abstract:In this paper, we study the spectrum of the complex Hill operator $L={d^2over dx^2}+q(x;tau)$ in $L^2(Bbb{R},Bbb{C})$ with the Darboux-Treibich-Verdier potential $$ q(x;tau):=-sum_{k=0}^{3}n_{k}(n_{k}+1)wpleft(x+z_0+{omega_kover 2};tauright), $$ where $n_kinBbb{Z}_{geq 0}$ with $max n_kgeq 1$ and $z_0inBbb{C}$ is chosen such that $q(x;tau)$ has no singularities on $Bbb{R}$. For any fixed $tauin iBbb{R}_{>0}$, we give a necessary and sufficient condition on $(n_0,n_1,n_2,n_3)$ to guarantee that the spectrum $sigma(L)$ is $$ sigma(L)=big(-infty, E_{2g}big]cupbig[E_{2g-1},E_{2g-2}big]cupcdotscup[E_1,E_0],quad E_jinBbb{R}, $$ and hence generalizes Ince's remarkable result in 1940 for the Lam'{e} potential to the Darboux-Treibich-Verdier potential. We also determine the number of (anti)periodic eigenvalues in each bounded interval $(E_{2j-1},E_{2j-2})$, which generalizes the recent result by Haese-Hill et al., who studied the Lam'{e} case $n_1=n_2=n_3=0$.
{"title":"A necessary and sufficient condition for the Darboux-Treibich-Verdier potential with its spectrum contained in ℝ","authors":"Zhijie Chen, Erjuan Fu, Changshou Lin","doi":"10.1353/ajm.2022.0017","DOIUrl":"https://doi.org/10.1353/ajm.2022.0017","url":null,"abstract":"abstract:In this paper, we study the spectrum of the complex Hill operator $L={d^2over dx^2}+q(x;tau)$ in $L^2(Bbb{R},Bbb{C})$ with the Darboux-Treibich-Verdier potential $$ q(x;tau):=-sum_{k=0}^{3}n_{k}(n_{k}+1)wpleft(x+z_0+{omega_kover 2};tauright), $$ where $n_kinBbb{Z}_{geq 0}$ with $max n_kgeq 1$ and $z_0inBbb{C}$ is chosen such that $q(x;tau)$ has no singularities on $Bbb{R}$. For any fixed $tauin iBbb{R}_{>0}$, we give a necessary and sufficient condition on $(n_0,n_1,n_2,n_3)$ to guarantee that the spectrum $sigma(L)$ is $$ sigma(L)=big(-infty, E_{2g}big]cupbig[E_{2g-1},E_{2g-2}big]cupcdotscup[E_1,E_0],quad E_jinBbb{R}, $$ and hence generalizes Ince's remarkable result in 1940 for the Lam'{e} potential to the Darboux-Treibich-Verdier potential. We also determine the number of (anti)periodic eigenvalues in each bounded interval $(E_{2j-1},E_{2j-2})$, which generalizes the recent result by Haese-Hill et al., who studied the Lam'{e} case $n_1=n_2=n_3=0$.","PeriodicalId":7453,"journal":{"name":"American Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2022-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42068992","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
abstract:We consider the Schr"odinger evolution of strongly localized wave packets under the magnetic Laplacian in the plane $Bbb{R}^2$. When the initial energy is low, we obtain a precise control, in Schwartz seminorms, of the propagated states for times of order $1/hbar$, where $hbar$ is Planck's constant. In this semiclassical regime, we prove that the initial particle will always split into multiple coherent states, each one following the average dynamics of the guiding center motion but at its own speed, demonstrating a purely quantum ``ubiquity'' phenomenon.
{"title":"Long-time dynamics of coherent states in strong magnetic fields","authors":"Gr'egory Boil, San Vũ Ngọc","doi":"10.1353/ajm.2021.0045","DOIUrl":"https://doi.org/10.1353/ajm.2021.0045","url":null,"abstract":"abstract:We consider the Schr\"odinger evolution of strongly localized wave packets under the magnetic Laplacian in the plane $Bbb{R}^2$. When the initial energy is low, we obtain a precise control, in Schwartz seminorms, of the propagated states for times of order $1/hbar$, where $hbar$ is Planck's constant. In this semiclassical regime, we prove that the initial particle will always split into multiple coherent states, each one following the average dynamics of the guiding center motion but at its own speed, demonstrating a purely quantum ``ubiquity'' phenomenon.","PeriodicalId":7453,"journal":{"name":"American Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2021-12-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46294862","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We correct a mistake in the statement and proof of Lemma 2.3(d) in [{it Amer. J. Math.} {bf 120} (1998), no. 1, 1--21]. This in turn implies a change in Table 2.14.
