Pub Date : 2023-11-11DOI: 10.2140/apde.2023.16.1989
Miguel Domínguez-Vázquez, Alberto Enciso, Daniel Peralta-Salas
We consider the overdetermined boundary problem for a general second-order semilinear elliptic equation on bounded domains of , where one prescribes both the Dirichlet and Neumann data of the solution. We are interested in the case where the data are not necessarily constant and where the coefficients of the equation can depend on the position, so that the overdetermined problem does not generally admit a radial solution. Our main result is that, nevertheless, under minor technical hypotheses nontrivial solutions to the overdetermined boundary problem always exist.
{"title":"Overdetermined boundary problems with nonconstant Dirichlet and Neumann data","authors":"Miguel Domínguez-Vázquez, Alberto Enciso, Daniel Peralta-Salas","doi":"10.2140/apde.2023.16.1989","DOIUrl":"https://doi.org/10.2140/apde.2023.16.1989","url":null,"abstract":"<p>We consider the overdetermined boundary problem for a general second-order semilinear elliptic equation on bounded domains of <math display=\"inline\" xmlns=\"http://www.w3.org/1998/Math/MathML\"><msup><mrow><mi>ℝ</mi></mrow><mrow><mi>n</mi></mrow></msup> </math>, where one prescribes both the Dirichlet and Neumann data of the solution. We are interested in the case where the data are not necessarily constant and where the coefficients of the equation can depend on the position, so that the overdetermined problem does not generally admit a radial solution. Our main result is that, nevertheless, under minor technical hypotheses nontrivial solutions to the overdetermined boundary problem always exist. </p>","PeriodicalId":49277,"journal":{"name":"Analysis & PDE","volume":null,"pages":null},"PeriodicalIF":2.2,"publicationDate":"2023-11-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138531078","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}
Pub Date : 2023-10-16DOI: 10.2140/apde.2023.16.1799
Norio Nawata
Combing Elliott, Gong, Lin and Niu's result and Castillejos and Evington's result, we see that if $A$ is a simple separable nuclear monotracial C$^*$-algebra, then $Aotimesmathcal{W}$ is isomorphic to $mathcal{W}$ where $mathcal{W}$ is the Razak-Jacelon algebra. In this paper, we give another proof of this. In particular, we show that if $mathcal{D}$ is a simple separable nuclear monotracial $M_{2^{infty}}$-stable C$^*$-algebra which is $KK$-equivalent to ${0}$, then $mathcal{D}$ is isomorphic to $mathcal{W}$ without considering tracial approximations of C$^*$-algebras with finite nuclear dimension. Our proof is based on Matui and Sato's technique, Schafhauser's idea in his proof of the Tikuisis-White-Winter theorem and properties of Kirchberg's central sequence C$^*$-algebra $F(mathcal{D})$ of $mathcal{D}$. Note that some results for $F(mathcal{D})$ is based on Elliott-Gong-Lin-Niu's stable uniqueness theorem. Also, we characterize $mathcal{W}$ by using properties of $F(mathcal{W})$. Indeed, we show that a simple separable nuclear monotracial C$^*$-algebra $D$ is isomorphic to $mathcal{W}$ if and only if $D$ satisfies the following properties:(i) for any $thetain [0,1]$, there exists a projection $p$ in $F(D)$ such that $tau_{D, omega}(p)=theta$,(ii) if $p$ and $q$ are projections in $F(D)$ such that $0
{"title":"A characterization of the Razak–Jacelon algebra","authors":"Norio Nawata","doi":"10.2140/apde.2023.16.1799","DOIUrl":"https://doi.org/10.2140/apde.2023.16.1799","url":null,"abstract":"Combing Elliott, Gong, Lin and Niu's result and Castillejos and Evington's result, we see that if $A$ is a simple separable nuclear monotracial C$^*$-algebra, then $Aotimesmathcal{W}$ is isomorphic to $mathcal{W}$ where $mathcal{W}$ is the Razak-Jacelon algebra. In this paper, we give another proof of this. In particular, we show that if $mathcal{D}$ is a simple separable nuclear monotracial $M_{2^{infty}}$-stable