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37
ON ASYMPTOTIC BEHAVIOR OF SOLUTIONS TO SCHRÖDINGER EQUATIONS WITH SINGULAR DIPOLE–TYPE POTENTIALS
, 2007
"... Abstract. Asymptotics of solutions to Schrödinger equations with singular dipoletype potentials is investigated. We evaluate the exact behavior near the singularity of solutions to elliptic equations with potentials which are purely angular multiples of radial inversesquare functions. Both the lin ..."
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Cited by 28 (20 self)
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Abstract. Asymptotics of solutions to Schrödinger equations with singular dipoletype potentials is investigated. We evaluate the exact behavior near the singularity of solutions to elliptic equations with potentials which are purely angular multiples of radial inversesquare functions. Both the linear and the semilinear (critical and subcritical) cases are considered. Dedicated to Prof. Norman Dancer on the occasion of his 60th birthday.
Absolutely continuous spectrum of Schrödinger operators with slowly decaying and oscillating potentials
 Comm. Math. Phys
"... ABSTRACT. The aim of this paper is to extend a class of potentials for which the absolutely continuous spectrum of the corresponding multidimensional Schrödinger operator is essentially supported by [0,∞). Our main theorem states that this property is preserved for slowly decaying potentials provi ..."
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Cited by 18 (1 self)
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ABSTRACT. The aim of this paper is to extend a class of potentials for which the absolutely continuous spectrum of the corresponding multidimensional Schrödinger operator is essentially supported by [0,∞). Our main theorem states that this property is preserved for slowly decaying potentials provided that there are some oscillations with respect to one of the variables. 1.
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 17 (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.
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 16 (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.
A Szegő 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 Szegő type condition. ..."
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Cited by 13 (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 Szegő type condition.
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 13 (7 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
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 8 (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.