Hill's spectral curves and the invariant measure of the periodic KdV equation

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作者：Gordon Blower^a ; ^{g.blower@lancaster.ac.uk" class="auth_mail" title="E-mail the corresponding author} ; Caroline Brett^a ; Ian Doust^b
关键词：37L55
刊名：Bulletin des Sciences Mathematiques
出版年：2016
出版时间：October 2016
年：2016
卷：140
期：7
页码：864-899
全文大小：626 K

文摘

This paper analyses the periodic spectrum of Schrödinger's equation n id="mmlsi1" class="mathmlsrc">n class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S000744971630032X&_mathId=si1.gif&_user=111111111&_pii=S000744971630032X&_rdoc=1&_issn=00074497&md5=31012ac290565197c8772a8887aa7e5e" title="Click to view the MathML source">−f^″+qf=λfn>n class="mathContainer hidden">n class="mathCode">

&minus; f^{″} + q f = λ f

n>n>n> when the potential is real, periodic, random and subject to the invariant measure n id="mmlsi2" class="mathmlsrc">nce?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S000744971630032X&_mathId=si2.gif&_user=111111111&_pii=S000744971630032X&_rdoc=1&_issn=00074497&md5=91df68437825a1e01782f4fe0e183c99">nlineImage" height="20" width="21" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S000744971630032X-si2.gif"><noscript>n:bottom" width="21" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S000744971630032X-si2.gif">noscript>n class="mathContainer hidden">n class="mathCode">

ν_{N}^{β}

n>n>n> of the periodic KdV equation. This n id="mmlsi2" class="mathmlsrc">nce?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S000744971630032X&_mathId=si2.gif&_user=111111111&_pii=S000744971630032X&_rdoc=1&_issn=00074497&md5=91df68437825a1e01782f4fe0e183c99">nlineImage" height="20" width="21" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S000744971630032X-si2.gif"><noscript>n:bottom" width="21" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S000744971630032X-si2.gif">noscript>n class="mathContainer hidden">n class="mathCode">

ν_{N}^{β}

n>n>n> is the modified canonical ensemble, as given by Bourgain (1994) n id="bbr0070">[7]n>, and n id="mmlsi2" class="mathmlsrc">nce?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S000744971630032X&_mathId=si2.gif&_user=111111111&_pii=S000744971630032X&_rdoc=1&_issn=00074497&md5=91df68437825a1e01782f4fe0e183c99">nlineImage" height="20" width="21" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S000744971630032X-si2.gif"><noscript>n:bottom" width="21" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S000744971630032X-si2.gif">noscript>n class="mathContainer hidden">n class="mathCode">

ν_{N}^{β}

n>n>n> satisfies a logarithmic Sobolev inequality. Associated concentration inequalities control the fluctuations of the periodic eigenvalues n id="mmlsi3" class="mathmlsrc">n class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S000744971630032X&_mathId=si3.gif&_user=111111111&_pii=S000744971630032X&_rdoc=1&_issn=00074497&md5=656df448ea453c418df7d21c1dfbc33e" title="Click to view the MathML source">(λ_n)n>n class="mathContainer hidden">n class="mathCode">

(λ_{n})

n>n>n>. For n id="mmlsi13" class="mathmlsrc">n class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S000744971630032X&_mathId=si13.gif&_user=111111111&_pii=S000744971630032X&_rdoc=1&_issn=00074497&md5=e02c075cf2e10bbfd34ab13bd6c90b71" title="Click to view the MathML source">β,N>0n>n class="mathContainer hidden">n class="mathCode">

β, N > n>0 n>

n>n>n> small, there exists a set of positive n id="mmlsi2" class="mathmlsrc">nce?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S000744971630032X&_mathId=si2.gif&_user=111111111&_pii=S000744971630032X&_rdoc=1&_issn=00074497&md5=91df68437825a1e01782f4fe0e183c99">nlineImage" height="20" width="21" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S000744971630032X-si2.gif"><noscript>n:bottom" width="21" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S000744971630032X-si2.gif">noscript>n class="mathContainer hidden">n class="mathCode">

ν_{N}^{β}

n>n>n> measure such that n id="mmlsi14" class="mathmlsrc">nce?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S000744971630032X&_mathId=si14.gif&_user=111111111&_pii=S000744971630032X&_rdoc=1&_issn=00074497&md5=cf16d5fffefa5eaeedcb94985b27c4f8">nlineImage" height="20" width="172" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S000744971630032X-si14.gif"><noscript>n:bottom" width="172" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S000744971630032X-si14.gif">noscript>n class="mathContainer hidden">n class="mathCode">

