Spectra of Linearized Operators for NLS Solitary Waves
Shu-Ming Chang∗, Stephen Gustafson†, Kenji Nakanishi‡, Tai-Peng Tsai§
Abstract. Nonlinear Schr¨odinger (NLS) equations with focusing power nonlinearities have solitary wave solutions. The spectra of the linearized operators around these solitary waves are intimately connected to stability properties of the solitary waves, and to the long-time dynamics of solutions of (NLS). We study these spectra in detail, both analytically and numerically.
Key words. Spectrum, linearized operator, NLS, solitary waves, stability. AMS subject classifications. 35Q55, 35P15.
1
Introduction
Consider the nonlinear Schr¨odinger equation (NLS) with focusing power nonlinearity, i∂tψ = −∆ψ − |ψ|p−1ψ, (1.1)
where ψ(t, x) : R × Rn→ C and 1 < p < ∞. Such equations arise in many physical settings,
including nonlinear optics, water waves, and quantum physics. Mathematically, nonlinear Schr¨odinger equations with various nonlinearities are studied as basic models of nonlinear dispersive phenomena. In this paper, we stick to the case of a pure power nonlinearity for the sake of simplicity.
For a certain range of the power p (see below), the NLS (1.1) has special solutions, of the form ψ(t, x) = Q(x) eit. These are called solitary waves. The aim of this paper is to
study the spectra of the linearized operators which arise when (1.1) is linearized around solitary waves. The main motivation for this study is that properties of these spectra are intimately related to the problem of the stability (orbital and asymptotic) of these solitary waves, and to the long-time dynamics of solutions of NLS.
Let us begin by recalling some well-known facts about (1.1). Standard references include [4, 29, 30]. Many basic results on the linearized operators we study here were proved by
∗Department of Mathematics, National Tsing Hua University, Hsinchu, Taiwan ([email protected])
†Department of Mathematics, University of British Columbia, Vancouver, BC V6T1Z2, Canada
‡Department of Mathematics, Kyoto University, Kyoto 606-8502, Japan ([email protected])
§Department of Mathematics, University of British Columbia, Vancouver, BC V6T1Z2, Canada
Weinstein [34, 35]. The Cauchy (initial value) problem for equation (1.1) is locally (in time) well-posed in H1(Rn) if 1 < p < pmax, where
pmax:= 1 + 4/(n − 2) if n ≥ 3; pmax:= ∞ if n = 1, 2.
Moreover, if 1 < p < pc where
pc := 1 + 4/n,
the problem is globally well-posed. For p ≥ pc, there exist solutions whose H1-norms go to
∞ (blow up) in finite time. In this paper, the cases p < pc, p = pc and p > pc are called
sub-critical, critical, and super-critical, respectively.
1.04 −1.39 1.1 −1 1.32 −0.5 1.82 −0.086 2 0 3 0.301 0.5 5 0.65 −1 0 −0.5 0 0.5 1 log10(p − 1) p
Figure 1: Spectra of L, L+and L−for n = 1. (solid line: purely imaginary eigenvalues of L;
dashed line: real eigenvalues of L; dotted line: eigenvalues of L+; dashdot line: eigenvalues
of L−)
The set of all solutions of (1.1) is invariant under the symmetries of translation, rotation, phase, Galilean transform and scaling: if ψ(t, x) is a solution, then so is
e ψ(t, x) := λ2/(p−1)ψ λ2t, λRx − λ2tv − x0 exp i λRx · v 2 − λ2tv2 4 + γ0
for any constant x0, v ∈ Rn, λ > 0, γ0 ∈ R and R ∈ O(n). When p = pc, there is an
additional symmetry called the “pseudo-conformal transform” (see [30, p.35]). We are interested here in solutions of (1.1) of the form
1.01 −2 1.03 −1.5 1.1 −1 1.32 −0.5 2 0 2.27 2.379 3 0.34 −1 −0.92 −0.5 0 0.5 0.92 1 re p la ce m en log10(p − 1) p
Figure 2: Spectra of L, L+ and L− restricted to radial functions for n = 2.
where Q(x) must therefore satisfy the nonlinear elliptic equation
−∆Q − |Q|p−1Q = −Q. (1.3)
Any such solution generates a family of solutions by the above-mentioned symmetries, called solitary waves. Solitary waves are special examples of nonlinear bound states, which, roughly speaking, are solutions that are spatially localized for all time. More precisely, one could define nonlinear bound states to be solutions ψ(t, x) which are non-dispersive in the sense that sup t∈R inf x0∈Rnk|x|ψ(t, x − x 0)kL2 x(Rn)< ∞.
Testing (1.3) with ¯Q and x.∇ ¯Q and taking real parts, one arrives at the Pohozaev identity ([22]) 1 2 Z |Q|2 = b 1 p + 1 Z |Q|p+1, 1 2 Z |∇Q|2 = a 1 p + 1 Z |Q|p+1 (1.4) where a = n(p − 1) 4 , b = n + 2 − (n − 2)p 4 .
The coefficients a and b must be positive, and hence a necessary condition for existence of non-trivial solutions is p ∈ (1, pmax).
For p ∈ (1, pmax), and for all space dimensions, there exists at least one non-trivial
1.01 −2 1.03 −1.5 1.1 −1 1.32 −0.5 1.94 0 2.046 0.18 −1 −0.841 −0.5 0 0.5 0.841 1 log10(p − 1) p 7 3
Figure 3: Spectra of L, L+ and L− restricted to radial functions for n = 3.
a nonlinear ground state, is smooth, decreases monotonically as a function of |x|, decays exponentially at infinity, and can be taken to be positive: Q(x) > 0. It is the unique positive solution. (See [30] for references for the various existence and uniqueness results for various nonlinearities.) The ground state can be obtained as the minimizer of several different variational problems. One such result we shall briefly use later is that, for all n ≥ 1 and p ∈ (1, pmax), the ground state minimizes the Gagliardo-Nirenberg quotient
J[u] := R
|∇u|2a Ru2b
R
up+1 (1.5)
among nonzero H1(Rn) radial functions (Weinstein [34]).
For n = 1, the ground state is the unique H1(R)-solution of (1.3) up to translation and phase [4, p.259, Theorem 8.1.6]. For n ≥ 2, this is not the case: there are countably infinitely many radial solutions (still real-valued), denoted in this paper by Q0,k,p(x), k = 0, 1, 2, 3, . . .,
each with exactly k positive zeros as a function of |x| (Strauss [28]; see also [2]). In this notation, Q0,0,p is the ground state.
There are also non-radial (and complex-valued) solutions, for example those suggested by P. L. Lions [16] with non-zero angular momenta,
n = 2, Q = φ(r) eimθ, in polar coordinates r, θ; n = 3, Q = φ(r, x3) eimθ, in cylindrical coordinates r, θ, x3,
and similarly defined for n ≥ 4. Some of these solutions are denoted here by Qm,k,p (see
1.01 −2 1.03 −1.5 1.1 −1 1.32 −0.5 1.75 1.841 2 0 2.12 0.05 −1 −0.77 −0.4 0 0.4 0.77 1 re p la ce m en log10(p − 1) p
Figure 4: Spectra of L, L+ and L− restricted to radial functions for n = 4.
We will refer to all the solitary waves generated by Q0,0,p as nonlinear ground states,
and all others as nonlinear excited states. We are not aware of a complete characterization of all solutions of (1.3), or of (1.1). For example, the uniqueness of Qm,k,p with m, k ≥ 1
is apparently open. Also, we do not know if there are “breather” solutions, analogous to those of the generalized KdV equations. In this paper we will mainly study radial solutions (and in particular the ground state), but we will also briefly consider non-radial solutions numerically in Section 5.
