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• W2963569439 abstract "We study high energy resonances for the operators <disp-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=minus normal upper Delta Subscript partial-differential normal upper Omega comma delta Baseline colon equals negative normal upper Delta plus delta Subscript partial-differential normal upper Omega Baseline circled-times upper V and minus normal upper Delta Subscript partial-differential normal upper Omega comma delta Sub Superscript prime Subscript Baseline colon equals negative normal upper Delta plus delta prime Subscript partial-differential normal upper Omega circled-times upper V partial-differential Subscript nu Baseline> <mml:semantics> <mml:mrow> <mml:mo>−<!-- − --></mml:mo> <mml:msub> <mml:mi mathvariant=normal>Δ<!-- Δ --></mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=normal>∂<!-- ∂ --></mml:mi> <mml:mi mathvariant=normal>Ω<!-- Ω --></mml:mi> <mml:mo>,</mml:mo> <mml:mi>δ<!-- δ --></mml:mi> </mml:mrow> </mml:msub> <mml:mo>:=</mml:mo> <mml:mo>−<!-- − --></mml:mo> <mml:mi mathvariant=normal>Δ<!-- Δ --></mml:mi> <mml:mo>+</mml:mo> <mml:msub> <mml:mi>δ<!-- δ --></mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=normal>∂<!-- ∂ --></mml:mi> <mml:mi mathvariant=normal>Ω<!-- Ω --></mml:mi> </mml:mrow> </mml:msub> <mml:mo>⊗<!-- ⊗ --></mml:mo> <mml:mi>V</mml:mi> <mml:mspace width=1em /> <mml:mtext>and</mml:mtext> <mml:mspace width=1em /> <mml:mo>−<!-- − --></mml:mo> <mml:msub> <mml:mi mathvariant=normal>Δ<!-- Δ --></mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=normal>∂<!-- ∂ --></mml:mi> <mml:mi mathvariant=normal>Ω<!-- Ω --></mml:mi> <mml:mo>,</mml:mo> <mml:msup> <mml:mi>δ<!-- δ --></mml:mi> <mml:mo>′</mml:mo> </mml:msup> </mml:mrow> </mml:msub> <mml:mo>:=</mml:mo> <mml:mo>−<!-- − --></mml:mo> <mml:mi mathvariant=normal>Δ<!-- Δ --></mml:mi> <mml:mo>+</mml:mo> <mml:msubsup> <mml:mi>δ<!-- δ --></mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=normal>∂<!-- ∂ --></mml:mi> <mml:mi mathvariant=normal>Ω<!-- Ω --></mml:mi> </mml:mrow> <mml:mo>′</mml:mo> </mml:msubsup> <mml:mo>⊗<!-- ⊗ --></mml:mo> <mml:mi>V</mml:mi> <mml:msub> <mml:mi mathvariant=normal>∂<!-- ∂ --></mml:mi> <mml:mi>ν<!-- ν --></mml:mi> </mml:msub> </mml:mrow> <mml:annotation encoding=application/x-tex>begin{equation*} -Delta _{partial Omega ,delta }:=-Delta +delta _{partial Omega }otimes Vquad text {and}quad -Delta _{partial Omega ,delta ’}:=-Delta +delta _{partial Omega }’otimes Vpartial _nu end{equation*}</mml:annotation> </mml:semantics> </mml:math> </disp-formula> where <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=normal upper Omega subset-of double-struck upper R Superscript d> <mml:semantics> <mml:mrow> <mml:mi mathvariant=normal>Ω<!-- Ω --></mml:mi> <mml:mo>⊂<!-- ⊂ --></mml:mo> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>R</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> </mml:mrow> <mml:annotation encoding=application/x-tex>Omega subset mathbb {R}^d</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is strictly convex with smooth boundary, <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper V colon upper L squared left-parenthesis partial-differential normal upper Omega right-parenthesis right-arrow upper L squared left-parenthesis partial-differential normal upper Omega right-parenthesis> <mml:semantics> <mml:mrow> <mml:mi>V</mml:mi> <mml:mo>:</mml:mo> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo stretchy=false>(</mml:mo> <mml:mi mathvariant=normal>∂<!-- ∂ --></mml:mi> <mml:mi mathvariant=normal>Ω<!-- Ω --></mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo stretchy=false>→<!-- → --></mml:mo> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo stretchy=false>(</mml:mo> <mml:mi mathvariant=normal>∂<!-- ∂ --></mml:mi> <mml:mi mathvariant=normal>Ω<!-- Ω --></mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mrow> <mml:annotation encoding=application/x-tex>V:L^2(partial Omega )to L^2(partial Omega )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> may depend on frequency, and <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=delta Subscript partial-differential normal upper Omega> <mml:semantics> <mml:msub> <mml:mi>δ<!-- δ --></mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=normal>∂<!-- ∂ --></mml:mi> <mml:mi mathvariant=normal>Ω<!-- Ω --></mml:mi> </mml:mrow> </mml:msub> <mml:annotation encoding=application/x-tex>delta _{partial Omega }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the surface measure on <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=partial-differential normal upper Omega> <mml:semantics> <mml:mrow> <mml:mi mathvariant=normal>∂<!-- ∂ --></mml:mi> <mml:mi mathvariant=normal>Ω<!