Groundstates for the nonlinear Schrödinger equation with potential vanishing at infinity

Groundstates of the stationary nonlinear Schrödinger equation $-\Delta u +V u =K u^{p-1}$ , are studied when the nonnegative function V and K are neither bounded away from zero, nor bounded from above. A special attention is paid in the case of a potential V that goes to 0 at infinity. Conditions on compact embeddings that allow to prove in particular the existence of groundstates are established. The fact that the solution is in ${L^2(\mathbb R^N)}Groundstates of the stationary nonlinear Schr?dinger equation

-Du +V u = K up-1-\Delta u +V u =K u^{p-1}      2. An inverse scattering problem for the Schrödinger equation with a potential having a periodic asymptotics      O. A. Anoshchenko 《Journal of Mathematical Sciences》1990,48(6):662-668 A theorem is proved regarding the expansion in the eigenfunctions of the one-dimensional Schrödinger equationL = –d z/dx 2+q(x)(–<x<)with a potential q(x), satisfying the condition       3. The nonlinear Schrödinger equation with a potential      Pierre Germain  Fabio Pusateri  Frédéric Rousset 《Annales de l'Institut Henri Poincaré (C) Analyse Non Linéaire》2018,35(6):1477-1530 We consider the cubic nonlinear Schrödinger equation with a potential in one space dimension. Under the assumptions that the potential is generic, sufficiently localized, with no bound states, we obtain the long-time asymptotic behavior of small solutions. In particular, we prove that, as time goes to infinity, solutions exhibit nonlinear phase corrections that depend on the scattering matrix associated to the potential. The proof of our result is based on the use of the distorted Fourier transform – the so-called Weyl–Kodaira–Titchmarsh theory – a precise understanding of the “nonlinear spectral measure” associated to the equation, and nonlinear stationary phase arguments and multilinear estimates in this distorted setting.      4. Scattering problem for the Schrödinger equation in the case of a potential linear in time and coordinate. I. Asymptotics in the shadow zone      V. M. Babich  V. P. Smyshlyaev 《Journal of Mathematical Sciences》1986,32(2):103-112 The formal asymptotics of the scattering problem for the Schrödinger equation with a linear potential as x+¦t¦+ is studied. In the shadow zone a formal asymptotic expansion is constructed which is matched with the known asymptotics as t– The expansion constructed loses asymptotic character near the curve x=1/6 t3 (in the so-called projector zone). An assumption regarding the analogous behavior of the asymptotic series in the projector zone makes it possible to construct an expansion in the post-projection zone which goes over into the formulas for creeping waves.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 140, pp. 6–17, 1984.In conclusion, the authors would like to bring to the reader's attention another approach to asymptotics in the projector zone proposed by M. M. Popov (see the present collection).      5. Dirichlet problem for Schrödinger equation with the boundary value in the BMO space      Jiang  Renjin  Li  Bo 《中国科学 数学(英文版)》2022,65(7):1431-1468 Science China Mathematics - Let (X, d, μ) be a metric measure space satisfying a Q-doubling condition (Q &gt; 1) and an L2-Poincaré inequality. Let $${\cal L} = {\cal L} + V$$ be a...      6. Space-time scattering for the Schrödinger equation      Arne Jensen 《Arkiv f?r Matematik》1998,36(2):363-377 Results are obtained on the scattering theory for the Schrödinger equation $i\partial _t u(t,x) = - \Delta _x u(t,x) + V(t,x)u(t,x) + F(u(t,x))$ in spacesL r (R;L q (R d )) for a certain range ofr, q, the so-called space-time scattering. In the linear case (i.e.F≡)) the relation with usual configuration space scattering is established.      7. The inverse problem for a nonlinear Schrödinger equation      G. Ya. Yagubov  M. A. Musaeva 《Journal of Mathematical Sciences》1999,97(2):3981-3984 Using variational methods, the solution of the inverse problem of finding the refractive index of a nonlinear medium in a multidimensional Schrödinger equation is studied. The correctness of the statement of the problem under consideration is investigated, and a necessary condition that must be satisfied by the solution of this problem is found. Bibliography: 8 titles.      8. The spectrum and the scattering problem for the Schrödinger operator in a magnetic field      Kh. Kh. Murtazin  A. N. Galimov 《Mathematical Notes》2008,83(3-4):364-377 The main result of this paper is the proof of a nonexistence theorem for solutions with nonzero real singularities to the problem of scattering theory for the Schrödinger operator with magnetic and electric potentials.      9. Inverse problem of scattering theory for a class one-dimensional Schrödinger equation      《Quaestiones Mathematicae》2013,36(7):841-856 Abstract In this work, direct and inverse scattering problem on the real axis for the Schrödinger equation with piecewise-constant coefficient are studied. Using the new integral representations for solutions, the scattering data is defined, the main integral equations of the inverse scattering problem are obtained, the spectral characteristics of the scattering data are investigated and uniqueness theorem for the solution of inverse problem is proved.      