To study the stability of a solitary wave solution (1.2), one considers solutions of (NLS) of the form
ψ(t, x) = [Q(x) + h(t, x)] eit. (1.6) For simplicity, let Q = Q0,0,p be the ground state for the remainder of this introduction (see
Section 5 for the general case). The perturbation h(t, x) satisfies an equation
∂th = Lh + (nonlinear terms) (1.7)
where L is the linearized operator around Q:
Lh = −in(−∆ + 1 − Qp−1)h − p−12 Qp−1(h + ¯h)
o
. (1.8)
It is convenient to write L as a matrix operator acting on Re h Im h , L = 0 L− −L+ 0 (1.9)
1.01 −2 1.03 −1.5 1.1 −1 1.32 −0.5 1.69 −0.161 2 0 3.19 0.34 −1 −0.5 0 0.5 1 re p la ce m en log10(p − 1) p
Figure 5: Spectra of L, L+ and L− in R2 restricted to functions of the form φ(r)eiθ.
where
L+= −∆ + 1 − pQp−1, L−= −∆ + 1 − Qp−1. (1.10)
Clearly the operators L− and L+ play a central role in the stability theory. They are
self-adjoint Schr¨odinger operators with continuous spectrum [1, ∞), and with finitely many eigenvalues below 1. In fact, when Q is the ground state, it is easy to see that L− is a nonnegative operator, while L+has exactly one negative eigenvalue (these facts follow from
Lemma 2.2 below).
Because of its connection to the stability problem, the object of interest to us in this paper is the spectrum of the non-self-adjoint operator L. The simplest properties of this spectrum are
1. for all p ∈ (1, pmax), 0 is an eigenvalue of L
2. the set Σc := {ir : r ∈ R, |r| ≥ 1} is the continuous spectrum of L.
(See the next section for the first statement. The second is easily checked.)
It is well-known that the exponent p = pc is critical for stability of the ground state
solitary wave (as well as for blow-up of solutions). For p < pc the ground state is orbitally
stable, while for p ≥ pc it is unstable (see [36, 10]). These facts have immediate spectral
counterparts: for p ∈ (1, pc], all eigenvalues of L are purely imaginary, while for p ∈
(pc, pmax), L has at least one eigenvalue with positive real part.
The goal of this paper is to get a more detailed understanding of the spectrum of L, using both analytical and numerical techniques. This finer information is essential for
1.01 −2 1.03 −1.5 1.1 −1 1.32 −0.5 3.24 −0.3497 2 0 3.19 0.34 −1 −0.5 0 0.5 1 re p la ce m en log10(p − 1) p
Figure 6: Spectra of L, L+ and L− in R2 restricted to functions of the form φ(r)ei2θ.
understanding the long-time dynamics of solutions of (NLS): for example, for proving as-ymptotic (rather than simply orbital) stability, for determining rates of relaxation to stable solitary waves, for constructing stable manifolds of unstable solitary waves, etc. (these are highly active areas of current research). Interesting questions with direct relevance to these stability-type problems include:
(i) Can one determine (or estimate) the number and locations of the eigenvalues of L lying on the segment between 0 and i?
(ii) Can ±i, the thresholds of the continuous spectrum Σc, be eigenvalues or resonances?
(iii) Can eigenvalues be embedded inside the continuous spectrum?
(iv) Can the linearized operator have eigenvalues with non-zero real and imaginary parts (this is already known not to happen for the ground state – see the next section – and so we pose this question with excited states in mind).
(v) Are there bifurcations, as p varies, of pairs of purely imaginary eigenvalues into pairs of eigenvalues with non-zero real part (a stability/instability transition)?
Let us now summarize the main results and observations of this paper:
1. Numerics. We present numerical computations of the spectra of L, L+ and L−
as functions of p, when Q is the ground state solitary wave. Figure 1 is the one-dimensional case. Figures 2, 3, and 4 are the spectra of these operators restricted
1.01 −2 1.03 −1.5 1.1 −1 −0.5 1.334 2 0 3.19 0.34 −1 −0.5 0 0.5 1 re p la ce m en log10(p − 1) p
Figure 7: Spectra of L, L+ and L− in R2 restricted to functions of the form φ(r)ei3θ.
to radial functions, for space dimensions n = 2, 3, 4, respectively. For p ∈ (1, pc), it
is the imaginary parts of the eigenvalues of L which are shown in the figures. For space dimension n = 2, Figures 5, 6, and 7 are the spectra restricted to functions of the form φ(r)eimθ, with m = 1, 2, 3, respectively. These pictures shed some light on
questions (i), (iv), and (v) above, and to a certain extent on question (ii).
2. One-dimensional phenomena. The case n = 1 is the easiest case to handle ana-lytically. In Section 3, we undertake a detailed study of the one-dimensional problem, giving rigorous proofs of a number of phenomena observed in Figure 1. One simple such phenomenon is the (actually classical) fact that the eigenvalues of L+ and L−
exactly coincide, with the exception of the first, negative, eigenvalue of L+ (note that
this appears to be a strictly one-dimensional phenomenon: the eigenvalues of L+ and
L− are different for n = 2, 3, 4, as Figures 2–7 indicate). In fact, we are able to prove sufficiently precise upper and lower bounds on the eigenvalues of L (lying outside the continuous spectrum) to determine their number, and estimate their positions, as functions of p (see Theorem 3.8). We use two basic techniques: an embedding of L+
and L− into a hierarchy of related operators, and a novel variational problem for the eigenvalues, in terms of a 4-th order self-adjoint differential operator (see Theorem 3.6). In this way, we get a fairly complete answer to question (i) above for n = 1. 3. Variational characterization of eigenvalues. We present self-adjoint variational
formulations of the eigenvalue problem for L in any dimension (see Theorem 2.5), including the novel n = 1 formulation mentioned above. In principle, these provide
a means of counting/estimating the eigenvalues of L (and hence addressing question (i) above in higher dimensions), though we only obtain detailed such information for n = 1.
4. Bifurcation at p = pc. In each of Figures 1–4, a pair of purely imaginary eigenvalues
for p < pc appears to collide at 0 at p = pc, and become a pair of real eigenvalues
for p > pc. This is exactly the stability/instability transition for the ground state.
We rigorously verify this picture, determining analytically the spectrum near 0 for p near pc, and making concrete a bifurcation picture suggested by M. I. Weinstein
(personal communication): see Theorem 2.6. This gives a partial answer to question (v) above. It is worth pointing out that for n = 1, the imaginary part of the (purely imaginary) eigenvalue bifurcating for p < pc is always larger than the third eigenvalue
of L+ (the first is negative and the second is zero) – this is proved analytically in
Theorem 3.8. For n ≥ 2, however, they intersect at p ≈ 2.379 for 2D, p ≈ 2.046 for 3D, and p ≈ 1.841 for 4D (see Figures 1–4).
5. Interlacing property. A numerical observation: in all the figures, the adjacent eigenvalues of L seem each to bound an eigenvalue of L+ and one of L− (at least for
p small enough). We are able to establish this “interlacing” property analytically in dimension one (see Theorem 3.8).
6. Threshold resonance. An interesting fact observed numerically (Figure 1) is that, in the 1D case, as p → 3, one eigenvalue curve converges to ±i, the threshold of the continuous spectrum. One might suspect that, at p = 3, ±i corresponds to a resonance or embedded eigenvalue. It is indeed a resonance: we find an explicit non spatially-decaying “eigenfunction”, and show numerically in Section 3.7 that the corresponding eigenfunctions converges, as p → 3, to this function. This observation addresses question (ii) above for n = 1.
7. Excited states. In Section 5 we consider the spectra of linearized operators around excited states with non-zero angular momenta. We observe that there are complex eigenvalues which are neither real nor purely imaginary (addressing question (iv) above). These complex eigenvalues may collide into the imaginary axis or the real axis (not at the origin), further addressing question (v) above (see Figures 12–15). It is worth mentioning some important questions we cannot answer:
1. We are so far unable to give precise rigorous estimates on the number and positions of the eigenvalues of L for n ≥ 2 (question (i) above).
