L p-analysis of one-dimensional repulsive Hamiltonian with a class of perturbations

Abstract: The spectrum of one-dimensional repulsive Hamiltonian with a class of perturbations Hp = − d2 dx2 − x2 + V(x) in Lp(R) (1 < p < ∞) is explicitly given. It is also proved that the domain of Hp is embedded into weighted Lq-spaces for some q > p. Additionally, non-existence of related Schrödinger (C0-)semigroup in Lp(R) is shown when V(x) ≡ 0.


Introduction
In this paper we consider in L p (R), where V ∈ C(R) is a real-valued and satisfies V(x) ≥ −a(1 + x 2 ) for some constant a ≥ 0 and The operator (1) describes the quantum particle affected by a strong repulsive force from the origin.In fact, in the classical sense the corresponding Hamiltonian (functional) is given by Ĥ(x, p) = p 2 − x 2 and then the particle satisfying ẋ = ∂ p Ĥ and ṗ = −∂ x Ĥ goes away much faster than that for the free Hamiltonian Ĥ0 (x, p) = p 2 .
In the case where p = 2, the essential selfadjointness of H, endowed with the domain C ∞ 0 (Ω), has been discussed by Ikebe and Kato [7].After that several properties of H is found out in a mount of subsequent papers (for studies of scattering theory e.g., Bony et al. [2], Nicoleau [10] and also Ishida [8]).
In contrast, if p is different from 2, then the situation becomes complicated.Actually, papers which deals with the properties of H is quite few because of absence of good properties like symmetricity.In the L p -framework, it is quite useful to consider the accretivity and sectoriality of the second-order differential operators.In fact, the case − d 2 dx 2 + V(x) with a nonnegative potential V is formally sectorial in L p , and therefore one can find many literature even N-dimensional case (e.g., Kato [9], Goldstein [6], Tanabe [14], Engel-Nagel [5]).However, it seems quite difficult to describe such a kind of nonaccretive operators in a certain unified theory in the literature.
The present paper is in a primary position to make a contribution for theory of non-accretive operators in L p as mentioned above.The aim of this paper is to give a spectral properties of for the case where V(x) can be regarded as a perturbation of the leading part admissible, which is same threshold as in the short range potential for − d 2 dx 2 − x 2 stated in Bony [2] and also Ishida [8].
Here we define the minimal realization H p,min of H in L p = L p (R) as Theorem 1.1.For every 1 < p < ∞, H p,min is closable and the spectrum of the closure H p is explicitly given as Moreover, for every 1 < p < q < ∞, one has consistence of the resolvent operators: Remark 1.1.If p = 2, then our assertion is nothing new.The crucial part is the case p 2 which is the case where the symmetricity of H breaks down.The similar consideration for − d 2 dx 2 + V (but in L 2 -setting) can be found in Dollard-Friedman [4].This paper is organized follows: In Section 2, we prepare two preliminary results.In Section 3, we consider the fundamental systems of λu + Hu = 0, and estimate the behavior of their solutions.By virtue of that estimates, we will describe the resolvent set of H p in Section 4. In section 5, we prove never to be generated C 0 -semigroups by ±iH p under the condition V = 0.

Preliminary results
First we state well-known results for the essentially selfadjointness of Schrödinger operators in L 2 which is firstly described in [7].We would like to refer also Okazawa [12].
Next we note the asymptotic behavior of solutions to second-order linear ordinary differential equations of the form in which the term Ψ(x)y(x) can be treated as a perturbation of the leading part Φ(x)y(x).
Theorem 2.2 (Olver [13, Theorem 6.2.2 (p.196)] ).In a given finite or infinite interval (a 1 , a 2 ), let a ∈ (a 1 , a 2 ), Ψ(x) a positive, real, twice continuously differentiable function, Ψ(x) a continuous real or complex function, and Then in this interval the differential equation provided that V a j ,x (F) < ∞ (where V a j ,x (F) = |F (t)| dt is the total variation of F).If Ψ(x) is real, then the solutions w 1 (x) and w 2 (x) are complex conjugates.
For the above theorem, see also 10.12,p.355].

Fundamental systems of λu
We consider the behavior of solutions to where λ ∈ R.

The case λ ∈ C \ R
We consider the behavior of solutions to where λ ∈ C \ R with Im λ > 0. The case Im λ < 0 can be reduced to the problem Im λ > 0 via complex conjugation.