10. Coordinate asymptotics for the Schrödinger equation with a rapidly oscillating potential      A. R. Its  V. B. Matveev 《Journal of Mathematical Sciences》1979,11(3):442-444 Coordinate asymptotics for solutions of the Schrödinger equation with a rapidly oscillating potential are considered. The character of the oscillations is such that the leading term in the asymptotic expression does not, in general, reduce to a plane wave and contains an additional phase shift which grows at infinity. The main asymptotic formula is constructed from solutions of an auxiliary problem with a purely periodic potential depending on two numerical parameters. The main formula is applicable, in particular, to potentials of the form xP(x1+), >,(x+1)=(x).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 51, pp. 119–122, 1975.      11. Scattering theory for the Schrödinger equation with repulsive potential      《Journal de Mathématiques Pures et Appliquées》2005,84(5):509-579       12. Semi-classical analysis for the nonlinear Schrödinger equation with potential      Rémi Carles 《PAMM》2007,7(1):1041001-1041002       13. Existence time for the semilinear Schrödinger equation      Wei Mingjun 《高校应用数学学报(英文版)》2003,18(1):30-34 Based on the methods introduced by Klainerman and Ponce, and Cohn, a lower hounded estimate of the existence time for a kind of semilinear Schrödinger equation is ohtained in this paper. The implementation of this method depends on the L p ? L q estimate and the energy estimate.      14. An optimal control problem for the Schrödinger equation with a real-valued factor      N. M. Makhmudov 《Russian Mathematics (Iz VUZ)》2010,54(11):27-35 We study an optimal control problem for the Schrödinger equation with a real-valued factor in its nonlinear part where the control function is square summable and the quality criterion is the Lions functional. First, we examine the correctness of the statement of the reduced problemand, second, we do that of the optimal control problem. We also study the differentiability of the Lions functional and obtain a necessary optimality condition in the form of a variational inequality.      15. Modified scattering for the critical nonlinear Schrödinger equation      Thierry Cazenave  Ivan Naumkin 《Journal of Functional Analysis》2018,274(2):402-432 We consider the nonlinear Schrödinger equation $i{u}_t+\mathrm{\Delta }u=\lambda |u{|}^{\frac{2}{N}}u$ in all dimensions $N\ge 1$, where $\lambda \in \mathbb{C}$ and $?\lambda \le 0$. We construct a class of initial values for which the corresponding solution is global and decays as $t\to \infty$, like $t^{?\frac{N}{2}}$ if $?\lambda =0$ and like ${(t\log ?t)}^{?\frac{N}{2}}$ if $?\lambda <0$. Moreover, we give an asymptotic expansion of those solutions as $t\to \infty$. We construct solutions that do not vanish, so as to avoid any issue related to the lack of regularity of the nonlinearity at $u=0$. To study the asymptotic behavior, we apply the pseudo-conformal transformation and estimate the solutions by allowing a certain growth of the Sobolev norms which depends on the order of regularity through a cascade of exponents.      16. Dispersive estimate for the 1D Schrödinger equation with a steplike potential      Piero DʼAncona  Sigmund Selberg 《Journal of Differential Equations》2012,252(2):1603-1634       17. Solution of the spectral problem for the Schrödinger equation with a degenerate polynomial potential of even power      A. S. Vshivzev  N. V. Norin  V. N. Sorokin 《Theoretical and Mathematical Physics》1996,109(1):1329-1341 We study the stationary Schrödinger equation with the degenerate potential U(x) = x2r, r Z+, which describes phase transitions in quantum systems. The symmetry of the problem is revealed and an analytical procedure is developed for finding the eigenvalues of such potentials. The eigenvalues and the lowest energy levels are found numerically for r=2, 3, ..., 18.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 109, No. 1, pp. 107–123, October, 1996.      18. Eigenfunction expansions for the Schrödinger equation with an inverse-square potential      A. G. Smirnov 《Theoretical and Mathematical Physics》2016,187(2):762-781 We consider the one-dimensional Schrödinger equation -f″ + qκf = Ef on the positive half-axis with the potential qκ(r) = (κ2 - 1/4)r-2. For each complex number ν, we construct a solution uνκ(E) of this equation that is analytic in κ in a complex neighborhood of the interval (-1, 1) and, in particular, at the “singular” point κ = 0. For -1 < κ < 1 and real ν, the solutions uνκ(E) determine a unitary eigenfunction expansion operator Uκ,ν: L2(0,∞) → L2(R, Vκ,ν), where Vκ,ν is a positive measure on R. We show that every self-adjoint realization of the formal differential expression -?r2 + qκ(r) for the Hamiltonian is diagonalized by the operator Uκ,ν for some ν ∈ R. Using suitable singular Titchmarsh–Weyl m-functions, we explicitly find the measures Vκ,ν and prove their continuity in κ and ν.      19. On a nonlinear Schrödinger equation with periodic potential      Thomas Bartsch  Yanheng Ding 《Mathematische Annalen》1999,313(1):15-37       20. Existence of solutions for a class of nonlinear Schrödinger equations with potential vanishing at infinity      Claudianor O. Alves  Marco A.S. Souto 《Journal of Differential Equations》2013,254(4):1977-1991