2. We cannot exclude the possible existence of embedded eigenvalues (question (iii) above).
3. We do not know a nice variational formulation for eigenvalues of L when Q is an excited state (this problem is also linked to question (i) above).
4. We do not have a complete characterization of solitary waves, or more generally of nonlinear bound states.
We end this introduction by describing some related numerical work. Buslaev-Grikurov [3, 8] study the linearized operators for solitary waves of the following 1D NLS with p < q,
iψt+ ψxx+ |ψ|pψ − α|ψ|qψ = 0.
They draw the bifurcation picture for eigenvalues near zero when the parameter α > 0 is near a critical value, with the frequency of the solitary wave fixed. This picture is similar to Weinstein’s picture which we study in Section 2.3.
Demanet and Schlag [7] consider the same linearization as us and study the super-critical case n = 3 and p ≤ 3 near 3. In this case, it is numerically shown that both L+ and L−
have no eigenvalues in (0, 1] and no resonance at 1, a condition which implies (see [27]) that L has no purely imaginary eigenvalues in [−i, 0) ∪ (0, i] and no resonance at ±i.
We outline the rest of the paper: in Section 2 we consider general results for all di-mensions. In Section 3 we consider one dimensional theory. In Section 4 we discuss the numerical methods. In Section 5 we discuss the spectra for excited states with angular momenta.
Notation: For an operator A, N (A) = φ ∈ L2| Aφ = 0 denotes the nullspace of A. Ng(A) = ∪∞k=1N (Ak) denotes the generalized nullspace of A. The L2-inner product in Rn
is (f, g) =RRnf g dx.¯
2
General theory
In this section we present results which are valid for all dimensions, for the ground state Q(x) = Q0,0,p(x).
We begin by recalling some well-known results for the linearized operator L defined by (1.8). As observed in [9] and probably known earlier, if λ is an eigenvalue, then so are −λ and ±¯λ. Hence if λ 6= 0 is real or purely imaginary, it comes in a pair. If it is complex with nonzero real and imaginary parts, it comes in a quadruple. It follows from nonlinear stability and instability results [36, 10] that all eigenvalues are purely imaginary if p ∈ (1, pc), and
that there is at least one eigenvalue with positive real part when p ∈ (pc, pmax). It is also
known (see e.g. [6]) that the set of isolated and embedded eigenvalues is finite, and the dimensions of the corresponding generalized eigenspaces are finite.
2.1 L+, L−, and the generalized nullspace of L
Here we recall the makeup of the generalized nullspace Ng(L) of L. Easy computations give
L+Q1= −2Q, L−Q = 0, where Q1 := (p−12 + x · ∇)Q, (2.1)
and
In the critical case p = pc, we also have
L−(|x|2Q) = −4Q1, L+ρ = |x|2Q (2.3)
for some radial function ρ(x) (for which we do not know an explicit formula in terms of Q). Denote
δppc =
1 p = pc
0 p 6= pc. (2.4)
For 1 < p < pmax, the generalized nullspace of L is given by
Ng(L) = span 0 Q , 0 xQ , δppc 0 |x|2Q , ∇Q 0 , Q1 0 , δppc ρ 0 . (2.5) In particular dim Ng(L) = 2n + 2 + 2δppc.
The fact that the vectors on the r.h.s of (2.5) lie in Ng(L) follows immediately from the
computations (2.1)-(2.3). That these vectors span Ng(L) is established in [35], Theorems
B.2 and B.3, which rely on the non-degeneracy of the kernel of L+:
Lemma 2.1 For all n ≥ 1 and p ∈ (1, pmax),
N (L+) = span {∇Q}
This lemma is proved in [35] for certain n and p (n = 1 and 1 < p < ∞, or n = 3 and 1 < p ≤ 3), and is completely proved later by a general result of [14]. We present here a direct proof of this lemma, without referring to [14], relying only on oscillation properties of Sturm-Liouville ODE eigenvalue problems. A similar argument (which in the present case, however, applies only for p ≤ 3) appears in [11], Appendix C. For completeness, we also include some arguments of [35].
Proof. We begin with the cases n ≥ 2. Since the potential in L+ is radial, any
solution of L+v = 0 can be decomposed as v =Pk≥0Pj∈Σkvk,j(r)Yk,j(ˆx), where r = |x|,
ˆ
x = xr is the spherical variable, and Yk,j denote spherical harmonics: −∆Sn−1Yk,j = λkYk,j
(a secondary multi-index j, appropriate to the dimension, runs over a finite set Σk for each
k). Then L+v = 0 can be written as Akvk,j= 0, where, for k = 0, 1, 2, 3, . . .,
A0 = −∂r2−n − 1r ∂r+ 1 − pQp−1(r), Ak= A0+ λkr−2, λk= k(k + n − 2).
Case 1: k = 1. Note ∇Q = Q′(r)ˆx. Since A1Q′ = 0 and Q′(r) < 0 (monotonicity of the
ground state) for r ∈ (0, ∞), Q′(r) is the unique ground state of A1 (up to a factor), and
so A1 ≥ 0, A1|{Q′}⊥ > 0.
Case 2: k ≥ 2. Since Ak= A1+ (λk− λ1)r−2 and λk> λ1, we have Ak > 0, and hence
Case 3: k = 0. Note that the first eigenvalue of A0 is negative because (Q, A0Q) =
(Q, −(p − 1)Qp) < 0. The second eigenvalue is non-negative due to (2.7) and the minimax principle. Hence, if there is a nonzero solution of A0v0 = 0, then 0 is the second eigenvalue.
By Sturm-Liouville theory, v0(r) can be taken to have only one positive zero, which we
denote by r0 > 0. By (2.1), A0Q = −(p − 1)Qp and A0Q1 = −2Q. Hence (Qp, v0) = 0 =
(Q, v0). Let α = (Q(r0))p−1. Since Q′(r) < 0 for r > 0, the function Qp−αQ = Q(Qp−1−α)
is positive for r < r0 and negative for r > r0. Thus v0(Qp − αQ) does not change sign,
contradicting (v0, Qp− αQ) = 0. Combining all these cases gives Lemma 2.1 for n ≥ 2.
Finally, consider n = 1. Suppose L+v = 0. Since L+ preserves oddness and evenness,
we may assume v is either odd or even. If it is odd, it vanishes at the origin, and so by linear ODE uniqueness, v is a multiple of Q′. So suppose v is even. As in Case 3 above, since L+ has precisely one negative eigenvalue, and has Q′ in its kernel, v(x) can be taken
to have two zeros, at x = ±x0, x0 6= 0. The argument of Case 3 above then applies on
[0, ∞) to yield a contradiction.