{"title":"Corrigendum to \"Classification of Varieties with canonical curve section via Gaussian maps on canonical curves\"","authors":"C. Ciliberto, A. Lopez, R. Miranda","doi":"10.1353/ajm.2021.0042","DOIUrl":"https://doi.org/10.1353/ajm.2021.0042","url":null,"abstract":"<p>abstract:</p><p>We correct a mistake in the statement and proof of <related-article href=\"/article/773\" related-article-type=\"corrected-article\" xmlns:xlink=\"http://www.w3.org/1999/xlink\">Lemma 2.3(d)</related-article> in [{it Amer. J. Math.} {bf 120} (1998), no. 1, 1--21]. This in turn implies a change in Table 2.14.</p>","PeriodicalId":7453,"journal":{"name":"American Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2021-12-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47472555","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract:We investigate the Vorono"{i} summation problem for ${rm GL}_n$ in the level aspect for $ngeq 2$. Of particular interest are those primes at which the level and modulus are jointly ramified, a common occurrence in analytic number theory when using techniques such as the Petersson trace formula. Building on previous legacies, our formula stands as the most general of its kind; in particular we extend the results of Ichino-Templier. We also give classical refinements of our formula and study the $p$-adic generalisations of the Bessel transform.
{"title":"Voronoï summation for GLn: collusion between level and modulus","authors":"A. Corbett","doi":"10.1353/ajm.2021.0034","DOIUrl":"https://doi.org/10.1353/ajm.2021.0034","url":null,"abstract":"Abstract:We investigate the Vorono\"{i} summation problem for ${rm GL}_n$ in the level aspect for $ngeq 2$. Of particular interest are those primes at which the level and modulus are jointly ramified, a common occurrence in analytic number theory when using techniques such as the Petersson trace formula. Building on previous legacies, our formula stands as the most general of its kind; in particular we extend the results of Ichino-Templier. We also give classical refinements of our formula and study the $p$-adic generalisations of the Bessel transform.","PeriodicalId":7453,"journal":{"name":"American Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2021-10-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44755952","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
abstract:This paper concerns with the asymptotic behavior of complete non-compact convex curves embedded in $Bbb{R}^2$ under the $alpha$-curve shortening flow for exponents $alpha>{1over 2}$. We show that any such curve having in addition its two ends asymptotic to two parallel lines, converges under $alpha$-curve shortening flow to the unique translating soliton whose ends are asymptotic to the same parallel lines. This is a new result even in the standard case $alpha=1$, and we prove for all exponents up to the critical case $alpha>{1over 2}$.
{"title":"Convergence of curve shortening flow to translating soliton","authors":"Beomjun Choi, K. Choi, P. Daskalopoulos","doi":"10.1353/ajm.2021.0027","DOIUrl":"https://doi.org/10.1353/ajm.2021.0027","url":null,"abstract":"abstract:This paper concerns with the asymptotic behavior of complete non-compact convex curves embedded in $Bbb{R}^2$ under the $alpha$-curve shortening flow for exponents $alpha>{1over 2}$. We show that any such curve having in addition its two ends asymptotic to two parallel lines, converges under $alpha$-curve shortening flow to the unique translating soliton whose ends are asymptotic to the same parallel lines. This is a new result even in the standard case $alpha=1$, and we prove for all exponents up to the critical case $alpha>{1over 2}$.","PeriodicalId":7453,"journal":{"name":"American Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2021-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1353/ajm.2021.0027","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"66914873","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
abstract:We study infinite time blow-up phenomenon for the half-harmonic map flow $$ casesno{ u_t=-(-Delta)^{{1over 2}}u+bigg({1over 2pi}int_{Bbb{R}}{|u(x)-u(s)|^2over |x-s|^2}dsbigg)u&quad {rm in} Bbb{R}times(0,infty),cr u(cdot,0)=u_0&quad {rm in} Bbb{R}, } $$ for a smooth function $u:Bbb{R}times [0,infty)toBbb{S}^1$. Let $q_1,ldots,q_k$ be distinct points in $Bbb{R}$, there exist a smooth initial datum $u_0$ and smooth functions $xi_j(t)to q_j$, $0
{"title":"Infinite time blow-up for half-harmonic map flow from R into S1","authors":"Y. Sire, Juncheng Wei, Youquan Zheng","doi":"10.1353/ajm.2021.0031","DOIUrl":"https://doi.org/10.1353/ajm.2021.0031","url":null,"abstract":"abstract:We study infinite time blow-up phenomenon for the half-harmonic map flow $$ casesno{ u_t=-(-Delta)^{{1over 2}}u+bigg({1over 2pi}int_{Bbb{R}}{|u(x)-u(s)|^2over |x-s|^2}dsbigg)u&quad {rm in} Bbb{R}times(0,infty),cr u(cdot,0)=u_0&quad {rm in} Bbb{R}, } $$ for a smooth function $u:Bbb{R}times [0,infty)toBbb{S}^1$. Let $q_1,ldots,q_k$ be distinct points in $Bbb{R}$, there exist a smooth initial datum $u_0$ and smooth functions $xi_j(t)to q_j$, $0<mu_j(t)to 0$, as $tto+infty$, $j=1,ldots,k$, such that there exists a smooth solution $u_q$ of Problem (0.1) of the form $$ u_q=omega_infty+sum_{j=1}^kBigg(omegabigg({x-xi_j(t)over mu_j(t)}bigg)-omega_inftyBigg)+theta(x,t), $$ where $omega$ is the canonical least energy half-harmonic map, $omega_infty=big({0atop 1}big)$, $theta(x,t)to 0$ as $tto+infty$, uniformly away from the points $q_j$. In addition, the parameter functions $mu_j(t)$ decay to $0$ exponentially.","PeriodicalId":7453,"journal":{"name":"American Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2021-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1353/ajm.2021.0031","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44364641","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
abstract:We establish results on unique continuation at the boundary for the solutions of real analytic elliptic partial differential equations of the form $Lu=Delta u+a(x,y)partial_{x}u+b(x,y)partial_{y}u+c(x,y)u=0$ in two dimensions.