C$^*$-algebra which is $KK$-equivalent to ${0}$, then $mathcal{D}$ is isomorphic to $mathcal{W}$ without considering tracial approximations of C$^*$-algebras with finite nuclear dimension. Our proof is based on Matui and Sato's technique, Schafhauser's idea in his proof of the Tikuisis-White-Winter theorem and properties of Kirchberg's central sequence C$^*$-algebra $F(mathcal{D})$ of $mathcal{D}$. Note that some results for $F(mathcal{D})$ is based on Elliott-Gong-Lin-Niu's stable uniqueness theorem. Also, we characterize $mathcal{W}$ by using properties of $F(mathcal{W})$. Indeed, we show that a simple separable nuclear monotracial C$^*$-algebra $D$ is isomorphic to $mathcal{W}$ if and only if $D$ satisfies the following properties:(i) for any $thetain [0,1]$, there exists a projection $p$ in $F(D)$ such that $tau_{D, omega}(p)=theta$,(ii) if $p$ and $q$ are projections in $F(D)$ such that $0<tau_{D, omega}(p)=tau_{D, omega}(q)$, then $p$ is Murray-von Neumann equivalent to $q$,(iii) there exists a homomorphism from $D$ to $mathcal{W}$.","PeriodicalId":49277,"journal":{"name":"Analysis & PDE","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136182101","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}
Pub Date : 2023-10-16DOI: 10.2140/apde.2023.16.1955
Umberto Guarnotta, Sunra Mosconi
For solutions of ${rm div},(DF(Du))=f$ we show that the quasiconformality of $zmapsto DF(z)$ is the key property leading to the Sobolev regularity of the stress field $DF(Du)$, in relation with the summability of $f$. This class of nonlinearities encodes in a general way the notion of uniform ellipticity and encompasses all known instances where the stress field is known to be Sobolev regular. We provide examples showing the optimality of this assumption and present three applications: the study of the strong locality of the operator ${rm div},(DF(Du))$, a nonlinear Cordes condition for equations in divergence form, and some partial results on the $C^{p'}$-conjecture.
{"title":"A general notion of uniform ellipticity and the regularity of the stress field for elliptic equations in divergence form","authors":"Umberto Guarnotta, Sunra Mosconi","doi":"10.2140/apde.2023.16.1955","DOIUrl":"https://doi.org/10.2140/apde.2023.16.1955","url":null,"abstract":"For solutions of ${rm div},(DF(Du))=f$ we show that the quasiconformality of $zmapsto DF(z)$ is the key property leading to the Sobolev regularity of the stress field $DF(Du)$, in relation with the summability of $f$. This class of nonlinearities encodes in a general way the notion of uniform ellipticity and encompasses all known instances where the stress field is known to be Sobolev regular. We provide examples showing the optimality of this assumption and present three applications: the study of the strong locality of the operator ${rm div},(DF(Du))$, a nonlinear Cordes condition for equations in divergence form, and some partial results on the $C^{p'}$-conjecture.","PeriodicalId":49277,"journal":{"name":"Analysis & PDE","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136077709","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}
Pub Date : 2023-10-16DOI: 10.2140/apde.2023.16.1885
Alessandro Olgiati, Nicolas Rougerie, Dominique Spehner
We study the ground state for many interacting bosons in a double-well potential, in a joint limit where the particle number and the distance between the potential wells both go to infinity. Two single-particle orbitals (one for each well) are macroscopically occupied, and we are concerned with deriving the corresponding effective Bose-Hubbard Hamiltonian. We prove (i) an energy expansion, including the two-modes Bose-Hubbard energy and two independent Bogoliubov corrections (one for each potential well), (ii) a variance bound for the number of particles falling inside each potential well. The latter is a signature of a correlated ground state in that it violates the central limit theorem.