We complete this section by summarizing some positivity estimates for the operators L+ and L−. These estimates are closely related to the stability/instability of the ground
state. Lemma 2.2 L−≥ 0, L−|{Q}⊥ > 0 (1 < p < pmax) (2.6) (Q, L+Q) < 0, L+|{Qp}⊥ ≥ 0 (1 < p < pmax), (2.7) L+|{Q}⊥ ≥ 0 (1 < p ≤ pc) (2.8) L+|{Q,xQ}⊥ > 0, L−| {Q1}⊥ > 0 (1 < p < pc) (2.9) L+|{Q,xQ,|x|2Q}⊥> 0, L−| {Q1,ρ}⊥ > 0 (p = pc). (2.10)
Proof. Estimates (2.6) follow from L−Q = 0, Q > 0, and the fact that a positive eigenfunction of a Schr¨odinger operator is its ground state ([23]). Estimates (2.9)–(2.10) are [35] Propositions 2.9 and 2.10. Estimate (2.8) is [35] Proposition 2.7. A refinement of its proof gives the second part of estimate (2.7) (the first part is a simple computation) as follows. Recall the ground state Q is obtained by the minimization problem (1.5). If a minimizer Q(x) is rescaled so that
R |∇Q|2 2a = R Q2 2b = R Qp+1 p + 1 = constant k > 0,
i.e., (1.4) is satisfied, then Q(x) satisfies (1.3). The minimization inequality dεd22
ε=0J[Q +
εη] ≥ 0 for all real functions η, is equivalent to k(η, L+η) ≥ 1 a( Z η∆Q)2+1 b( Z Qη)2− ( Z Qpη)2. (2.11) Thus (η, L+η) ≥ 0 if η ⊥ Qp. Note that, if η ⊥ Q, by (1.3) the right side of (2.11) is
2.2 Variational formulations of the eigenvalue problem for L
In this subsection we summarize various variational formulations for eigenvalues of L. The generalized nullspace is given by (2.5). Suppose λ 6= 0 is a (complex) eigenvalue of L with corresponding eigenfunction [wu] ∈ L2, 0 L− −L+ 0 u w = λ u w . (2.12)
The functions u and w satisfy
L+u = −λw, L−w = λu. (2.13)
Therefore
L−L+u = µu, µ = −λ2. (2.14)
Since (µu, Q) = (L−L+u, Q) = (L+u, L−Q) = 0 and µ 6= 0, we have u ⊥ Q.
Denote by Π the L2-orthogonal projection onto Q⊥. We can write L+u = ΠL+u + αQ.
Eq. (2.14) implies L−ΠL+u = µu and hence, using u ⊥ Q and (2.6), ΠL+u = L−1− µu. Thus
(u, Q) = 0, L+u = µL−1− u + αQ. (2.15)
Since (2.14) is also implied by Eq. (2.15), these two equations are equivalent.
If Q(x) is a general solution of (1.3), µ = −λ2 may not be real. However, it must be
real for the nonlinear ground state Q = Q0,0,p. This fact is already known (see [25]). We
will give a different proof.
Claim. For Q = Q0,0,p, every eigenvalue µ of (2.14) is real.
Proof. Multiply (2.13) by ¯u and ¯w respectively and integrate. Then we get
(u, L+u) = −λ(u, w), (w, L−w) = λ(w, u) = λ(u, w). (2.16)
Taking the product, we get
(u, L+u)(w, L−w) = −λ2|(u, w)|2= µ|(u, w)|2.
If µ 6= 0, w is not a multiple of Q, and so by (2.6), (w, L−w) > 0. Hence (u, w) 6= 0 by
(2.16). Thus
µ = (u, L+u)(w, L−w) |(u, w)|2 ∈ R.
This argument does not work when Q is an excited state, since (u, w) may be zero (see e.g. [33, Eq.(2.63)]). The fact µ ∈ R implies that eigenvalues λ of L are either real or purely imaginary. Thus L has no complex eigenvalues with nonzero real and imaginary parts. This is not the case for excited states (see Section 5, also [33]).
The proof of reality of µ in [25] uses the following formulation. For the nonlinear ground state Q, L− is nonnegative and the operator L1/2− is defined on L2 and invertible on Q⊥. A nonzero µ ∈ C is an eigenvalue of (2.14) if and only if it is also an eigenvalue of the following problem:
L1/2− L+L1/2− g = µg, (2.17)
with g = L−1/2− u. The operator L1/2− L+L1/2− already appeared in [32]. Since it can be
realized as a self-adjoint operator, µ must be real.
Furthermore, the eigenvalues of L1/2− L+L1/2− can be counted using the minimax
princi-ple. Note that Q is an eigenfunction with eigenvalue 0. For easy comparison with other formulations, we formulate the principle on Q⊥. Let
µj := inf
g⊥Q,gk,k=1,...,j−1
(g, L1/2− L+L1/2− g)
(g, g) , (j = 1, 2, 3, . . .) (2.18) with a suitably normalized minimizer denoted by gj (if it exists – the definition terminates
once a minimizer fails to exist). The corresponding definition for (2.15) is µj := inf
u⊥Q, (u, L−1− uk)=0, k=1,...,j−1
(u, L+u)
(u, L−1− u), (j = 1, 2, 3, . . .) (2.19) with a suitably normalized minimizer denoted by uj (if it exists). In fact, the minimizer uj
satisfies
L+uj = µjL−1− uj+ αjQ + β1L−1− u1+ · · · + βj−1L−1− uj−1, (2.20)
for some Lagrange multipliers β1, . . . βj−1. Testing (2.20) with uk with k < j, we get
(uk, βkL−1− uk) = (uk, L+uj) = (L+uk, uj) = 0 by (2.20) for uk and the orthogonality
condi-tions. Thus βk= 0 and L+uj = µjL−1− uj+ αjQ and hence uj satisfies (2.15).
Lemma 2.3 The eigenvalues of (2.18) and (2.19) are the same, and if 1 < p < pc : µ1 = · · · = µn= 0, µn+1 > 0.
if p = pc : µ1 = · · · = µn+1= 0, µn+2> 0.
if pc < p < pmax: µ1 < 0, µ2 = · · · = µn+1= 0, µn+2> 0.
The 0-eigenspaces are span L−1/2− {∇Q, δppcQ1} for (2.18) and span{∇Q, δ
p
pcQ1} for (2.19),
where δppc is defined in (2.4).
Proof. The eigenvalues of (2.18) and (2.19) are seen to be the same by taking g = L−1/2− u up to a factor. By estimate (2.8), µ1 ≥ 0 for p ∈ (1, pc]. For p ∈ (pc, pmax), using
(1.4), ΠQ1= Q1−(Q(Q,Q)1,Q)Q, and elementary computations (such as (2.22) below), one finds
(ΠQ1, L+ΠQ1) = n2(p − 1) 4 (pc− p) 1 p + 1 Z Qp+1
which is negative for p > pc. Thus µ1< 0. By estimate (2.7), µ2 ≥ 0 for p ∈ (1, pmax).
It is clear that u = ∂x∂
jQ, j = 1, . . . , n, provides n 0-eigenfunctions. For p = pc, another
0-eigenfunction is u = Q1 since Q1 ⊥ Q (see again (2.22) below), L−1− ∇Q = −12xQ, and
(Q1, L+Q1) = 0. It remains to show that µn+1 > 0 for p ∈ (1, pc) and µn+2 > 0 for
p ∈ [pc, pmax). If µn+2 = 0 for p ∈ (pc, pmax), the argument after (2.20) shows the existence
of a function un+26= 0 satisfying
L+un+2= αQ for some α ∈ R, un+2 ⊥ Q, L−−1u1, L−1− ∇Q = −12xQ.
By Lemma 2.1, un+2+α2Q1= c · ∇Q for some c ∈ Rd. The orthogonality conditions imply
un+2 = 0. The cases p ∈ (1, pc] are proved similarly.
Remark 2.4 The formulation (2.19) for µ1 has been used for the stability problem, see
e.g. [30, p.73, (4.1.9)], which can be used to prove that µ1 < 0 if and only if p ∈ (pc, pmax)
by a different argument. The later fact also follows from [35, 10] indirectly. We summarize our previous discussion in the following theorem.
Theorem 2.5 Let Q(x) be the unique positive radial ground state solution of (1.3), and let L, L+ and L− be as in (1.8) and (1.10). The eigenvalue problems (2.14), (2.15), and
(2.17) for µ 6= 0 are equivalent, and the eigenvalues µ must be real. These eigenvalues can be counted by either (2.18) or (2.19). µ1 < 0 if and only if p ∈ (pc, pmax). Furthermore, all
eigenvalues of L are purely imaginary except for an additional real pair when p ∈ (pc, pmax).