Properties of solutions to an auxiliary problem
We start with the following function ϕ λ : Then by a direct computation we have where g λ (x) : , then ϕ λ is nothing but a solution of the original equation ( 6) with V = 0.
Next we construct another solution of (8) which is linearly independent of ϕ λ .Before construction, we prepare the following lemma.Lemma 3.3.Let λ satisfy Im λ > 0 and let ϕ λ be given in (7).Then for every a > 0, there exists a is independent of a.Moreover, for every x > 0, where Proof.By integration by part, we have Noting that t λi−3 e −it 2 is integrable in (a, ∞), we have This is nothing but the desired inequality.

Fundamental system of the original problem
Next we consider λw − w − x 2 w + g λ w = gλ h, x > 0 (10) with a given function h, where g λ is given as in Lemma 3.2 and gλ := g λ − V. To construct solutions of (6), we will define two types of solution maps h → w and consider their fixed points.
First we construct a solution of (6) which behaves like ψ λ at infinity.
for h belonging to a Banach space Remark 3.3.For arbitrary fixed b > 0, all solutions of (10) can be described as follows: In Definition 3.5 we deal with such a solution with c 2 = 1.
Well-definedness of U in Definition 3.5 and its contractivity are proved in next lemma.
Lemma 3.6.The following assertions hold: and then U has a unique fixed point w 1 ∈ X λ (b λ ); (iii) w 1 can be extended to a solution of (6) in R satisfying Proof.(i) By Lemma 3.4 we have ψ λ ∈ X λ (b).Therefore to prove well-definedness of U, it suffices to show that the second term in the definition of U belongs to

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Volume 3, Issue 1, 21-34 Proceeding the same computation as above, we deduce Choosing b large enough, we obtain Uh . By contraction mapping principle, we obtain that U has a unique fixed point w 1 ∈ X λ (b).
(iii) Since w 1 satisfies (10) with h = w 1 , w 1 is a solution of the original equation ( 6) in [b, ∞).As in the last part of the proof of Proposition 3.1, we can extend w 1 as a solution of (6) in R. Since Uw 1 = w 1 and U0 = ψ λ , it follows from the contractivity of U that Consequently, we have w 1 − ψ λ X ≤ 4 −1 ψ λ X ≤ 4 −1 and then for x ≥ b, Next we construct another solution of ( 6) which behaves like ϕ λ at infinity.x Lemma 3.8.The following assertions hold: and then U has a unique fixed point w1 ∈ Y λ (b λ ); (iii) w1 can be extended to a solution of (6) in R satisfying 1 2 x Proof.The proof is similar to the one of Lemma 3.6.
Considering the equation ( 6) for x < 0, we also obtain the following lemma.
Lemma 3.9.For every λ ∈ C with Im λ > 0, there exist a fundamental system (w 1 , w 2 ) of (6) and positive constants c λ , C λ , R λ such that and Proof.In view of Lemma 3.6, it suffices to find w 2 satisfying the conditions above.Let w * and w * be given as in Lemmas 3.6 and 3.8 with V(x) replaced with V(−x).Noting that w 1 can be rewritten as w 1 (x) = c 1 w * (−x) + c 2 w * (−x), we see from Lemma 3.6 and 3.8 that (11) and the first half of (13) are satisfied.Set w 2 (x) = w * (−x) for x ∈ R. As in the same way, we can verify (12).

Resolvent estimates in L p
The following lemma, verified by the variation of parameters, gives a possibility of representation of the Green function for resolvent operator H in L p .Lemma 4.1.Assume that λ ∈ ρ( H) in L p , where H is a realization of H in L p .Then for every u can be extended to a bounded operator on L p .More precisely, there exists M λ > 0 such that In particular, H p,min is closable and its closure H p satisfies We divide the proof of u 1 ∈ L p (R) into two cases x ≥ 0 and x < 0; since the proof of u 2 ∈ L p (R) is similar, this part is omitted.
The case u 1 for x ≥ 0, it follows from Lemma 3.9 and Hölder inequality that with 0 < α < Imλ+1 2 + 1/p .By the triangle inequality we have and The case u 1 for x < 0, by the same way as the case x > 0, we have where 0 Proceeding the same argument for u 2 and combining the estimates for u 1 and u 2 , we obtain (14).
Proof.Let f ∈ C ∞ 0 (R) and set u 1 and u 2 as in the proof of Proposition 4.2.Since the proof for u 1 and u 2 are similar, we only show the estimate of u 1 .From (15), we have for x ≥ 0, (1 + |x|) More precisely, there exists a constant C p,q > 0 such that x 1 p − 1 q u L q ≤ C p,q H p u L p + u L p , u ∈ D(H p ). Proof.The assertion follows from Proposition 4.
14. H. Tanabe, Functional Analytic Methods for Partial Differential Equations, Pure and Applied Mathematics, 204, Marcel Dekker, New York, 1997.c 2018 the Author(s), licensee AIMS Press.This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)