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27
Sharp inequalities for heat kernels of Schrödinger operators and applications to spectral gaps
"... This paper derives inequalities for multiple integrals from which sharp inequalities for ratios of heat kernels and integrals of heat kernels of certain Schrodinger operators follow. Such ratio inequalities imply sharp inequalities for spectral gaps. The multiple integral inequalities, although very ..."
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Cited by 20 (4 self)
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This paper derives inequalities for multiple integrals from which sharp inequalities for ratios of heat kernels and integrals of heat kernels of certain Schrodinger operators follow. Such ratio inequalities imply sharp inequalities for spectral gaps. The multiple integral inequalities, although very di#erent, are motivated by the now classical Brascamp LiebLuttinger rearrangement inequalities. Contents 1. Introduction and statement of results 2. An inequality for multiple integrals 3. Proof of Theorem 1: Inequalities for integrals of heat kernels 4. Proofs of Corollaries 1 and 2 and consequences for nodal lines 5. A Schr odinger operator with a closed nodal line 6. Pointwise inequalities for heat kennels # supported in part by NSF grant # 9700585DMS + supported in part by Purdue Research Foundation grant # 69013953149 1 <E918> 1 Introduction and statement of results. Let D be a bounded domain in R n and let V be a nonnegative bounded potential in D. It is we...
A unified approach to universal inequalities for eigenvalues of elliptic operators
"... We present an abstract approach to universal inequalities for the discrete spectrum of a selfadjoint operator, based on commutator algebra, the Rayleigh–Ritz principle, and one set of “auxiliary ” operators. The new proof unifies classical inequalities of Payne–Pólya–Weinberger, Hile–Protter, and H ..."
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Cited by 15 (5 self)
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We present an abstract approach to universal inequalities for the discrete spectrum of a selfadjoint operator, based on commutator algebra, the Rayleigh–Ritz principle, and one set of “auxiliary ” operators. The new proof unifies classical inequalities of Payne–Pólya–Weinberger, Hile–Protter, and H.C. Yang and provides a Yang type strengthening of Hook’s bounds for various elliptic operators with Dirichlet boundary conditions. The proof avoids the introduction of the “free parameters ” of many previous authors and relies on earlier works of Ashbaugh and Benguria, and, especially, Harrell (alone and with Michel), in addition to those of the other authors listed above. The Yang type inequality is proved to be stronger under general conditions on the operator and the auxiliary operators. This approach provides an alternative route to recent results obtained by Harrell and Stubbe. 1.
ON THE SHAPE OF THE GROUND STATE EIGENFUNCTION FOR STABLE PROCESSES
, 2004
"... MÉNDEZHERNÁNDEZ Abstract. We prove that the ground state eigenfunction for symmetric stable processes of order α ∈ (0, 2) killed upon leaving the interval (−1, 1) is concave on (−1 1 ..."
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Cited by 13 (6 self)
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MÉNDEZHERNÁNDEZ Abstract. We prove that the ground state eigenfunction for symmetric stable processes of order α ∈ (0, 2) killed upon leaving the interval (−1, 1) is concave on (−1 1
Spectral estimates and nonselfadjoint perturbations of spheroidal wave operators,” mathph/0405010
, 2006
"... We derive a spectral representation for the oblate spheroidal wave operator which is holomorphic in the aspherical parameter Ω in a neighborhood of the real line. For real Ω, estimates are derived for all eigenvalue gaps uniformly in Ω. The proof of the gap estimates is based on detailed estimates f ..."
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Cited by 11 (9 self)
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We derive a spectral representation for the oblate spheroidal wave operator which is holomorphic in the aspherical parameter Ω in a neighborhood of the real line. For real Ω, estimates are derived for all eigenvalue gaps uniformly in Ω. The proof of the gap estimates is based on detailed estimates for complex solutions of the Riccati equation. The spectral representation for complex Ω is obtained using the theory of slightly nonselfadjoint perturbations. 1
An estimate of the gap of the first two eigenvalues in the Schrödinger operator. Lectures on partial differential equations
 223–235, New Stud. Adv. Math
, 2003
"... In my previous paper [2] with I. M. Singer, B. Wong and Stephen Yau, I gave a lower estimate of the gap of the first 2 eigenvalues of the Schrödinger operator in case the potential is convex. In this note we note that the estimate can be improved if we assume the potential is strongly convex. In par ..."