The last statement follows from the relation µ = −λ2 in (2.14).
2.3 Spectrum near 0 for p near pc
We now consider eigenvalues of L near 0 when p is near pc. It was suggested by M.I.
Weinstein that as p approaches pc from below, a pair of purely imaginary eigenvalues will
collide at the origin, and split into a pair of real eigenvalues for p > pc. Although this picture
is well-known, there does not seem to be a written proof. In the following theorem and corollary we prove this picture rigorously and identify the leading terms of the eigenvalues and eigenfunctions.
Theorem 2.6 There are small constants µ∗ > 0 and ε∗ > 0 so that for every p ∈ (pc−
ε∗, pc+ ε∗), there is a solution of L+L−w = µw (2.21) of the form w = w0+ (p − pc)2g, w0= Q + a(p − pc)|x|2Q, g ⊥ Q, µ = 8a(p − pc) + (p − pc)2η, a = a(p) = n(Q1, Q p) 4(Q1, x2Q) < 0,
with kgkL2, |η|, |a| and 1/|a| uniformly bounded in p. Moreover, for p 6= pc, this is the
Proof. Set ε := p − pc. Computations yield (Q1, Q) = 2 p − 1 − n 2 (Q, Q) = − εn 2(p − 1)(Q, Q), (2.22) (Q1, Qp) = − 1 p − 1(L+Q, Q1) = − 1 p − 1(Q, L+Q1) = 2 p − 1(Q, Q), (2.23) and (Q1, |x|2Q) = 2 p − 1 − n + 2 2 (Q, |x|2Q) = −(1 + εn 2(p − 1))(Q, |x| 2Q). (2.24) Since by (2.21) with µ 6= 0, (Q1, w) = µ−1(Q1, L+L−w) = µ−1(L−L+Q1, w) = 0,
we require the leading term (Q1, w0) = 0, which decides the value of a using (2.22) and
(2.24). Thus we also need (Q1, g) = 0. That a < 0 (at least for ε sufficiently small) follows
from (2.23) and (2.24). Using the computations
L−|x|2Q = [L−, |x|2]Q = −4x · ∇Q − 2nQ = −4Q1− 2n p − 1εQ (2.25) and L+Q = [L−− (p − 1)Qp−1]Q = −(p − 1)Qp, we find L+L−w0= aεL+[−4Q1− 2n p − 1εQ] = aε[8Q + 2nεQ p].
Thus µ = 8aε + o(ε) and we need to solve
0 = [L+L−− 8aε − ε2η][w0+ ε2g]
which yields our main equation for g and η:
L+L−g = 8a2(|x|2Q) − 2an(Qp) + ηw0+ (8aε + ε2η)g. (2.26)
Recall that on radial functions (we will only work on radial functions here) ker[(L+L−)∗] = ker[L−L+] = span{Q1}.
Let P denote the L2-orthogonal projection onto Q1 and ¯P := 1 − P . It is necessary that
P [8a2(|x|2Q) − 2an(Qp) + ηw0+ (8aε + ε2η)g] = 0
for (2.26) to be solvable. This solvability condition holds since (Q1, g) = (Q1, w0) = 0, and,
Consider the restriction (on radial functions)
T = L+L−: [ker L−]⊥= Q⊥−→ Ran( ¯P ) = Q⊥1.
Its inverse T−1 = (L
−)−1(L+)−1 is bounded because (L+)−1 : Q⊥1 → Q⊥ and (L−)−1 :
Q⊥→ Q⊥ are bounded. So our strategy is to solve (2.26) as
g = T−1P [8a¯ 2(|x|2Q) − 2an(Qp) + ηw0+ (8aε + ε2η)g] (2.27)
by a contraction mapping argument, with η chosen so that (Q1, g) = 0. Specifically, we
define a sequence g0= 0, η0= 0, and
gk+1 = ¯P T−1P [8a¯ 2(|x|2Q) − 2an(Qp) + ηkw0+ (8aε + ε2ηk)gk],
ηk+1 = −
1 (Q1, T−1w0)
(Q1, T−1P [8a¯ 2(|x|2Q) − 2an(Qp) + (8aε + ε2ηk)gk]).
We need to check (Q1, T−1w0) is of order one. Since w0 = Q + O(ε) and L+Q1 = −2Q,
we have (L+)−1w0 = −12ΠQ1+ O(ε) where Π denotes the orthogonal projection onto Q⊥.
Thus, using (2.25) and (2.22), (Q1, T−1w0) = − 1 2(Q1, (L−)−1ΠQ1) + O(ε) = 1 8(Q1, Π|x| 2Q) + O(ε) = 1 8(Q1, |x| 2Q) + O(ε),
which is of order one because of (2.24). One may then check that Nk := kgk+1− gkkL2 +
ε1/2|η
k+1− ηk| satisfies Nk+1 ≤ Cε1/2Nk, and hence (gk, ηk) is indeed a Cauchy sequence.
Finally, the uniqueness follows from the invariance of the total dimension of generalized
eigenspaces near 0 under perturbations.
Remark 2.7 To understand heuristically the leading terms in w and µ, consider the fol-lowing analogy. Let Aε =
0 1 0 ε
, which corresponds to L+L−. One has Aε[10] = [00],
Aε[01] = [1ε] and Aε[1ε] = ε [1ε]. The vectors [10], [01] and [1ε] correspond to Q, |x|2Q and w,
respectively.
The theorem yields an eigenvalue µ with the same sign as pc− p. Since the eigenvalues
of L are given by λ = ±√−µ, we have the following corollary.
Corollary 2.8 With notations as in Theorem 2.6, L has a pair of eigenvalues λ = ±√−µ = ±p8|a|(p − pc) − (p − pc)2η with corresponding eigenvectors [wu] solving (2.12) and
u = λ−1L−w = ∓p2|a|(p − pc) Q1+ O((p − pc)3/2).
When p ∈ (pc− ε∗, pc) (stable case), λ and u are purely imaginary.
When p ∈ (pc, pc+ ε∗) (unstable case), λ and u are real.
In deriving the leading term of u we have used (2.25). We solved for w before u simply because w is larger than u.
3
One dimensional theory
In this section we focus on the one dimensional theory. For n = 1, the ground state Q(x) has an explicit formula for all p ∈ (1, ∞),
Q(x) = cpch−β(x/β), cp := ( p + 1 2 ) 1 p−1, β := 2 p − 1, (3.1) where ch(x) denotes the hyperbolic cosine function cosh(x). The function Q(x) satisfies (1.3) and is the unique H1(R)-solution of (1.3) up to translation and phase [4, p.259, Theorem 8.1.6].
3.1 Eigenfunctions of L+ and L−
We first consider eigenvalues and eigenfunctions of L+ and L−. For n = 1,
L+= −∂xx+ 1 − pQp−1, L−= −∂xx+ 1 − Qp−1. (3.2)
By (3.1), these operators are both of the form
−∂xx+ 1 − C sech2(x/β).
Such operators have essential spectrum [1, ∞), and finitely many eigenvalues below 1. A lot of information about such operators is available in the classical book [31], p. 103:
• all eigenvalues are simple, and can be computed explicitly, as zeros and poles of an explicit meromorphic function;
• all eigenfunctions can be expressed in terms of the hypergeometric function.
We begin by presenting another way to derive the eigenvalues, as well as different for-mulas for the eigenfunctions. We will not prove right here that this set contains all of the eigenvalues/eigenfunctions. This fact is a consequence of the more general Theorem 3.4, proved later (and see also [31]).
Define λm := 1 − km2, km:= p + 1 2 − m(p − 1) 2 , pm:= m + 1 m − 1 for m > 1, p1 = ∞. (3.3)
The following theorem agrees with the numerical observation Figure 1.
