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Let\Sigmabe a strictly convex, compact patch of aC^{2}hypersurface in\mathbb{R}^{n}, with non-vanishing Gaussian curvature and surface measured\sigmainduced by the Lebesgue measure in\mathbb{R}^{n}. The Mizohata–Takeuchi conjecture states that \int |\widehat{g d\sigma}|^{2} w \leq C \|Xw\|_{\infty} \int |g|^{2} for allg\in L^{2}(\Sigma)and all weightsw \colon \mathbb{R}^{n}\rightarrow [0,+\infty), whereXdenotes theX-ray transform. As partial progress towards the conjecture, we show, as a straightforward consequence of recently-established decoupling inequalities, that for every\varepsilon>0, there exists a positive constantC_{\varepsilon}, which depends only on\Sigmaand\varepsilon, such that for allR \geq 1and all weightsw \colon \mathbb{R}^{n}\rightarrow [0,+\infty), we have \int_{B_R}|\widehat{g d\sigma}|^{2} w \leq C_{\varepsilon} R^{\varepsilon} \sup_{T} \Big(\int_{T} w^{(n+1)/2}\Big)^{2/(n+1)}\int |g|^{2}, whereTranges over the family of tubes in\mathbb{R}^{n}of dimensionsR^{1/2}\times \cdots \times R^{1/2}\times R. From this we deduce the Mizohata–Takeuchi conjecture with anR^{(n-1)/(n+1)}-loss; i.e., that \int_{B_R}|\widehat{g d\sigma}|^{2} w \leq C_{\varepsilon} R^{\frac{n-1}{n+1}+ \varepsilon}\|Xw\|_{\infty} \int |g|^{2} for any ballB_{R}of radiusRand any\varepsilon>0. The power(n-1)/(n+1)here cannot be replaced by anything smaller unless properties of\widehat{g d\sigma}beyond ‘decoupling axioms’ are exploited. We also provide estimates which improve this inequality under various conditions on the weight, and discuss some new cases where the conjecture holds.
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This content will become publicly available on April 29, 2027
Near invariance of quasi-energy spectrum of Floquet Hamiltonians
The spectral analysis of the unitary monodromy operator, associated with a time-periodically (parametrically) forced Schrödinger equation, is a question of longstanding interest. Here, we consider this question for Hamiltonians of the form H^{\varepsilon}(t)=H^{0} + \varepsilon^{a} W(\varepsilon^{a} t, -i\nabla) , whereH^{0}is an unperturbed autonomous Hamiltonian,a\geq 1, andW(T,\cdot)has a period ofT_{\mathrm{per}}>0. In particular, in the small\varepsilon>0regime, we seek a comparison between the spectral properties of the monodromy operator, the one-period flow map associated with theH^{\varepsilon}(t)dynamics, and that of the autonomous (unforced) flow,\exp[-iH^{0} T_{\mathrm{per}}\varepsilon^{-a}]. We considerH^{0}which is spatially periodic on\R^{n}with respect to a lattice. Using the decomposition ofH^{0}andH^{\varepsilon}(t)into their actions on spaces (Floquet–Bloch fibers) of pseudo-periodic functions, we establish a spectral near-invariance property for the monodromy operator, when acting on data which are\varepsilon-localized in energy and quasi-momentum. Our analysis requires the following steps: (i) spectrally-localized data are approximated by -emphband-limited (Floquet–Bloch) wavepackets; (iii) the envelope dynamics of such wavepackets is well approximated by an effective (homogenized) PDE; and (iii) an exact invariance property for band-limited Floquet–Bloch wavepackets, which follows from the effective dynamics. We apply our general results to a number of periodic Hamiltonians,H^{0}, of interest in the study of photonic and quantum materials.
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- Award ID(s):
- 2508811
- PAR ID:
- 10702003
- Publisher / Repository:
- European Mathematical Society
- Date Published:
- Journal Name:
- Journal of Spectral Theory
- Volume:
- 16
- Issue:
- 2
- ISSN:
- 1664-039X
- Page Range / eLocation ID:
- 425 to 458
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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