-- Ω --></mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>partial Omega</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. These operators are model Hamiltonians for the quantum corrals studied in Aligia and Lobos (2005), Barr, Zaletel, and Heller (2010), and Crommie et al. (1995) and for leaky quantum graphs in Exner (2008). We give a quantum version of the Sabine Law (Sabine, 1964) from the study of acoustics for both <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=minus normal upper Delta Subscript partial-differential normal upper Omega comma delta> <mml:semantics> <mml:mrow> <mml:mo>−<!-- − --></mml:mo> <mml:msub> <mml:mi mathvariant=normal>Δ<!-- Δ --></mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=normal>∂<!-- ∂ --></mml:mi> <mml:mi mathvariant=normal>Ω<!-- Ω --></mml:mi> <mml:mo>,</mml:mo> <mml:mi>δ<!-- δ --></mml:mi> </mml:mrow> </mml:msub> </mml:mrow> <mml:annotation encoding=application/x-tex>-Delta _{partial Omega ,delta }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=minus normal upper Delta Subscript partial-differential normal upper Omega comma delta prime> <mml:semantics> <mml:mrow> <mml:mo>−<!-- − --></mml:mo> <mml:msub> <mml:mi mathvariant=normal>Δ<!-- Δ --></mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=normal>∂<!-- ∂ --></mml:mi> <mml:mi mathvariant=normal>Ω<!-- Ω --></mml:mi> <mml:mo>,</mml:mo> <mml:msup> <mml:mi>δ<!-- δ --></mml:mi> <mml:mo>′</mml:mo> </mml:msup> </mml:mrow> </mml:msub> </mml:mrow> <mml:annotation encoding=application/x-tex>-Delta _{partial Omega ,delta ’}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. It characterizes the decay rates (imaginary parts of resonances) in terms of the system’s ray dynamics. In particular, the decay rates are controlled by the average reflectivity and chord length of the barrier. For <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=minus normal upper Delta Subscript partial-differential normal upper Omega comma delta> <mml:semantics> <mml:mrow> <mml:mo>−<!-- − --></mml:mo> <mml:msub> <mml:mi mathvariant=normal>Δ<!-- Δ --></mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=normal>∂<!-- ∂ --></mml:mi> <mml:mi mathvariant=normal>Ω<!-- Ω --></mml:mi> <mml:mo>,</mml:mo> <mml:mi>δ<!-- δ --></mml:mi> </mml:mrow> </mml:msub> </mml:mrow> <mml:annotation encoding=application/x-tex>-Delta _{partial Omega ,delta }</mml:annotation> </mml:semantics> </mml:math> </inline-formula> with <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=normal upper Omega> <mml:semantics> <mml:mi mathvariant=normal>Ω<!-- Ω --></mml:mi> <mml:annotation encoding=application/x-tex>Omega</mml:annotation> </mml:semantics> </mml:math> </inline-formula> smooth and strictly convex, our results improve those given for general <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=partial-differential normal upper Omega> <mml:semantics> <mml:mrow> <mml:mi mathvariant=normal>∂<!-- ∂ --></mml:mi> <mml:mi mathvariant=normal>Ω<!-- Ω --></mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>partial Omega</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in Galkowski and Smith (2015) and are generically optimal. Indeed, we show that for generic domains and potentials there are infinitely many resonances arbitrarily close to the resonance free region found by our theorem. In the case of <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=minus normal upper Delta Subscript partial-differential normal upper Omega comma delta prime> <mml:semantics> <mml:mrow> <mml:mo>−<!-- − --></mml:mo> <mml:msub> <mml:mi mathvariant=normal>Δ<!-- Δ --></mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=normal>∂<!-- ∂ --></mml:mi> <mml:mi mathvariant=normal>Ω<!-- Ω --></mml:mi> <mml:mo>,</mml:mo> <mml:msup> <mml:mi>δ<!-- δ --></mml:mi> <mml:mo>′</mml:mo> </mml:msup> </mml:mrow> </mml:msub> </mml:mrow> <mml:annotation encoding=application/x-tex>-Delta _{partial Omega ,delta ’}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, the quantum Sabine law gives the existence of a resonance free region that converges to the real axis at a fixed polynomial rate. The size of this resonance free region is optimal in the case of the unit disk in <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=double-struck upper R squared> <mml:semantics> <mml:msup> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=double-struck>R</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> <mml:annotation encoding=application/x-tex>mathbb {R}^2</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. As far as the author is aware, this is the only class of examples that is known to have resonances converging to the real axis at a fixed polynomial rate but no faster. The proof of our theorem requires several new technical tools. We adapt intersecting Lagrangian distributions from Melrose and Uhlman (1979) to the semiclassical setting and give a description of the kernel of the free resolvent as such a distribution. We also construct a semiclassical version of the Melrose–Taylor parametrix (Melrose and Taylor, unpublished manuscript) for complex energies. We use these constructions to give a complete microlocal description of the single, double, and derivative double layer operators in the case that <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=partial-differential normal upper Omega> <mml:semantics> <mml:mrow> <mml:mi mathvariant=normal>∂<!