Bochner-Riesz profile of anharmonic oscillator

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• 作者：Peng Chen^a ; ^{chenpeng3@mail.sysu.edu.cn" class="auth_mail" title="E-mail the corresponding author} ; Waldemar Hebisch^b ; ^{hebisch@math.uni.wroc.pl" class="auth_mail" title="E-mail the corresponding author} ; Adam Sikora^c ; ^{adam.sikora@mq.edu.au" class="auth_mail" title="E-mail the corresponding author}
• 关键词：42B15 ; 42B20 ; 47F05
• 刊名：Journal of Functional Analysis
• 出版年：2016
• 出版时间：1 December 2016
• 年：2016
• 卷：271
• 期：11
• 页码：3186-3241
• 全文大小：752 K

文摘

We investigate spectral multipliers, Bochner–Riesz means and the convergence of eigenfunction expansion corresponding to the Schrödinger operator with anharmonic potential pan id="mmlsi1" class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123616302269&_mathId=si1.gif&_user=111111111&_pii=S0022123616302269&_rdoc=1&_issn=00221236&md5=95189206224ef9b38c76dcf7efed1910">class="imgLazyJSB inlineImage" height="23" width="108" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022123616302269-si1.gif">pan class="mathContainer hidden">pan class="mathCode">$\mathcal{L}=-\frac{d^2}{d{x}^2}+|x|$pan>pan>pan>. We show that the Bochner–Riesz profile of the operator pan id="mmlsi3" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123616302269&_mathId=si3.gif&_user=111111111&_pii=S0022123616302269&_rdoc=1&_issn=00221236&md5=213d62ced4029dd51f4ddc3074ad6481" title="Click to view the MathML source">Lpan>pan class="mathContainer hidden">pan class="mathCode">$\mathcal{L}$pan>pan>pan> completely coincides with such profile of the harmonic oscillator pan id="mmlsi14" class="mathmlsrc">title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123616302269&_mathId=si14.gif&_user=111111111&_pii=S0022123616302269&_rdoc=1&_issn=00221236&md5=4a0f0c56286894fef4119bae3ec2208a">class="imgLazyJSB inlineImage" height="23" width="109" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022123616302269-si14.gif">pan class="mathContainer hidden">pan class="mathCode">$\mathcal{H}=-\frac{d^2}{d{x}^2}+x^2$pan>pan>pan>. It is especially surprising because the Bochner–Riesz profile of the one-dimensional standard Laplace operator is known to be essentially different and the case of operators pan id="mmlsi5" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123616302269&_mathId=si5.gif&_user=111111111&_pii=S0022123616302269&_rdoc=1&_issn=00221236&md5=d5e37b5b17350a667b9fdf3f97b41d53" title="Click to view the MathML source">Hpan>pan class="mathContainer hidden">pan class="mathCode">$\mathcal{H}$pan>pan>pan> and pan id="mmlsi3" class="mathmlsrc">pan class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022123616302269&_mathId=si3.gif&_user=111111111&_pii=S0022123616302269&_rdoc=1&_issn=00221236&md5=213d62ced4029dd51f4ddc3074ad6481" title="Click to view the MathML source">Lpan>pan class="mathContainer hidden">pan class="mathCode">$\mathcal{L}$pan>pan>pan> resembles more the profile of multidimensional Laplace operators. Another surprising element of the main obtained result is the fact that the proof is not based on restriction type estimates and instead an entirely new perspective has to be developed to obtain the critical exponent for Bochner–Riesz means convergence.