Theorem 3.1 In one space dimension with 1 < p < ∞, let Q(x) be defined by (3.1), L+
and L− be defined by (3.2), and λm be defined by (3.3). Suppose for j ∈ Z+,
Then the operator L+ has eigenvalues λm, m = 0, 1, 2, 3, . . . , j, with eigenfunctions ϕm to
be defined by (3.6) and (3.7), which are odd for odd m and even for even m. The operator L− has eigenvalues λm, m = 1, 2, 3, . . . , j, with eigenfunctions ψm to be defined by (3.9),
which are odd for even m and even for odd m.
In particular, all eigenvalues of L− are eigenvalues of L+, and L+ always has one more
eigenvalue (λ0) than L−. Note that λ0 < 0 = λ1 and λm> 0 for 2 ≤ m ≤ j.
Proof. We first consider even eigenfunctions of L+. By the explicit formula (3.1) of
Q(x), we have Qp−1= p + 1 2 ch −2(x/β), Qx= −Q tanh(x/β), Q2x= Q2(1 − 1 ch2) = Q 2 (1 −p + 12 Qp−1). Thus Q−k∂xxQk= (k2− k)Q−2Q2x+ kQ−1Qxx = (k2− k)(1 − 2 p + 1Q p−1) + k(1 − Qp−1). By (3.2), Q−kL+Qk = −(k2− k)(1 − 2 p + 1Q p−1) − k(1 − Qp−1) + 1 − pQp−1 = fp(k)Qp−1+ (1 − k2), (3.5) where fp(k) = 2 p + 1(k + p) (k − p + 1 2 ). Let k0 = p + 1 2 , ϕ0= Q k0, λ 0 = 1 − k20 < 0. We have fp(k0) = 0. Thus L+ϕ0 = λ0ϕ0. If 1 < p < 3, let k2= k0− (p − 1) = 3 − p 2 , ϕ2= Q k2 + c2 0Qk0, λ2 = 1 − k22.
Here and after, cjm denotes constants to be chosen. We have k2 > 0, 0 < λ2 < 1 for p < 3
and, by (3.5), L+ϕ2= [fp(k2)Qk0 + λ2Qk2] + c20λ0Qk0. If we choose c20= fp(k2) λ2− λ0 = fp(k2) k20− k22 = − 3 + p 2(p + 1),
we have L+ϕ2= λ2ϕ2, ϕ2 = Qk2+ fp(k2) λ2− λ0Q k0. If 1 < p < 5/3, let k4 = k2− (p − 1) = 5 − 3p 2 , ϕ4 = Q k4+ c4 2Qk2+ c40Qk0, λ4 = 1 − k42. We have k4 > 0 and L+ϕ4 = [fp(k4)Qk2 + λ4Qk4] + c42[fp(k2)Qk0 + λ2Qk2] + c40λ0Qk0. If we choose c42 = fp(k4) λ4− λ2 , c40= c 4 2· fp(k2) λ4− λ0 , we have L+ϕ4= λ4ϕ4, ϕ4 = Qk4+ fp(k4) λ4− λ2 Qk2 + fp(k4) λ4− λ2 · fp(k2) λ4− λ0 Qk0.
For general integer m ≥ 0, let km be defined by (3.3), which is positive if m = 0, 1, or if
p < m+1m−1. If k2j > 0, the j-th even eigenfunction of L+ is given by
ϕ2j = j
X
m=0
c2j2mQk2m, (3.6)
with eigenvalue λ2j = 1 − k2j2 and
c2j2j = 1, c2j2m= c2j2m+2·fp(k2m+2) λ2j − λ2m
, (0 ≤ m ≤ j − 1). We next consider odd eigenfunctions of L+. Since
∂x2(Qk)x= 3k(k − 1)Qk−2QxQxx+ k(k − 1)(k − 2)Qk−3Q3x+ kQk−1Qxxx, we have [(Qk)x]−1∂2x(Qk)x= 3(k − 1)Q−1Qxx+ (k − 1)(k − 2)Q−2Q2x+ Q−1x Qxxx = 3(k − 1)(1 − Qp−1) + (k − 1)(k − 2)(1 −p + 12 Qp−1) + (1 − pQp−1) = k2+ Qp−1[−3k + 3 − 2 p + 1(k 2− 3k + 2) − p]. Thus [(Qk)x]−1L+(Qk)x = gp(k)Qp−1+ 1 − k2, where gp(k) = 2k + 3p − 1 p + 1 (k − 1).
The only zero of gp(k) = 0 with k > 0 is k = k1 = 1. The first odd eigenfunction of L+
is given by
ϕ1= (Qk1)x, k1 = 1, L+ϕ1 = 0.
If k2j−1 > 0, the j-th odd eigenfunction of L+ is
ϕ2j−1 = j X m=1 c2j−12m−1(Qk2m−1) x, (3.7)
with eigenvalue λ2j−1 = 1 − k22j−1 and coefficients
c2j−12j−1= 1, c2j−12m−1= c2j−12m+1· gp(k2m+1)
λ2j−1− λ2m−1, (0 < m ≤ j − 1).
In conclusion, if pm+1 ≤ p < pm (p1 = ∞), we have found m + 1 eigenvalues for L+.
We now consider L−, which is similar. We have
Q−kL−Qk= fp−(k)Qp−1+ (1 − k2), (3.8) [(Qk)x]−1L−(Qk)x= g−p(k)Qp−1+ (1 − k2), where fp−(k) = fp(k) + p − 1 = 1 p + 1(k − 1)(2k + p + 1), gp−(k) = gp(k) + p − 1 = 1 p + 1(k + p)(2k + p − 3).
The only zero for fp− is k = 1 = k1. The only zero for g−p is k = 3−p2 = k2. Thus the
first even eigenfunction for L− is Qk1 and the first odd eigenfunction for L
− is (Qk2)x. By
an argument similar to those for L+, if pm+1 ≤ p < pm, we can find m eigenfunctions
ψ1, . . . , ψm for L− of the form
ψ2j−1 = j X m=1 d2j−12m−1Qk2m−1, ψ 2j = j X m=1 d2j2m(Qk2m) x, (3.9)
with eigenvalues λ1, . . . , λm. The coefficients djm can be defined recursively in a similar
fashion. Note that these thresholds for the exponent p are the same as those for L+. In
particular, the number of eigenvalues of L−is always that of L+minus one. This completes
3.2 Connection between L+ and L− and their factorizations
In light of Theorem 3.1, it is natural to ask why all eigenvalues of L− are also eigenvalues of L+. Is there a simple connection between their eigenfunctions? In this section we prove
this is indeed so.
We first look for an operator U of the form
U = ∂x+ R(x), ( so U∗= −∂x+ R(x)),
such that
L−U = U L+, ( so U∗L−= L+U∗). (3.10)
It turns out that there is a unique choice of R(x): R(x) = −p + 12 QQx = p + 1
2 tanh(
(p − 1)x 2 ). In fact, with this choice of R(x),
U = ϕ0∂xϕ−10 , ( so U∗= −ϕ−10 ∂xϕ0), (3.11)
where ϕ0 = Q
p+1
2 is the ground state of L+, and is considered here as a multiplication
operator: U f = ϕ0∂x(ϕ−10 f ).
Suppose now ψ is an eigenfunction of L− with eigenvalue λ: L−ψ = λψ. By (3.10), 0 = U∗(L−− λ)ψ = (L+− λ)U∗ψ.
Thus U∗ψ is an eigenfunction of L+with same eigenvalue λ (provided U∗ψ ∈ L2). Therefore,
the map
ψ 7→ U∗ψ
sends an eigenfunction of L− to an eigenfunction of L+ with same eigenvalue. This map is
not onto because U∗ is not invertible. Specifically, the ground state ϕ0 is not in the range.