-- ∂ --></mml:mi> <mml:mi mathvariant=normal>Ω<!-- Ω --></mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>partial Omega</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is smooth and strictly convex. These operators are given respectively for <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=x element-of partial-differential normal upper Omega> <mml:semantics> <mml:mrow> <mml:mi>x</mml:mi> <mml:mo>∈<!-- ∈ --></mml:mo> <mml:mi mathvariant=normal>∂<!-- ∂ --></mml:mi> <mml:mi mathvariant=normal>Ω<!-- Ω --></mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>xin partial Omega</mml:annotation> </mml:semantics> </mml:math> </inline-formula> by <disp-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=StartLayout 1st Row 1st Column upper G left-parenthesis lamda right-parenthesis f left-parenthesis x right-parenthesis 2nd Column a m p semicolon colon equals integral Underscript partial-differential normal upper Omega Endscripts upper R 0 left-parenthesis lamda right-parenthesis left-parenthesis x comma y right-parenthesis f left-parenthesis y right-parenthesis d upper S left-parenthesis y right-parenthesis comma 2nd Row 1st Column ModifyingAbove upper N With tilde left-parenthesis lamda right-parenthesis f left-parenthesis x right-parenthesis 2nd Column a m p semicolon colon equals integral Underscript partial-differential normal upper Omega Endscripts partial-differential Subscript nu Sub Subscript y Subscript Baseline upper R 0 left-parenthesis lamda right-parenthesis left-parenthesis x comma y right-parenthesis f left-parenthesis y right-parenthesis d upper S left-parenthesis y right-parenthesis 3rd Row 1st Column partial-differential Subscript nu Baseline script upper D script l left-parenthesis lamda right-parenthesis f left-parenthesis x right-parenthesis 2nd Column a m p semicolon colon equals integral Underscript partial-differential normal upper Omega Endscripts partial-differential Subscript nu Sub Subscript x Subscript Baseline partial-differential Subscript nu Sub Subscript y Subscript Baseline upper R 0 left-parenthesis lamda right-parenthesis left-parenthesis x comma y right-parenthesis f left-parenthesis y right-parenthesis d upper S left-parenthesis y right-parenthesis period EndLayout> <mml:semantics> <mml:mtable columnalign=right left right left right left right left right left right left rowspacing=3pt columnspacing=0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em side=left displaystyle=true> <mml:mtr> <mml:mtd> <mml:mi>G</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>λ<!-- λ --></mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mi>f</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mtd> <mml:mtd> <mml:mi>a</mml:mi> <mml:mi>m</mml:mi> <mml:mi>p</mml:mi> <mml:mo>;</mml:mo> <mml:mo>:=</mml:mo> <mml:msub> <mml:mo>∫<!-- ∫ --></mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=normal>∂<!-- ∂ --></mml:mi> <mml:mi mathvariant=normal>Ω<!-- Ω --></mml:mi> </mml:mrow> </mml:msub> <mml:msub> <mml:mi>R</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo stretchy=false>(</mml:mo> <mml:mi>λ<!-- λ --></mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo stretchy=false>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>y</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mi>f</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>y</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mi>d</mml:mi> <mml:mi>S</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>y</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mspace width=thinmathspace /> <mml:mo>,</mml:mo> </mml:mtd> </mml:mtr> <mml:mtr> <mml:mtd> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mover> <mml:mi>N</mml:mi> <mml:mo stretchy=false>~<!-- ~ --></mml:mo> </mml:mover> </mml:mrow> <mml:mo stretchy=false>(</mml:mo> <mml:mi>λ<!-- λ --></mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mi>f</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mtd> <mml:mtd> <mml:mi>a</mml:mi> <mml:mi>m</mml:mi> <mml:mi>p</mml:mi> <mml:mo>;</mml:mo> <mml:mo>:=</mml:mo> <mml:msub> <mml:mo>∫<!-- ∫ --></mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=normal>∂<!-- ∂ --></mml:mi> <mml:mi mathvariant=normal>Ω<!-- Ω --></mml:mi> </mml:mrow> </mml:msub> <mml:msub> <mml:mi mathvariant=normal>∂<!-- ∂ --></mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mi>ν<!-- ν --></mml:mi> <mml:mi>y</mml:mi> </mml:msub> </mml:mrow> </mml:msub> <mml:msub> <mml:mi>R</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo stretchy=false>(</mml:mo> <mml:mi>λ<!