d\mu\right)^2,\quad r>0, \]
for some decreasing \(\beta_0:(0,\infty)\to (0,\infty)\).
The Schrödinger semigroup generated by \(L_0-V\) for a class of (unbounded below) potentials \(V\) is proved to be \(L^2(\mu)\)-compact provided that \(\mu(V\leq N) < \infty\) for all \(N>0\). This condition is sharp at least in the context of countable Markov chains, and considerably improves known ones on, e.g., \(\mathbb R^d\) under the condition that \(V(x)\to\infty\) as \(| x| \to\infty\). Concrete examples are provided to illustrate the main result.
\[ \int_Ef^2\,d\mu\leq r{\mathcal E}_0(f,f)+\beta_0(r)\left(\int_E| f| \,d\mu\right)^2,\quad r>0, \]
for some decreasing \(\beta_0:(0,\infty)\to (0,\infty)\).
The Schrödinger semigroup generated by \(L_0-V\) for a class of (unbounded below) potentials \(V\) is proved to be \(L^2(\mu)\)-compact provided that \(\mu(V\leq N) < \infty\) for all \(N>0\). This condition is sharp at least in the context of countable Markov chains, and considerably improves known ones on, e.g., \(\mathbb R^d\) under the condition that \(V(x)\to\infty\) as \(| x| \to\infty\). Concrete examples are provided to illustrate the main result.
Reviewer: Michael Perelmuter (Kyïv)
MSC:
| 47D08 | Schrödinger and Feynman-Kac semigroups |
| 60G51 | Processes with independent increments; Lévy processes |
| 60J10 | Markov chains (discrete-time Markov processes on discrete state spaces) |
| 60J35 | Transition functions, generators and resolvents |
Keywords:
Schrödinger semigroup; Poincaré inequality; compactness; Feynman-Kac formula; super Poincaré inequality; intrinsic super Poincaré inequalityReferences:
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