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Cited by 11 (1 self)
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In my previous paper [2] with I. M. Singer, B. Wong and Stephen Yau, I gave a lower estimate of the gap of the first 2 eigenvalues of the Schrödinger operator in case the potential is convex. In this note we note that the estimate can be improved if we assume the potential is strongly convex. In particular if the Hessian of the potential is bounded from below by a positive constant, the gap
Geometrical structure of Laplacian eigenfunctions
, 2013
"... We summarize the properties of eigenvalues and eigenfunctions of the Laplace operator in bounded Euclidean domains with Dirichlet, Neumann, or Robin boundary condition. We keep the presentation at a level accessible to scientists from various disciplines ranging from mathematics to physics and com ..."
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Cited by 10 (3 self)
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We summarize the properties of eigenvalues and eigenfunctions of the Laplace operator in bounded Euclidean domains with Dirichlet, Neumann, or Robin boundary condition. We keep the presentation at a level accessible to scientists from various disciplines ranging from mathematics to physics and computer sciences. The main focus is placed onto multiple intricate relations between the shape of a domain and the geometrical structure of eigenfunctions.
The AllegrettoPiepenbrink Theorem for Strongly Local Dirichlet Forms
 DOCUMENTA MATH.
, 2009
"... The existence of positive weak solutions is related to spectral information on the corresponding partial differential operator. ..."
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Cited by 8 (6 self)
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The existence of positive weak solutions is related to spectral information on the corresponding partial differential operator.
Generalized eigenfunctions and spectral theory for strongly local Dirichlet forms
, 2009
"... We present an introduction to the framework of strongly local Dirichlet forms and discuss connections between the existence of certain generalized eigenfunctions and spectral properties within this framework. The range of applications is illustrated by a list of examples. ..."
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Cited by 8 (5 self)
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We present an introduction to the framework of strongly local Dirichlet forms and discuss connections between the existence of certain generalized eigenfunctions and spectral properties within this framework. The range of applications is illustrated by a list of examples.
Optimal bounds for ratios of eigenvalues of onedimensional Schrödinger operators with Dirichlet boundary conditions and positive
, 1989
"... Abstract. Consider the Schrodinger equation — u " + V(x)u = λu on the interval la [R, where K(x)^0 for xel and where Dirichlet boundary conditions are imposed at the endpoints of /. We prove the optimal boundf^n2 for n = 2,3,4,... λ1 on the ratio of the nih eigenvalue to the first eigenvalue ..."
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Cited by 6 (0 self)
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Abstract. Consider the Schrodinger equation — u " + V(x)u = λu on the interval la [R, where K(x)^0 for xel and where Dirichlet boundary conditions are imposed at the endpoints of /. We prove the optimal boundf^n2 for n = 2,3,4,... λ1 on the ratio of the nih eigenvalue to the first eigenvalue for this problem. This leads to a complete treatment of bounds on ratios of eigenvalues for such problems. Extensions of these results to singular problems are also presented. A modified Priifer transformation and comparison techniques are the key elements of the proof. 1.
GEOMETRIC ANALYSIS
, 2005
"... This was a talk I gave in the occasion of the seventieth anniversary of the Chinese Mathematical Society. I dedicated it in memory of my teacher S. S. Chern who passed away half a year ago. During my graduate study, I was rather free in picking research topics. I [538] worked on fundamental groups o ..."
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Cited by 4 (0 self)
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This was a talk I gave in the occasion of the seventieth anniversary of the Chinese Mathematical Society. I dedicated it in memory of my teacher S. S. Chern who passed away half a year ago. During my graduate study, I was rather free in picking research topics. I [538] worked on fundamental groups of manifolds with nonpositive curvature. But in the second year of my study, I started to look into differential equations on manifolds. While Chern did not express much opinions on this part of my research, he started to appreciate it a few years later. In fact, after Chern gave a course on Calabi’s works on affine geometry in 1972 in Berkeley, Cheng told me these inspiring lectures. By 1973, Cheng and I started to work on some problems mentioned in his lectures. We did not realize that great geometers Pogorelov, Calabi and Nirenberg were also working on them. We were excited that we solved some of the conjectures of Calabi on improper affine spheres. But soon we found out that Pogorelov [398] published it right before us by different arguments. Nevertheless our ideas are useful to handle other problems in