In fact, U ϕ0 = ϕ0∂xϕ−10 ϕ0 = 0. If ϕ0 = U∗ψ, then (ϕ0, ϕ0) = (ϕ0, U∗ψ) = (U ϕ0, ψ) = 0, a
contradiction. We summarize our finding as the following proposition.
Proposition 3.2 Under the same assumptions and notation as Theorem 3.1, the eigen-functions ϕm and ψm of L+ and L− satisfy
ϕm= U∗ψm, (m = 1, . . . , j),
up to constant factors. Note that U∗ sends even functions to odd functions and vice versa.
Proof. We only need to verify that U∗ψm ∈ L2. This is the case since U∗ = −∂x+ p+1
2 tanh(x/β), ψm(x) are sums of powers of Q and Qx, and that tanh(x/β), Qx/Q, and
Analogous to the definition of U , we define S := Q∂xQ−1= ∂x− Qx Q, ( so S ∗ = −Q−1∂ xQ). (3.12)
Clearly SQ = 0. Recall that λ0 is the first eigenvalue of L+ with eigenfunction ϕ0. Hence
L+− λ0 is a nonnegative operator. In fact we have the following factorizations.
Lemma 3.3 Let U and S be defined by (3.11) and (3.12), respectively. One has
L+− λ0 = U∗U, L−− λ0 = U U∗. (3.13)
L−= S∗S, SS∗= −∂x2+ 1 + p − 3 p + 1Q
p−1. (3.14)
Moreover, SS∗ > 0.
The last fact is due to SS∗ = L−+2(p−1)p+1 Qp−1. The formula L−= S∗S was known, see e.g. [30, p.73, (4.1.8)].
3.3 Hierarchy of Operators
In this subsection we generalize Theorem 3.1 and Lemma 3.3 to a family of operators containing L+ and L−. As a reminder, we have
Q′′/Q = 1 − Qp−1, (Q′/Q)2 = 1 − 2
p + 1Qp−1, (Q′/Q)′= Q′′/Q − (Q′/Q)2 = −p − 1
p + 1Qp−1.
(3.15)
Let S(a) := Qa∂xQ−a. We have
S(a) = ∂x− aQ′/Q, S(a)∗ = −∂x− aQ′/Q, S(a)∗S(a) = −∂2x+ a2− a a +p − 1 2 2 p + 1Q p−1. (3.16)
Define the following hierarchy of operators:
Sj := S(kj), where recall kj = 1 − (j − 1)p − 1 2 , Lj := Sj−1Sj−1∗ + λj−1 = Sj∗Sj+ λj, where recall λj = 1 − k2j. (3.17) Then we have S0= U, S1= S, . . . L0 = L+, L1 = L−, L2 = SS∗, . . . SjLj = Lj+1Sj, LjSj∗= Sj∗Lj+1. (3.18)
More explicitly,
Lj = −∂x2+ 1 − kj−1kj
2 p + 1Q
p−1. (3.19)
Note that j here can be any real number.
Recall the definition pj := 1 + 2/(j − 1) for j > 1, and set pj = ∞ for j ≤ 1. Then pj
is a monotone decreasing function of j, kj > 0 for p < pj, kj = 0 for p = pj and kj < 0 for
p > pj. Let λ′j := λj (1 < p ≤ pj), 1 (pj < p < pj−1), λj−1 (pj−1≤ p). (3.20) By the second identity of (3.17), and (3.19) together with the fact kj−1kj < 0 for pj < p <
pj−1, we have the lower bound
Lj ≥ λ′j. (3.21)
In fact, this estimate is sharp: for p ∈ (1, pj) ∪ (pj−1, ∞), the ground state is obvious from
the second identity of (3.17): ( LjQj = λjQj, (1 < p < pj), LjQ∗j−1= λj−1Q∗j−1, (pj−1 < p), (3.22) where we denote Qj := Qkj, Q∗j := Q−kj. (3.23)
For p ∈ [pj, pj−1], there is no ground state. Thus we have completely determined the ground
state of Lj for all p > 1. The complete spectrum, together with explicit eigenfunctions, are
derived using the third identity of (3.18) as follows.
Theorem 3.4 For any j ∈ R and p > 1, the point spectrum of Lj consists of simple
eigenvalues
specp(Lj) ={λk| p < pk, k ∈ {j, j + 1, j + 2, . . . }}
∪ {λk| p > pk, k ∈ {j − 1, j − 2, j − 3, . . . }},
(3.24) and the eigenfunction for the eigenvalue λk is given uniquely up to constant multiple by
( S∗
j · · · Sk−1∗ Qk (k ∈ {j, j + 1, . . . }),
Sj−1· · · Sk+1Q∗k (k ∈ {j − 1, j − 2, . . . }),
(3.25) each of which is a linear combination of
Qj, Qj+2, . . . Qk (k ∈ {j, j + 2, . . . }), Qj+1R, Qj+3R, . . . QkR (k ∈ {j + 1, j + 3, . . . }), Q∗j−1, Q∗j−3, . . . Q∗k (k ∈ {j − 1, j − 3, . . . }), Q∗ j−2R, Q∗j−4R, . . . Q∗kR (k ∈ {j − 2, j − 4, . . . }) (3.26) where R := Q′/Q.
Proof. The ground states have been determined. The third identity of (3.18) implies that (3.25) belong to the eigenspace of Lj with eigenvalue λk. Moreover, each function is
nonzero because S∗
k is injective for p < pk and so is Skfor pk< p. Since Sj annihilates only
the ground state Qj for p < pj and Sj−1∗ annihilates only the ground state Q∗j−1 for p > pj,
all the excited states of Lj for p < pj are mapped injectively by Sj to bound states of Lj+1,
and for p > pj by Sj−1∗ to those of Lj−1. Hence we have (3.24) and all the eigenvalues are
simple because the ground states are so. (3.26) follows from the fact that Sj and Sj∗ act on
Qa like C(a, j)R, while SjSj−1 and Sj−1∗ Sj∗ act on Qa like C1(a, j) + C2(a, j)Qp−1.
3.4 Mirror conjugate identity
The following remarkable identity has application to estimating eigenvalues of L (see Sec-tion 3.6):
Sj(Lj−1− λj)Sj∗= Sj∗(Lj+2− λj)Sj. (3.27)
To prove this, start with the formula (∂x+ R)(∂x2+ V )(∂x− R)
= ∂x4+ (−3R′− R2+ V )∂x2+ (−3R′− R2+ V )′∂x
− R′′′− (V R)′− RR′′− R2V, (3.28) which implies that (∂x+ R)(∂x2+ V+)(∂x− R) = (∂x− R)(∂x2+ V−)(∂x+ R) is equivalent to
V±= −R′′/R ± 3R′− R2+ C/R. (3.29) Now set R := aQ′/Q. Plugging the following identities
R2= a2(1 − 2 p + 1Q p−1), R′= −ap − 1 p + 1Q p−1, R′′/R = −(p − 1) 2 p + 1 Q p−1. (3.30)
into (3.29), we get, for C = 0, V±= −a2+ 2
p + 1(a ± (p − 1))(a ± (p − 1)/2)Q
p−1. (3.31)
Hence for a = kj we have
V±= −kj2+ 2
p + 1kj±2kj±1, (3.32) which gives the desired identity (3.27). The above proof also shows that Lj−1 and Lj+2 are
the unique choice for the identity to hold with Sj (modulo a constant multiple of Q/Qx,
3.5 Variational formulations for eigenvalues of L
We considered two variational formulations for nonzero eigenvalues of L in general dimen-sions in Section 2.2. Here we present a new variational formulation for 1-D. Define the selfadjoint operator
H := SL+S∗. (3.33)
This is a fourth-order differential operator, with essential spectrum [1, ∞). By a direct check, we have
HQ = SL+S∗Q = SL+(−2Qx) = 0.