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An inverse scattering problem for the Schrödinger equation with a potential having a periodic asymptotics

A theorem is proved regarding the expansion in the eigenfunctions of the one-dimensional Schrödinger equationL = –d ^z/dx ²+q(x)(–<x<)with a potential q(x), satisfying the condition

The nonlinear Schrödinger equation with a potential

We consider the cubic nonlinear Schrödinger equation with a potential in one space dimension. Under the assumptions that the potential is generic, sufficiently localized, with no bound states, we obtain the long-time asymptotic behavior of small solutions. In particular, we prove that, as time goes to infinity, solutions exhibit nonlinear phase corrections that depend on the scattering matrix associated to the potential. The proof of our result is based on the use of the distorted Fourier transform – the so-called Weyl–Kodaira–Titchmarsh theory – a precise understanding of the “nonlinear spectral measure” associated to the equation, and nonlinear stationary phase arguments and multilinear estimates in this distorted setting.     

Scattering problem for the Schrödinger equation in the case of a potential linear in time and coordinate. I. Asymptotics in the shadow zone

The formal asymptotics of the scattering problem for the Schrödinger equation with a linear potential as x+¦t¦+ is studied. In the shadow zone a formal asymptotic expansion is constructed which is matched with the known asymptotics as t– The expansion constructed loses asymptotic character near the curve x=1/6 t³ (in the so-called projector zone). An assumption regarding the analogous behavior of the asymptotic series in the projector zone makes it possible to construct an expansion in the post-projection zone which goes over into the formulas for creeping waves.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 140, pp. 6–17, 1984.In conclusion, the authors would like to bring to the reader's attention another approach to asymptotics in the projector zone proposed by M. M. Popov (see the present collection).     

Dirichlet problem for Schrödinger equation with the boundary value in the BMO space

Science China Mathematics - Let (X, d, μ) be a metric measure space satisfying a Q-doubling condition (Q &gt; 1) and an L2-Poincaré inequality. Let $${\cal L} = {\cal L} + V$$ be a...     

Space-time scattering for the Schrödinger equation

Results are obtained on the scattering theory for the Schrödinger equation $i\partial _t u(t,x) = - \Delta _x u(t,x) + V(t,x)u(t,x) + F(u(t,x))$ in spacesL ^r (R;L ^q (R ^d )) for a certain range ofr, q, the so-called space-time scattering. In the linear case (i.e.F≡)) the relation with usual configuration space scattering is established.     

The inverse problem for a nonlinear Schrödinger equation

Using variational methods, the solution of the inverse problem of finding the refractive index of a nonlinear medium in a multidimensional Schrödinger equation is studied. The correctness of the statement of the problem under consideration is investigated, and a necessary condition that must be satisfied by the solution of this problem is found. Bibliography: 8 titles.     

