Pointwise Bounds for Joint Eigenfunctions of Quantum Completely Integrable Systems
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Abstract
Abstract Let ( M , g ) be a compact Riemannian manifold of dimension n and $$P_1:=-h^2\Delta _g+V(x)-E_1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>P</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>:</mml:mo><mml:mo>=</mml:mo><mml:mo>-</mml:mo><mml:msup><mml:mi>h</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:msub><mml:mi>Δ</mml:mi><mml:mi>g</mml:mi></mml:msub><mml:mo>+</mml:mo><mml:mi>V</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>-</mml:mo><mml:msub><mml:mi>E</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:mrow></mml:math> so that $$dp_1\ne 0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>d</mml:mi><mml:msub><mml:mi>p</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>≠</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math> on $$p_1=0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>p</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math> . We assume that $$P_1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>P</mml:mi><mml:mn>1</mml:mn></mml:msub></mml:math> is quantum completely integrable (ACI) in the sense that there exist functionally independent pseuodifferential operators $$P_2,\dots P_n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>P</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:mo>⋯</mml:mo><mml:msub><mml:mi>P</mml:mi><mml:mi>n</mml:mi></mml:msub></mml:mrow></mml:math> with $$[P_i,P_j]=0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>[</mml:mo><mml:msub><mml:mi>P</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>P</mml:mi><mml:mi>j</mml:mi></mml:msub><mml:mo>]</mml:mo><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math> , $$i,j=1,\dots n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>i</mml:mi><mml:mo>,</mml:mo><mml:mi>j</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mo>⋯</mml:mo><mml:mi>n</mml:mi></mml:mrow></mml:math> . We study the pointwise bounds for the joint eigenfunctions, $$u_h$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>u</mml:mi><mml:mi>h</mml:mi></mml:msub></mml:math> of the system $$\{P_i\}_{i=1}^n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msubsup><mml:mrow><mml:mo>{</mml:mo><mml:msub><mml:mi>P</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>}</mml:mo></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:mrow><mml:mi>n</mml:mi></mml:msubsup></mml:math> with $$P_1u_h=E_1u_h+o(1)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>P</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:msub><mml:mi>u</mml:mi><mml:mi>h</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mi>E</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:msub><mml:mi>u</mml:mi><mml:mi>h</mml:mi></mml:msub><mml:mo>+</mml:mo><mml:mi>o</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mn>1</mml:mn><mml:mo>)</mml:mo></mml:mrow></mml:mrow></mml:math> . In Theorem 1, we first give polynomial improvements over the standard Hörmander bounds for typical points in M . In two and three dimensions, these estimates agree with the Hardy exponent $$h^{-\frac{1-n}{4}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>h</mml:mi><mml:mrow><mml:mo>-</mml:mo><mml:mfrac><mml:mrow><mml:mn>1</mml:mn><mml:mo>-</mml:mo><mml:mi>n</mml:mi></mml:mrow><mml:mn>4</mml:mn></mml:mfrac></mml:mrow></mml:msup></mml:math> and in higher dimensions we obtain a gain of $$h^{\frac{1}{2}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi>h</mml:mi><mml:mfrac><mml:mn>1</mml:mn><mml:mn>2</mml:mn></mml:mfrac></mml:msup></mml:math> over the Hörmander bound. In our second main result (Theorem 3), under a real-analyticity assumption on the QCI system, we give exponential decay estimates for joint eigenfunctions at points outside the projection of invariant Lagrangian tori; that is at points $$x\in M$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>x</mml:mi><mml:mo>∈</mml:mo><mml:mi>M</mml:mi></mml:mrow></mml:math> in the “microlocally forbidden” region $$p_1^{-1}(E_1)\cap \dots \cap p_n^{-1}(E_n)\cap T^*_xM=\emptyset .$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msubsup><mml:mi>p</mml:mi><mml:mn>1</mml:mn><mml:mrow><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:mrow><mml:mo>(</mml:mo><mml:msub><mml:mi>E</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>)</mml:mo></mml:mrow><mml:mo>∩</mml:mo><mml:mo>⋯</mml:mo><mml:mo>∩</mml:mo><mml:msubsup><mml:mi>p</mml:mi><mml:mi>n</mml:mi><mml:mrow><mml:mo>-</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:mrow><mml:mo>(</mml:mo><mml:msub><mml:mi>E</mml:mi><mml:mi>n</mml:mi></mml:msub><mml:mo>)</mml:mo></mml:mrow><mml:mo>∩</mml:mo><mml:msubsup><mml:mi>T</mml:mi><mml:mi>x</mml:mi><mml:mo>∗</mml:mo></mml:msubsup><mml:mi>M</mml:mi><mml:mo>=</mml:mo><mml:mi>∅</mml:mi><mml:mo>.</mml:mo></mml:mrow></mml:math> These bounds are sharp locally near the projection of the invariant tori.
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|---|---|---|
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