{"title":"Bosons in a double well: two-mode approximation and fluctuations","authors":"Alessandro Olgiati, Nicolas Rougerie, Dominique Spehner","doi":"10.2140/apde.2023.16.1885","DOIUrl":"https://doi.org/10.2140/apde.2023.16.1885","url":null,"abstract":"We study the ground state for many interacting bosons in a double-well potential, in a joint limit where the particle number and the distance between the potential wells both go to infinity. Two single-particle orbitals (one for each well) are macroscopically occupied, and we are concerned with deriving the corresponding effective Bose-Hubbard Hamiltonian. We prove (i) an energy expansion, including the two-modes Bose-Hubbard energy and two independent Bogoliubov corrections (one for each potential well), (ii) a variance bound for the number of particles falling inside each potential well. The latter is a signature of a correlated ground state in that it violates the central limit theorem.","PeriodicalId":49277,"journal":{"name":"Analysis & PDE","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136078199","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}
Pub Date : 2023-10-16DOI: 10.2140/apde.2023.16.1825
Katya Krupchyk, Gunther Uhlmann
We study the inverse boundary problem for a nonlinear magnetic Schrodinger operator on a conformally transversally anisotropic Riemannian manifold of dimension $nge 3$. Under suitable assumptions on the nonlinearity, we show that the knowledge of the Dirichlet-to-Neumann map on the boundary of the manifold determines the nonlinear magnetic and electric potentials uniquely. No assumptions on the transversal manifold are made in this result, whereas the corresponding inverse boundary problem for the linear magnetic Schrodinger operator is still open in this generality.
{"title":"Inverse problems for nonlinear magnetic Schrödinger equations on conformally transversally anisotropic manifolds","authors":"Katya Krupchyk, Gunther Uhlmann","doi":"10.2140/apde.2023.16.1825","DOIUrl":"https://doi.org/10.2140/apde.2023.16.1825","url":null,"abstract":"We study the inverse boundary problem for a nonlinear magnetic Schrodinger operator on a conformally transversally anisotropic Riemannian manifold of dimension $nge 3$. Under suitable assumptions on the nonlinearity, we show that the knowledge of the Dirichlet-to-Neumann map on the boundary of the manifold determines the nonlinear magnetic and electric potentials uniquely. No assumptions on the transversal manifold are made in this result, whereas the corresponding inverse boundary problem for the linear magnetic Schrodinger operator is still open in this generality.","PeriodicalId":49277,"journal":{"name":"Analysis & PDE","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136078196","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}
Pub Date : 2023-10-16DOI: 10.2140/apde.2023.16.1869
Leif Arkeryd, Anne Nouri
The paper proves existence of mild solutions to normal discrete velocity Boltzmann equations sin the plane with no pair of colinear interacting velocities, and given ingoing boundary values. A key property is L1 compactness of the integrated collision frequency for a sequence of approximations. This is proved using the Kolmogorov-Riesz theorem, which here replaces the L1 compactness of velocity averages, not available when the velocities are discrete.
{"title":"Discrete velocity Boltzmann equations in the plane: stationary solutions","authors":"Leif Arkeryd, Anne Nouri","doi":"10.2140/apde.2023.16.1869","DOIUrl":"https://doi.org/10.2140/apde.2023.16.1869","url":null,"abstract":"The paper proves existence of mild solutions to normal discrete velocity Boltzmann equations sin the plane with no pair of colinear interacting velocities, and given ingoing boundary values. A key property is L1 compactness of the integrated collision frequency for a sequence of approximations. This is proved using the Kolmogorov-Riesz theorem, which here replaces the L1 compactness of velocity averages, not available when the velocities are discrete.","PeriodicalId":49277,"journal":{"name":"Analysis & PDE","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136077519","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}
Pub Date : 2023-10-16DOI: 10.2140/apde.2023.16.1745
Michele Correggi, Marco Falconi, Marco Olivieri
We study the ground state energy and ground states of systems coupling non-relativistic quantum particles and force-carrying Bose fields, such as radiation, in the quasi-classical approximation. The latter is very useful whenever the force-carrying field has a very large number of excitations,and thus behaves in a semiclassical way, while the non-relativistic particles, on the other hand, retain their microscopic features. We prove that the ground state energy of the fully microscopic model converges to the one of a nonlinear quasi-classical functional depending on both the particles' wave function and the classical configuration of the field. Equivalently, this energy can be interpreted as the lowest energy of a Pekar-like functional with an effective nonlinear interaction for the particles only. If the particles are confined, the ground state of the microscopic system converges as well, to a probability measure concentrated on the set of minimizers of the quasi-classical energy.
