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Schrödinger operators in the twentyfirst century. www.math. caltech.edu/papers/bsimon/r40.ps
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Evolution of a Model Quantum System under Time Periodic Forcing
 Conditions for Complete Ionization, Comm. Math. Phys. 221, n
, 2001
"... Abstract: We analyze the time evolution of a onedimensional quantum system with an attractive delta function potential whose strength is subjected to a time periodic (zero mean) parametric variation η(t). We show that for generic η(t), which includes the sum of any finite number of harmonics, the s ..."
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Cited by 13 (5 self)
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Abstract: We analyze the time evolution of a onedimensional quantum system with an attractive delta function potential whose strength is subjected to a time periodic (zero mean) parametric variation η(t). We show that for generic η(t), which includes the sum of any finite number of harmonics, the system, started in a bound state will get fully ionized as t →∞. This is irrespective of the magnitude or frequency (resonant or not) of η(t). There are however exceptional, very nongeneric η(t), that do not lead to full ionization, which include rather simple explicit periodic functions. For these η(t) the system evolves to a nontrivial localized stationary state which is related to eigenfunctions of the Floquet operator.
A Szegö Condition for a Multidimensional Schrödinger Operator
, 2002
"... We consider spectral properties of a Schrödinger operator perturbed by a potential vanishing at infinity and prove that the corresponding spectral measure satisfies a Szegö type condition. ..."
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Cited by 9 (0 self)
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We consider spectral properties of a Schrödinger operator perturbed by a potential vanishing at infinity and prove that the corresponding spectral measure satisfies a Szegö type condition.
Recent developments in quantum mechanics with magnetic fields
 Proc. of Symposia in Pure Math. Vol 76 Spectral Theory and Mathematical Physics: A Festschrift in Honor of Barry Simon’s 60th Birthday Part
, 2006
"... We present a review on the recent developments concerning rigorous mathematical results on Schrödinger operators with magnetic fields. This paper is dedicated to the sixtieth birthday of Barry Simon. ..."
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Cited by 8 (0 self)
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We present a review on the recent developments concerning rigorous mathematical results on Schrödinger operators with magnetic fields. This paper is dedicated to the sixtieth birthday of Barry Simon.
Maximal inequalities and Riesz transform estimates on L p spaces for Schrödinger operators with nonnegative potentials
, 2006
"... We show various L p estimates for Schrödinger operators −∆+V on R n and their square roots. We assume reverse Hölder estimates on the potential, and improve some results of Shen [Sh1]. Our main tools are improved FeffermanPhong inequalities and reverse Hölder estimates for weak solutions of − ∆ + ..."
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Cited by 7 (3 self)
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We show various L p estimates for Schrödinger operators −∆+V on R n and their square roots. We assume reverse Hölder estimates on the potential, and improve some results of Shen [Sh1]. Our main tools are improved FeffermanPhong inequalities and reverse Hölder estimates for weak solutions of − ∆ + V and their gradients.
ON SOME SCHRÖDINGER AND WAVE EQUATIONS WITH TIME DEPENDENT POTENTIALS
"... Abstract. The existence and uniqueness of the initial value problem for Schrödinger and wave equations in the presence of a (large) time dependent potential is studied. The usual Strichartz estimates for such linear evolutions are shown to hold true with optimal assumptions on the potentials. As a b ..."
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Cited by 4 (0 self)
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Abstract. The existence and uniqueness of the initial value problem for Schrödinger and wave equations in the presence of a (large) time dependent potential is studied. The usual Strichartz estimates for such linear evolutions are shown to hold true with optimal assumptions on the potentials. As a byproduct, one obtains a counterexample to the two dimensional double endpoint inhomogeneous Strichartz estimate. 1.
Essential selfadjointness of Schrödinger type operators on manifolds
 RUSS. MATH. SURVEYS
, 2002
"... We obtain several essential selfadjointness conditions for the Schrödinger type operator HV = D ∗ D + V, where D is a first order elliptic differential operator acting on the space of sections of a hermitian vector bundle E over a manifold M with positive smooth measure dµ, and V is a Hermitian bu ..."
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Cited by 4 (1 self)
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We obtain several essential selfadjointness conditions for the Schrödinger type operator HV = D ∗ D + V, where D is a first order elliptic differential operator acting on the space of sections of a hermitian vector bundle E over a manifold M with positive smooth measure dµ, and V is a Hermitian bundle endomorphism. These conditions are expressed in terms of completeness of certain metrics on M naturally associated with HV. These results generalize the
Global stability of travelling fronts for a damped wave equation with bistable nonlinearity
, 2007
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unknown title
, 2006
"... Topics in the theory of positive solutions of secondorder elliptic and parabolic partial differential equations ..."
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Topics in the theory of positive solutions of secondorder elliptic and parabolic partial differential equations