A semilinear Schrödinger equation with zero on the boundary of the spectrum and exponential growth in ℝ2

2019 ◽  
Vol 21 (06) ◽  
pp. 1850037
Author(s):  
Marcius Petrúcio Cavalcante ◽  
Claudianor O. Alves ◽  
Everaldo Medeiros
Keyword(s):  
Schrödinger Equation ◽  
Schrödinger Operator ◽  
Exponential Growth ◽  
Schrodinger Equation ◽  
Spectral Gap ◽  
Existence Of Solutions ◽  
Type Inequality ◽  
Schrodinger Operator ◽  
Semilinear Schrödinger Equation

The main purpose of this paper is to establish the existence of solutions for the semilinear Schrödinger equation [Formula: see text] where the potential [Formula: see text] is periodic, [Formula: see text] lies on the boundary of a spectral gap of the Schrödinger operator [Formula: see text] and the nonlinearity [Formula: see text] is periodic and has subquadratic exponential growth. The proofs rely on a linking-type argument and a Trudinger–Moser type inequality proved in this paper.

scholarly journals A Nonlinear Schrödinger Equation Resonating at an Essential Spectrum

2016 ◽  
Vol 2016 ◽  
pp. 1-10
Author(s):  
Shaowei Chen ◽  
Haijun Zhou
Keyword(s):  
Schrödinger Equation ◽  
Essential Spectrum ◽  
Nonlinear Schrödinger Equation ◽  
Schrödinger Operator ◽  
Schrodinger Equation ◽  
Nonlinear Schrodinger Equation ◽  
Infinitely Many Solutions ◽  
Schrodinger Operator ◽  
Nonlinear Schrödinger ◽  
Negative Eigenvalues

We consider the nonlinear Schrödinger equation-Δu+f(u)=V(x)u  in  RN. The potential functionVsatisfies that the essential spectrum of the Schrödinger operator-Δ-Vis[0,+∞)and this Schrödinger operator has infinitely many negative eigenvalues accumulating at zero. The nonlinearityfsatisfies the resonance type conditionlimt→∞f(t)/t=0. Under some additional conditions onVandf, we prove that this equation has infinitely many solutions.

The existence of eigenvalues of infinite multiplicity for the Schrödinger operator

1980 ◽  
Vol 86 (1-2) ◽  
pp. 61-64 ◽  
Author(s):  
Christine R. Thurlow ◽  
Michael S.P. Eastham
Keyword(s):  
Schrödinger Equation ◽  
Spectral Theory ◽  
Schrödinger Operator ◽  
Schrodinger Equation ◽  
Dimensional Space ◽  
Schrodinger Operator ◽  
Inverse Spectral Theory ◽  
Inverse Spectral ◽  
Infinite Multiplicity

SynopsisIt is shown that eigenvalues of infinite multiplicity can exist for the Schrödinger equation holding in the whole N-dimensional space RN(N ≧ 2). In the example which is constructed, the potential is separable and bounded in RN, and the method is an application of inverse spectral theory.

scholarly journals Characterization of Spectrum and Eigenvectors of the Schrödinger Operator with Chaotic Potentials

2014 ◽  
Vol 15 (2) ◽  
Author(s):  
Weslley Florentino de Oliveira ◽  
Giancarlo Queiroz Pellegrino
Keyword(s):  
Schrödinger Equation ◽  
Schrödinger Operator ◽  
Schrodinger Equation ◽  
Dimensional Space ◽  
Tight Binding ◽  
System Size ◽  
Schrodinger Operator ◽  
Tight Binding Approximation ◽  
Chaotic Sequences

<em>Chaotic sequences are sequences generated by chaotic maps. A particle moving in a one-dimensional space has its behavior modeled according to the time-independent Schrödinger equation. The tight-binding approximation enables the use of chaotic sequences as the simulation of quantum potentials in the discretized version of the Schrödinger equation. The present work consists of the generation and characterization of spectral curves and eigenvectors of the Schrödinger operator with potentials generated by chaotic sequences, as  well as their comparison with the curves generated by periodic, peneperiodic and random sequences.</em> <em>This comparison is made by calculating in each case the inverse participation ratio as a function of the system size.</em>

scholarly journals An Existence Result for a Class of Magnetic Problems in Exterior Domains

2021 ◽  
Author(s):  
Claudianor O. Alves ◽  
Vincenzo Ambrosio ◽  
César E. Torres Ledesma
Keyword(s):  
Schrödinger Equation ◽  
Vector Potential ◽  
Exterior Domain ◽  
Schrodinger Equation ◽  
Existence Result ◽  
Existence Of Solutions ◽  
Exterior Domains ◽  
Continuous Vector ◽  
Semilinear Schrödinger Equation

AbstractIn this paper we deal with the existence of solutions for the following class of magnetic semilinear Schrödinger equation $$\begin{aligned} (P) \qquad \qquad \left\{ \begin{aligned}&(-i\nabla + A(x))^2u +u = |u|^{p-2}u,\;\;\text{ in }\;\;\Omega ,\\&u=0\;\;\text{ on }\;\;\partial \Omega , \end{aligned} \right. \end{aligned}$$ ( P ) ( - i ∇ + A ( x ) ) 2 u + u = | u | p - 2 u , in Ω , u = 0 on ∂ Ω , where $$N \ge 3$$ N ≥ 3 , $$\Omega \subset {\mathbb {R}}^N$$ Ω ⊂ R N is an exterior domain, $$p\in (2, 2^*)$$ p ∈ ( 2 , 2 ∗ ) with $$2^*=\frac{2N}{N-2}$$ 2 ∗ = 2 N N - 2 , and $$A: {\mathbb {R}}^N\rightarrow {\mathbb {R}}^N$$ A : R N → R N is a continuous vector potential verifying $$A(x) \rightarrow 0\;\;\text{ as }\;\;|x|\rightarrow \infty .$$ A ( x ) → 0 as | x | → ∞ .

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