{"title":"Ground state properties in the quasiclassical regime","authors":"Michele Correggi, Marco Falconi, Marco Olivieri","doi":"10.2140/apde.2023.16.1745","DOIUrl":"https://doi.org/10.2140/apde.2023.16.1745","url":null,"abstract":"We study the ground state energy and ground states of systems coupling non-relativistic quantum particles and force-carrying Bose fields, such as radiation, in the quasi-classical approximation. The latter is very useful whenever the force-carrying field has a very large number of excitations,and thus behaves in a semiclassical way, while the non-relativistic particles, on the other hand, retain their microscopic features. We prove that the ground state energy of the fully microscopic model converges to the one of a nonlinear quasi-classical functional depending on both the particles' wave function and the classical configuration of the field. Equivalently, this energy can be interpreted as the lowest energy of a Pekar-like functional with an effective nonlinear interaction for the particles only. If the particles are confined, the ground state of the microscopic system converges as well, to a probability measure concentrated on the set of minimizers of the quasi-classical energy.","PeriodicalId":49277,"journal":{"name":"Analysis & PDE","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136077705","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}
Pub Date : 2023-09-21DOI: 10.2140/apde.2023.16.1547
Malte Litsgård, Kaj Nyström
We consider the operators [ nabla_Xcdot(A(X)nabla_X), nabla_Xcdot(A(X)nabla_X)-partial_t, nabla_Xcdot(A(X)nabla_X)+Xcdotnabla_Y-partial_t, ] where $Xin Omega$, $(X,t)in Omegatimes mathbb R$ and $(X,Y,t)in Omegatimes mathbb R^mtimes mathbb R$, respectively, and where $Omegasubsetmathbb R^m$ is a (unbounded) Lipschitz domain with defining function $psi:mathbb R^{m-1}tomathbb R$ being Lipschitz with constant bounded by $M$. Assume that the elliptic measure associated to the first of these operators is mutually absolutely continuous with respect to the surface measure $mathrm{d} sigma(X)$, and that the corresponding Radon-Nikodym derivative or Poisson kernel satisfies a scale invariant reverse H"older inequalities in $L^p$, for some fixed $p$, $1
{"title":"A structure theorem for elliptic and parabolic operators with applications to homogenization of operators of Kolmogorov type","authors":"Malte Litsgård, Kaj Nyström","doi":"10.2140/apde.2023.16.1547","DOIUrl":"https://doi.org/10.2140/apde.2023.16.1547","url":null,"abstract":"We consider the operators [ nabla_Xcdot(A(X)nabla_X), nabla_Xcdot(A(X)nabla_X)-partial_t, nabla_Xcdot(A(X)nabla_X)+Xcdotnabla_Y-partial_t, ] where $Xin Omega$, $(X,t)in Omegatimes mathbb R$ and $(X,Y,t)in Omegatimes mathbb R^mtimes mathbb R$, respectively, and where $Omegasubsetmathbb R^m$ is a (unbounded) Lipschitz domain with defining function $psi:mathbb R^{m-1}tomathbb R$ being Lipschitz with constant bounded by $M$. Assume that the elliptic measure associated to the first of these operators is mutually absolutely continuous with respect to the surface measure $mathrm{d} sigma(X)$, and that the corresponding Radon-Nikodym derivative or Poisson kernel satisfies a scale invariant reverse H\"older inequalities in $L^p$, for some fixed $p$, $1<p<infty$, with constants depending only on the constants of $A$, $m$ and the Lipschitz constant of $psi$, $M$. Under this assumption we prove that then the same conclusions are also true for the parabolic measures associated to the second and third operator with $mathrm{d} sigma(X)$ replaced by the surface measures $mathrm{d} sigma(X)mathrm{d} t$ and $mathrm{d} sigma(X)mathrm{d} Ymathrm{d} t$, respectively. This structural theorem allows us to reprove several results previously established in the literature as well as to deduce new results in, for example, the context of homogenization for operators of Kolmogorov type. Our proof of the structural theorem is based on recent results established by the authors concerning boundary Harnack inequalities for operators of Kolmogorov type in divergence form with bounded, measurable and uniformly elliptic coefficients.","PeriodicalId":49277,"journal":{"name":"Analysis & PDE","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136236991","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}
Pub Date : 2023-09-21DOI: 10.2140/apde.2023.16.1485
Alex Iosevich, Akos Magyar
Let $De$ be a non-degenerate simplex on $k$ vertices. We prove that there exists a threshold $s_k
设$De$为$k$顶点上的非退化单纯形。我们证明了存在一个阈值$s_k
{"title":"Simplices in thin subsets of Euclidean spaces","authors":"Alex Iosevich, Akos Magyar","doi":"10.2140/apde.2023.16.1485","DOIUrl":"https://doi.org/10.2140/apde.2023.16.1485","url":null,"abstract":"Let $De$ be a non-degenerate simplex on $k$ vertices. We prove that there exists a threshold $s_k<k$ such that any set $Asubs R^k$ of Hausdorff dimension $dim,Ageq s_k$ necessarily contains a similar copy of the simplex $De$.","PeriodicalId":49277,"journal":{"name":"Analysis & PDE","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136101918","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}
Pub Date : 2023-09-21DOI: 10.2140/apde.2023.16.1497
Christiansen, T. J.