{(&plusmn; \sqrt{n>2 n> (λ_{n>2 n> n} + λ_{n>2 n> n &minus; n>1 n>})})}_{n = n>0 n>}^{&infin;}

n>n>n> gives a sampling sequence for Paley–Wiener space n id="mmlsi15" class="mathmlsrc">n class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S000744971630032X&_mathId=si15.gif&_user=111111111&_pii=S000744971630032X&_rdoc=1&_issn=00074497&md5=8862befd4f5e11be4c2f46e5021d6c2d" title="Click to view the MathML source">PW(π)n>n class="mathContainer hidden">n class="mathCode">

nt="italic">PW (π)

n>n>n> and the reproducing kernels give a Riesz basis. Let n id="mmlsi7" class="mathmlsrc">nce?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S000744971630032X&_mathId=si7.gif&_user=111111111&_pii=S000744971630032X&_rdoc=1&_issn=00074497&md5=c7d50c7aa6e91ac63afc13a8d19fb1f7">nlineImage" height="19" width="52" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S000744971630032X-si7.gif"><noscript>n:bottom" width="52" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S000744971630032X-si7.gif">noscript>n class="mathContainer hidden">n class="mathCode">

{(μ_{j})}_{j = n>1 n>}^{&infin;}

n>n>n> be the tied spectrum; then n id="mmlsi8" class="mathmlsrc">nce?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S000744971630032X&_mathId=si8.gif&_user=111111111&_pii=S000744971630032X&_rdoc=1&_issn=00074497&md5=51df1ff91aaec49945e78c8c439d6f22">nlineImage" height="19" width="78" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S000744971630032X-si8.gif"><noscript>n:bottom" width="78" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S000744971630032X-si8.gif">noscript>n class="mathContainer hidden">n class="mathCode">

(n>2 n> \sqrt{μ_{j}} &minus; j)

n>n>n> belongs to a Hilbert cube in n id="mmlsi162" class="mathmlsrc">n class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S000744971630032X&_mathId=si162.gif&_user=111111111&_pii=S000744971630032X&_rdoc=1&_issn=00074497&md5=7d68694686abec7e53e67f3fbc64fa62" title="Click to view the MathML source">ℓ²n>n class="mathContainer hidden">n class="mathCode">

ℓ^{n>2 n>}

n>n>n> and is distributed according to a measure that satisfies Gaussian concentration for Lipschitz functions. The sampling sequence n id="mmlsi10" class="mathmlsrc">nce?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S000744971630032X&_mathId=si10.gif&_user=111111111&_pii=S000744971630032X&_rdoc=1&_issn=00074497&md5=3b1c5234ee0bb939d4cdcb34bfa5b0c6">nlineImage" height="19" width="65" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S000744971630032X-si10.gif"><noscript>n:bottom" width="65" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S000744971630032X-si10.gif">noscript>n class="mathContainer hidden">n class="mathCode">

{(\sqrt{μ_{j}})}_{j = n>1 n>}^{&infin;}

n>n>n> arises from a divisor on the spectral curve, which is hyperelliptic of infinite genus. The linear statistics n id="mmlsi11" class="mathmlsrc">nce?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S000744971630032X&_mathId=si11.gif&_user=111111111&_pii=S000744971630032X&_rdoc=1&_issn=00074497&md5=a994b98e7ac30b213cac13f0066285d6">nlineImage" height="22" width="87" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S000744971630032X-si11.gif"><noscript>n:bottom" width="87" alt="View the MathML source" title="View the MathML source" src="http://origin-ars.els-cdn.com/content/image/1-s2.0-S000744971630032X-si11.gif">noscript>n class="mathContainer hidden">n class="mathCode">

\sum_{j} g (\sqrt{λ_{n>2 n> j}})

n>n>n> with test function n id="mmlsi12" class="mathmlsrc">n class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S000744971630032X&_mathId=si12.gif&_user=111111111&_pii=S000744971630032X&_rdoc=1&_issn=00074497&md5=0204ee31d291018209e8d7a42bd85d08" title="Click to view the MathML source">g∈PW(π)n>n class="mathContainer hidden">n class="mathCode">

g &isin; nt="italic">PW (π)

n>n>n> satisfy Gaussian concentration inequalities.

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