Classical attenuation versus uniform semiclassical local decay
Let $V_1, V_2 \in C^\infty(\mathbb{R}^d; \mathbb{R})$ with $V_2 \ge 0$ and $V_2 \not\equiv 0$, and suppose that for some $\rho > 0$ and every multi-index $\beta$ $$|\partial^\beta V_j(x)| \le C_\beta \langle x \rangle^{-\rho - |\beta|} \qquad (j = 1, 2),$$ where $\langle x \rangle = (1 + |x|^2)^{1/2}$. For $h > 0$ let $P_h = -h^2 \Delta + V_1 - ih V_2$ and let $S_h(t) = e^{-itP_h/h}$, $t \ge 0$, the associated contraction semigroup on $L^2(\mathbb{R}^d)$. Let $x(t; y, \eta)$ be the position component of the Hamiltonian flow of $|\xi|^2 + V_1(x)$ started at $(y, \eta)$, and fix $s > 0$. Wang observes that the classical condition below is necessary for the decay in the conclusion. Royer proved semiclassical resolvent estimates for the dissipative Helmholtz equation at energies away from zero when the trapped trajectories meet the set where $V_2 > 0$.
Problem. Suppose $$\lim_{t \to \infty} \sup_{(y, \eta) \in \mathbb{R}^{2d}} \langle x(t; y, \eta) \rangle^{-s} \exp\!\Bigl(-2 \int_0^t V_2\bigl(x(\tau; y, \eta)\bigr)\, d\tau\Bigr) \langle y \rangle^{-s} = 0.$$ Must there exist $h_0 > 0$ such that $$\lim_{t \to \infty} \sup_{0 < h \le h_0} \bigl\| \langle x \rangle^{-s} S_h(t) \langle x \rangle^{-s} \bigr\|_{L^2 \to L^2} = 0?$$ No energy cutoff is inserted, so the conclusion includes the zero-energy threshold.
Source: Wang (2010), open problem 2010-12 of the ESF meeting on non-self-adjoint operators, published in Integral Equations Operator Theory (2012): whether his classical condition (2) suffices for the uniform decay estimate (1). Stated in this form by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- MP, SP
- Keywords
- AIM Open Applied Problems, dissipative Schrödinger operators, local decay, semiclassical analysis
References
@article{wang2012,
author = {Wang, Xue Ping},
title = {Uniform time-decay of semigroups of contractions},
journal = {Integral Equations Operator Theory},
volume = {73},
number = {1},
year = {2012},
pages = {3--4},
doi = {10.1007/s00020-011-1930-z},
note = {Open problem 2010-12 of the 2010 ESF meeting, https://nsa.fjfi.cvut.cz/problems.html},
}
@article{royer2010,
author = {Royer, Julien},
title = {Limiting absorption principle for the dissipative {H}elmholtz equation},
journal = {Comm. Partial Differential Equations},
volume = {35},
number = {8},
year = {2010},
pages = {1458--1489},
doi = {10.1080/03605302.2010.490287},
eprint = {0905.0355},
archivePrefix = {arXiv},
}
@misc{aim-dissipative-schrodinger-uniform-decay,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Classical attenuation versus uniform semiclassical local decay}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/a153f469ee0ea64e876b2ad97c79d252c19a3e22/problems/430-dissipative-schrodinger-uniform-decay.md},
note = {Entry 430 at commit a153f46, 2026-10-13},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00425, "Classical attenuation versus uniform semiclassical local decay", clopen.solutions.