We study the resonances of Schr"odinger operators on the infinite product $X=mathbb{R}^dtimes mathbb{S}^1$, where $d$ is odd, $mathbb{S}^1$ is the unit circle, and the potential $Vin L^infty_c(X)$. This paper shows that at high energy, resonances of the Schr"odinger operator $-Delta +V$ on $X=mathbb{R}^dtimes mathbb{S}^1$ which are near the continuous spectrum are approximated by the resonances of $-Delta +V_0$ on $X$, where the potential $V_0$ given by averaging $V$ over the unit circle. These resonances are, in turn, given in terms of the resonances of a Schr"odinger operator on $mathbb{R}^d$ which lie in a bounded set. If the potential is smooth, we obtain improved localization of the resonances, particularly in the case of simple, rank one poles of the corresponding scattering resolvent on $mathbb{R}^d$. In that case, we obtain the leading order correction for the location of the corresponding high energy resonances. In addition to direct results about the location of resonances, we show that at high energies away from the resonances, the resolvent of the model operator $-Delta+V_0$ on $X$ approximates that of $-Delta+V$ on $X$. If $d=1$, in certain cases this implies the existence of an asymptotic expansion of solutions of the wave equation. Again for the special case of $d=1$, we obtain a resonant rigidity type result for the zero potential among all real-valued potentials.
{"title":"Resonances for Schrödinger operators on infinite cylinders and other products","authors":"Christiansen, T. J.","doi":"10.2140/apde.2023.16.1497","DOIUrl":"https://doi.org/10.2140/apde.2023.16.1497","url":null,"abstract":"We study the resonances of Schr\"odinger operators on the infinite product $X=mathbb{R}^dtimes mathbb{S}^1$, where $d$ is odd, $mathbb{S}^1$ is the unit circle, and the potential $Vin L^infty_c(X)$. This paper shows that at high energy, resonances of the Schr\"odinger operator $-Delta +V$ on $X=mathbb{R}^dtimes mathbb{S}^1$ which are near the continuous spectrum are approximated by the resonances of $-Delta +V_0$ on $X$, where the potential $V_0$ given by averaging $V$ over the unit circle. These resonances are, in turn, given in terms of the resonances of a Schr\"odinger operator on $mathbb{R}^d$ which lie in a bounded set. If the potential is smooth, we obtain improved localization of the resonances, particularly in the case of simple, rank one poles of the corresponding scattering resolvent on $mathbb{R}^d$. In that case, we obtain the leading order correction for the location of the corresponding high energy resonances. In addition to direct results about the location of resonances, we show that at high energies away from the resonances, the resolvent of the model operator $-Delta+V_0$ on $X$ approximates that of $-Delta+V$ on $X$. If $d=1$, in certain cases this implies the existence of an asymptotic expansion of solutions of the wave equation. Again for the special case of $d=1$, we obtain a resonant rigidity type result for the zero potential among all real-valued potentials.","PeriodicalId":49277,"journal":{"name":"Analysis & PDE","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136102136","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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