The spectrum and the scattering problem for the Schrödinger operator in a magnetic field

The main result of this paper is the proof of a nonexistence theorem for solutions with nonzero real singularities to the problem of scattering theory for the Schrödinger operator with magnetic and electric potentials.     

Inverse problem of scattering theory for a class one-dimensional Schrödinger equation

Abstract

In this work, direct and inverse scattering problem on the real axis for the Schrödinger equation with piecewise-constant coefficient are studied. Using the new integral representations for solutions, the scattering data is defined, the main integral equations of the inverse scattering problem are obtained, the spectral characteristics of the scattering data are investigated and uniqueness theorem for the solution of inverse problem is proved.     

Coordinate asymptotics for the Schrödinger equation with a rapidly oscillating potential

Coordinate asymptotics for solutions of the Schrödinger equation with a rapidly oscillating potential are considered. The character of the oscillations is such that the leading term in the asymptotic expression does not, in general, reduce to a plane wave and contains an additional phase shift which grows at infinity. The main asymptotic formula is constructed from solutions of an auxiliary problem with a purely periodic potential depending on two numerical parameters. The main formula is applicable, in particular, to potentials of the form xP(x¹⁺), >,(x+1)=(x).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 51, pp. 119–122, 1975.     

Existence time for the semilinear Schrödinger equation

Based on the methods introduced by Klainerman and Ponce, and Cohn, a lower hounded estimate of the existence time for a kind of semilinear Schrödinger equation is ohtained in this paper. The implementation of this method depends on the L ^p ? L ^q estimate and the energy estimate.     

An optimal control problem for the Schrödinger equation with a real-valued factor

We study an optimal control problem for the Schrödinger equation with a real-valued factor in its nonlinear part where the control function is square summable and the quality criterion is the Lions functional. First, we examine the correctness of the statement of the reduced problemand, second, we do that of the optimal control problem. We also study the differentiability of the Lions functional and obtain a necessary optimality condition in the form of a variational inequality.     

Modified scattering for the critical nonlinear Schrödinger equation

We consider the nonlinear Schrödinger equation

$i{u}_t+\mathrm{\Delta }u=\lambda |u{|}^{\frac{2}{N}}u$

in all dimensions $N\ge 1$, where $\lambda \in \mathbb{C}$ and $?\lambda \le 0$. We construct a class of initial values for which the corresponding solution is global and decays as $t\to \infty$, like $t^{?\frac{N}{2}}$ if $?\lambda =0$ and like ${(t\log ?t)}^{?\frac{N}{2}}$ if $?\lambda <0$. Moreover, we give an asymptotic expansion of those solutions as $t\to \infty$. We construct solutions that do not vanish, so as to avoid any issue related to the lack of regularity of the nonlinearity at $u=0$. To study the asymptotic behavior, we apply the pseudo-conformal transformation and estimate the solutions by allowing a certain growth of the Sobolev norms which depends on the order of regularity through a cascade of exponents.     

Solution of the spectral problem for the Schrödinger equation with a degenerate polynomial potential of even power

We study the stationary Schrödinger equation with the degenerate potential U(x) = x^2r, r Z₊, which describes phase transitions in quantum systems. The symmetry of the problem is revealed and an analytical procedure is developed for finding the eigenvalues of such potentials. The eigenvalues and the lowest energy levels are found numerically for r=2, 3, ..., 18.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 109, No. 1, pp. 107–123, October, 1996.     

Eigenfunction expansions for the Schrödinger equation with an inverse-square potential

We consider the one-dimensional Schrödinger equation -f″ + q_κf = Ef on the positive half-axis with the potential q_κ(r) = (κ² - 1/4)r^-2. For each complex number ν, we construct a solution u_ν^κ(E) of this equation that is analytic in κ in a complex neighborhood of the interval (-1, 1) and, in particular, at the “singular” point κ = 0. For -1 < κ < 1 and real ν, the solutions u_ν^κ(E) determine a unitary eigenfunction expansion operator U_κ,ν: L₂(0,∞) → L₂(R, Vκ,ν), where Vκ,ν is a positive measure on R. We show that every self-adjoint realization of the formal differential expression -?_r² + q_κ(r) for the Hamiltonian is diagonalized by the operator U_κ,ν for some ν ∈ R. Using suitable singular Titchmarsh–Weyl m-functions, we explicitly find the measures V_κ,ν and prove their continuity in κ and ν.