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21
Schrödinger Operators In The TwentyFirst Century
 Mathematical physics
, 2000
"... Introduction Yogi Berra is reputed to have said, "Prediction is difficult, especially about the future." Lists of open problems are typically lists of problems on which you expect progress in a reasonable time scale and so they involve an element of prediction. We have seen remarkable progress in ..."
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Cited by 21 (1 self)
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Introduction Yogi Berra is reputed to have said, "Prediction is difficult, especially about the future." Lists of open problems are typically lists of problems on which you expect progress in a reasonable time scale and so they involve an element of prediction. We have seen remarkable progress in the past fifty years in our understanding of Schrodinger operators, as I discussed in Simon [1]. In this companion piece, I present fifteen open problems. In 1984, I presented a list of open problem in Mathematical Physics, including thirteen in Schrodinger operators. Depending on how you count (since some are multiple), five have been solved. We will focus on two main areas: anomalous transport (Section 2) where I expect progress in my lifetime, and Coulomb energies where some of the problems are so vast and so far from current technology that I do not expect them to be solved in my lifetime. (There is a story behind the use of this phrase. I have heard that when Jeans lectured in
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.
Maximal inequalities and Riesz transform estimates on L p spaces for Schrödinger operators with nonnegative potentials, Ann. Inst. Fourier, Grenoble 57 no 6
, 2007
"... The paper concerns the magnetic Schrödinger operator H(a, V) = ..."
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Cited by 9 (3 self)
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The paper concerns the magnetic Schrödinger operator H(a, V) =
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
May 2006A Wavelet Based Sparse Grid Method for the Electronic Schrödinger Equation
"... Abstract. We present a direct discretization of the electronic Schrödinger equation. It is based on onedimensional Meyer wavelets from which we build an anisotropic multiresolution analysis for general particle spaces by a tensor product construction. We restrict these spaces to the case of antisym ..."
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Abstract. We present a direct discretization of the electronic Schrödinger equation. It is based on onedimensional Meyer wavelets from which we build an anisotropic multiresolution analysis for general particle spaces by a tensor product construction. We restrict these spaces to the case of antisymmetric functions. To obtain finitedimensional subspaces we first discuss semidiscretization with respect to the scale parameter by means of sparse grids which relies on mixed regularity and decay properties of the electronic wave functions. We then propose different techniques for a discretization with respect to the position parameter. Furthermore we present the results of our numerical experiments using this new generalized sparse grid methods for Schrödinger’s equation.
The extraordinary spectral properties of radially periodic Schrödinger operators
, 2002
"... Since it became clear that the band structure of the spectrum of periodic SturmLiouville operators t =(d )+q(r) does not survive a spherically symmetric extension to Schrödinger operators T =#+VwithV(x)=q(x) for x N\{1}, a wealth of detailed information about the spectrum of suc ..."
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Since it became clear that the band structure of the spectrum of periodic SturmLiouville operators t =(d )+q(r) does not survive a spherically symmetric extension to Schrödinger operators T =#+VwithV(x)=q(x) for x N\{1}, a wealth of detailed information about the spectrum of such operators has been acquired. The observation of eigenvalues embedded in the essential spectrum of T with exponentially decaying eigenfunctions provided evidence for the existence of intervals of dense point spectrum, eventually proved by spherical separation into perturbed SturmLiouville operators t c ). Subsequently, a numerical approach was employed to investigate the distribution of eigenvalues of T more closely. An eigenvalue was discovered below the essential spectrum in the case d 2, and it turned out that there are in fact infinitely many, accumulating at 0 . Moreover, a method based on oscillation theory made it possible to count eigenvalues of t c contributing to an interval of dense point spectrum of T . We gained evidence that an asymptotic formula, valid for c does in fact produce correct numbers even for small values of the coupling constant, such that a rather precise picture of the spectrum of radially periodic Schrödinger operators has now been obtained.