-- λ --></mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo stretchy=false>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>y</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mi>f</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>y</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mi>d</mml:mi> <mml:mi>S</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>y</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mtd> </mml:mtr> <mml:mtr> <mml:mtd> <mml:msub> <mml:mi mathvariant=normal>∂<!-- ∂ --></mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>ν<!-- ν --></mml:mi> </mml:mrow> </mml:msub> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>D</mml:mi> </mml:mrow> <mml:mi>ℓ<!-- ℓ --></mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>λ<!-- λ --></mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mi>f</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy=false>)</mml:mo> </mml:mtd> <mml:mtd> <mml:mi>a</mml:mi> <mml:mi>m</mml:mi> <mml:mi>p</mml:mi> <mml:mo>;</mml:mo> <mml:mo>:=</mml:mo> <mml:msub> <mml:mo>∫<!-- ∫ --></mml:mo> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi mathvariant=normal>∂<!-- ∂ --></mml:mi> <mml:mi mathvariant=normal>Ω<!-- Ω --></mml:mi> </mml:mrow> </mml:msub> <mml:msub> <mml:mi mathvariant=normal>∂<!-- ∂ --></mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mi>ν<!-- ν --></mml:mi> <mml:mi>x</mml:mi> </mml:msub> </mml:mrow> </mml:msub> <mml:msub> <mml:mi mathvariant=normal>∂<!-- ∂ --></mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:msub> <mml:mi>ν<!-- ν --></mml:mi> <mml:mi>y</mml:mi> </mml:msub> </mml:mrow> </mml:msub> <mml:msub> <mml:mi>R</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo stretchy=false>(</mml:mo> <mml:mi>λ<!-- λ --></mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mo stretchy=false>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mi>y</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mi>f</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>y</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mi>d</mml:mi> <mml:mi>S</mml:mi> <mml:mo stretchy=false>(</mml:mo> <mml:mi>y</mml:mi> <mml:mo stretchy=false>)</mml:mo> <mml:mspace width=thinmathspace /> <mml:mo>.</mml:mo> </mml:mtd> </mml:mtr> </mml:mtable> <mml:annotation encoding=application/x-tex>begin{align*} G(lambda )f(x)&:=int _{partial Omega }R_0(lambda )(x,y)f(y)dS(y),, tilde {N}(lambda )f(x)&:=int _{partial Omega }partial _{nu _y}R_0(lambda )(x,y)f(y)dS(y) partial _{nu }mathcal {D}ell (lambda )f(x)&:=int _{partial Omega }partial _{nu _x}partial _{nu _y}R_0(lambda )(x,y)f(y)dS(y),. end{align*}</mml:annotation> </mml:semantics> </mml:math> </disp-formula> This microlocal description allows us to prove sharp high energy estimates on <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper G> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding=application/x-tex>G</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper N overTilde> <mml:semantics> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mover> <mml:mi>N</mml:mi> <mml:mo stretchy=false>~<!-- ~ --></mml:mo> </mml:mover> </mml:mrow> <mml:annotation encoding=application/x-tex>tilde {N}</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, and <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=partial-differential Subscript nu Baseline script upper D script l> <mml:semantics> <mml:mrow> <mml:msub> <mml:mi mathvariant=normal>∂<!-- ∂ --></mml:mi> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi>ν<!-- ν --></mml:mi> </mml:mrow> </mml:msub> <mml:mrow class=MJX-TeXAtom-ORD> <mml:mi class=MJX-tex-caligraphic mathvariant=script>D</mml:mi> </mml:mrow> <mml:mi>ℓ<!-- ℓ --></mml:mi> </mml:mrow> <mml:annotation encoding=application/x-tex>partial _{nu }mathcal {D}ell</mml:annotation> </mml:semantics> </mml:math> </inline-formula> when <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=normal upper Omega> <mml:semantics> <mml:mi mathvariant=normal>Ω<!-- Ω --></mml:mi> <mml:annotation encoding=application/x-tex>Omega</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is smooth and strictly convex, removing the log losses from the estimates for <inline-formula content-type=math/mathml> <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML alttext=upper G> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding=application/x-tex>G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> in Galkowski and Smith (2015) and Han and Tacy (2015) and proving a conjecture from Appendix A in Han and Tacy (2015)." @default.
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• W2963569439 date "2019-05-01" @default.
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• W2963569439 title "Distribution of Resonances in Scattering by Thin Barriers" @default.
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