{"title":"Boundary unique continuation for a class of elliptic equations","authors":"S. Berhanu","doi":"10.1353/AJM.2021.0019","DOIUrl":"https://doi.org/10.1353/AJM.2021.0019","url":null,"abstract":"abstract:We establish results on unique continuation at the boundary for the solutions of real analytic elliptic partial differential equations of the form $Lu=Delta u+a(x,y)partial_{x}u+b(x,y)partial_{y}u+c(x,y)u=0$ in two dimensions.","PeriodicalId":7453,"journal":{"name":"American Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2021-06-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1353/AJM.2021.0019","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43029825","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Jingjun Han, Zhan Li, Lu Qi, Riccardo Brasca, Giovanni Rosso, Shanlin Huang, A. Soffer, S. Berhanu, H. Fan, Lei Ni, Qingsong Wang, F. Zheng, H. Grobner, Jie Lin, H. Gimperlein, M. Goffeng, P. Freitas, R. Laugesen, Richard Aoun, Cagri Sert
abstract:We show that the log canonical threshold polytopes of varieties with log canonical singularities satisfy the ascending chain condition.
文摘：我们证明了具有对数正则奇点的变种的对数正则阈值多面体满足升链条件。
{"title":"In Memoriam: J. Michael Boardman 1938–2021","authors":"Jingjun Han, Zhan Li, Lu Qi, Riccardo Brasca, Giovanni Rosso, Shanlin Huang, A. Soffer, S. Berhanu, H. Fan, Lei Ni, Qingsong Wang, F. Zheng, H. Grobner, Jie Lin, H. Gimperlein, M. Goffeng, P. Freitas, R. Laugesen, Richard Aoun, Cagri Sert","doi":"10.1353/ajm.2021.0015","DOIUrl":"https://doi.org/10.1353/ajm.2021.0015","url":null,"abstract":"abstract:We show that the log canonical threshold polytopes of varieties with log canonical singularities satisfy the ascending chain condition.","PeriodicalId":7453,"journal":{"name":"American Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2021-06-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1353/ajm.2021.0015","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44668853","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abstract:In this article we study the first eigenvalues of closed Riemann surfaces for large genus. We show that for every closed Riemann surface $X_g$ of genus $g$ $(ggeq 2)$, the first eigenvalue of $X_g$ is greater than ${cal L}_1(X_g)over g^2$ up to a uniform positive constant multiplication. Where ${cal L}_1(X_g)$ is the shortest length of multi closed curves separating $X_g$. Moreover,we also show that this new lower bound is optimal as $gtoinfty$.
{"title":"Optimal lower bounds for first eigenvalues of Riemann surfaces for large genus","authors":"Yunhui Wu, Yuhao Xue","doi":"10.1353/ajm.2022.0024","DOIUrl":"https://doi.org/10.1353/ajm.2022.0024","url":null,"abstract":"Abstract:In this article we study the first eigenvalues of closed Riemann surfaces for large genus. We show that for every closed Riemann surface $X_g$ of genus $g$ $(ggeq 2)$, the first eigenvalue of $X_g$ is greater than ${cal L}_1(X_g)over g^2$ up to a uniform positive constant multiplication. Where ${cal L}_1(X_g)$ is the shortest length of multi closed curves separating $X_g$. Moreover,we also show that this new lower bound is optimal as $gtoinfty$.","PeriodicalId":7453,"journal":{"name":"American Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2021-03-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42532870","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
abstract:This brief note describes an error in the following paper: Multiparameter Riesz commutators, {it Amer. J. Math.} {bf 131} (2009), no. 3, 731--769.A correction of this error seems to require new ideas, and has not been produced as of this note.
{"title":"Notification of error: Multiparameter Riesz commutators","authors":"M. Lacey, S. Petermichl, J. Pipher, B. Wick","doi":"10.1353/AJM.2021.0009","DOIUrl":"https://doi.org/10.1353/AJM.2021.0009","url":null,"abstract":"abstract:This brief note describes an error in the following paper: Multiparameter Riesz commutators, {it Amer. J. Math.} {bf 131} (2009), no. 3, 731--769.A correction of this error seems to require new ideas, and has not been produced as of this note.","PeriodicalId":7453,"journal":{"name":"American Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":1.7,"publicationDate":"2021-03-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1353/AJM.2021.0009","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46691386","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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