Resolvent estimates in amalgam spaces and asymptotic expansions for Schrodinger equations

• Galtbayar Artbazar

  School of Mathematics and Computer Science, National University of Mongolia

• Yajima Kenji

  Department of Mathematics, Gakushuin University

書誌事項

タイトル別名

• Resolvent estimates in amalgam spaces and asymptotic expansions for Schrödinger equations

抄録

We consider Schrödinger equations i∂_tu = (−Δ + V)u in ℝ³ with a real potential V such that, for an integer k ≥ 0, ⟨x⟩^kV(x) belongs to an amalgam space ℓ^p(L^q) for some 1 ≤ p < 3/2 < q ≤ ∞, where ⟨x⟩ = (1 + |x|²)^1/2. Let H = −Δ + V and let P_ac be the projector onto the absolutely continuous subspace of L²(ℝ³) for H. Assuming that zero is not an eigenvalue nor a resonance of H, we show that solutions u(t) = exp(−itH)P_acφ admit asymptotic expansions as t → ∞ of the form<br/> $$\bigg\| \langle x \rangle^{-k-\varepsilon} \bigg( u(t)- \sum_{j=0}^{[k/2]}t^{-\frac32-j}P_j \varphi \bigg) \bigg\|_{\infty}\leq C |t|^{-\frac{k+3+\varepsilon}2} \big\| \langle x \rangle^{k+\varepsilon}\varphi \big\|_1$$<br/> for 0 < ε < 3 (1/p − 2/3), where P₀, …, P_[k/2] are operators of finite rank and [k/2] is the integral part of k/2. The proof is based upon estimates of boundary values on the reals of the resolvent (−Δ − λ²)⁻¹ as an operator-valued function between certain weighted amalgam spaces.

収録刊行物

• Ｊｏｕｒｎａｌ　ｏｆ　ｔｈｅ　Ｍａｔｈｅｍａｔｉｃａｌ　Ｓｏｃｉｅｔｙ　ｏｆ　Ｊａｐａｎ

  Ｊｏｕｒｎａｌ　ｏｆ　ｔｈｅ　Ｍａｔｈｅｍａｔｉｃａｌ　Ｓｏｃｉｅｔｙ　ｏｆ　Ｊａｐａｎ 65 (2), 563-605, 2013

  一般社団法人 日本数学会

キーワード

• Schrödinger equation
• asymptotic expansions
• resolvent estimates

詳細情報

• CRID: 1390282680092828288
• NII論文ID: 10031177291
• NII書誌ID: AA0070177X
• DOI: 10.2969/jmsj/06520563
• ISSN: 18811167
• ISSN: 18812333
• ISSN: 00255645
• NDL書誌ID: 024429106
• Web Site: https://ndlsearch.ndl.go.jp/books/R000000004-I024429106
• 本文言語コード: en
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