Thus Q is an eigenfunction with eigenvalue 0. Since (Q, S∗f ) = (SQ, f ) = 0 for any f , we have
Range S∗ ⊥ Q. (3.34)
In particular, since L+|Q⊥ is nonnegative for p ≤ 5 by Lemma 2.2, so is H.
Lemma 3.5 The null space of H is
N (H) = spanQ, δppcxQ , where, recall, δppc is 0 if p 6= pc, and 1 if p = pc.
Remark. Note that dim N (H) = 1 + δppc which is different from dim N (L
1/2
− L+L1/2− ) =
2 + δppc. We will show below that H and L
1/2
− L+L1/2− have the same nonzero eigenvalues.
Proof. If Hf = 0, then L+S∗f = −2cQ and S∗f = cQ1 + dQx for some c, d ∈ R
by Lemma 2.1. We have Q1 ⊥ Q iff p = pc = 5. Thus, if p 6= 5, c = 0 by (3.34), and
S∗(f +d2Q) = 0. We conclude f = −d2Q.
When p = 5, we have S∗xQ = −Q−1∂x(QxQ) = −2Q1. Thus S∗(f + c
2Q1 +d2Q) = 0
and f = −c2Q1−d2Q.
Define eigenvalues of H as follows: ˜
µj := inf f ⊥fk,k<j
(f, Hf )
(f, f ) , (j = 1, 2, 3, . . .) (3.35) with a suitably normalized minimizer denoted by fj, if it exists. By standard variational
arguments, if ˜µj < 1, then a minimizer fj exists. By convention, if µkis the first of the µj’s
to hit 1 (and so fk may not be defined), we set µj := 1 for all j > k.
We can expand Theorem 2.5 to the following.
Theorem 3.6 (Equivalence) Let n = 1. Let µj be defined as in Theorem 2.5 and ˜µj be
defined by (3.35). Then µj = ˜µj. When µj 6= 0 and µj < 1, the eigenfunctions of (2.19)
and (3.35) can be chosen to satisfy
uj = S∗fj, fj =
1 µj
Proof. First we establish the equivalence of nonzero eigenvalues. Suppose f = fj is an
eigenfunction of (3.35) with eigenvalue ˜µ 6= 0, then SL+S∗f = ˜µf . Let u := S∗f 6= 0 and
apply S∗ on both sides. By L
− = S∗S we get L−L+u = ˜µu. Thus u is an eigenfunction
satisfying (2.14) with µ = ˜µ. On the other hand, suppose u satisfies L−L+u = µu with
µ 6= 0. Applying SL+ on both sides and using L− = S∗S, we get SL+S∗SL+u = µSL+u,
i.e., Hf = µf for f = µ−1SL+u.
Now use Lemmas 2.3 and 3.5. If p ∈ (1, 5), then µ1 = ˜µ1 = 0, corresponding to Qx and
Q, and µ2 = ˜µ2 > 0. If p = 5, then µ1 = µ2 = ˜µ1 = ˜µ2 = 0, corresponding to Qx, Q1, and
Q, xQ, and µ3 = ˜µ3 = 1. If p ∈ (5, ∞), then µ1 = ˜µ1 < 0, µ2 = ˜µ2 = 0, corresponding to
Qx and Q, and µ3= µ3 = 1. We have shown ˜µj = µj.
In the following we will make no distinction between µj and ˜µj. By the minimax
principle, (3.35) has the following equivalent formulations: µj = inf
dim M =j f ∈Msup
(f, Hf )
(f, f ) =dim M =j−1sup f ⊥Minf
(f, Hf )
(f, f ) . (3.36) Here M runs over all linear subspaces of L2(R) with the specified dimension.
3.6 Estimates of eigenvalues of L
In this subsection we prove lower and upper bounds for eigenvalues of L, confirming some aspects of the numerical computations shown in Figure 1. Recall that, by Lemma 2.3, the first positive µj is µ2 for p ∈ (1, pc) and µ3 for p ∈ [pc, pmax). The first theorem concerns
upper bounds for µ1 and µ2.
- 0.699 - 0.6 - 0.086 0 0.301 0.602 - 8.44 - 0.5 0 0 0.5 1 1 7 5 3 2 1.82 1 1.2 log10(p − 1) µj (p ) p µ1 µ1 µ2 µ2 µ2 µ3 µ4 µ5 Figure 8: p vs. µj.
Theorem 3.7 Suppose n = 1 and 1 < p < ∞.
(a) If p 6= 3, then µ2 ≤ Cp for some explicitly computable Cp < 1. In particular f2
exists.
(b) µ1 < 0 if and only if p > 5. For any C > 0, we have µ1(p) ≤ −Cp3 for p sufficiently
Proof. For part (a), we already know µ2 = 0 for p ≥ 5. Assume p ∈ (1, 5). Consider test
functions of the form f = SQk with k > 0. f is odd and hence f ⊥ Q, the 0-eigenfunction of H. Since H = SL+S∗ and S∗S = L−, we have
µ2≤ (f, Hf ) (f, f ) = (L−Qk, L+L−Qk) (Qk, L −Qk) . By formulas (3.5) and (3.8), L−Qk= aQk+p−1+ bQk, a = fp−(k) = 1 p + 1(k − 1)(2k + p + 1), b = 1 − k 2. L+Qk+p−1= σQk+2p−2+ dQk+p−1, σ = fp(k + p − 1) = 1 p + 1(k + 2p − 1)(2k + p − 3), d = 1 − (k + p − 1) 2. L+Qk= cQk+p−1+ bQk, c = fp(k) = 1 p + 1(k + p)(2k − p − 1). Thus (f, Hf ) (f, f ) = a2σJ3+ a(ad + bc + bσ)J2+ b(ad + ab + bc)J1+ b3J0 aJ1+ bJ0 (3.37) where Jm= Z R Q2k+m(p−1)(x) dx, (m = 0, 1, 2, 3),
which are always positive. If k → 0+, then Jm converges to
R
RQm(p−1)dx for m > 0, and
J0 = O(k−1). The above quotient can be written as
(3.37) = b2+ J aJ1+ bJ0 where J = a2σJ3+ a(ad + bc + bσ)J2+ b(ad + bc)J1. Note that Jm|k=0 = (p+12 )mp−12 R R sech 2m(y) dy with R R sech 2m(y) dy = 2,4 3, 16 15 for m =
1, 2, 3, respectively. Also, as k → 0+, a → −1, b → 1, c → −p, σ → (2p−1)(p−3)p+1 , and
d → 1 − (p − 1)2. Direct calculation shows
lim k→0+J = − 2 15(p − 1)(p + 1) 2 (p − 3)2.
Also note b2 < 1 for k > 0. Thus, if 1 < p < ∞ and p 6= 3, then J < 0 and the quotient
(3.37) is less than 1 for k sufficiently small. (If p = 3, the sign of J is unclear and (3.37) may not be less than 1.) This proves µ2 < 1 and provides an upper bound less than 1 for
For statement (b), the fact that µ1 < 0 if and only if p > 5 is part of Lemma 2.3. We
now consider the behavior of µ1 for p large. Fix k > 1 to be chosen later. As p → ∞,
Jm = ( p + 1 2 ) 2k p−1+m· 2 p − 1 · Z R ( sechx)p−14k +2m dx ∼ C mpm−1, with Cm = 21−m R R( sechx) 2mdx = 2,2
3,154 for m = 1, 2, 3, respectively, and
a ∼ k − 1, b = 1 − k2, c ∼ −p, σ ∼ 2p, d ∼ −p2. Thus, by (3.37), (f, Hf ) (f, f ) ∼ aσJ3+ adJ2 J1 ∼ 1 − k 15 p 3 as p → ∞.
By choosing k > 1 sufficiently large, we have shown that for any C, µ1 ≤ −Cp3 for p
sufficiently large.
The next theorem bounds eigenvalues of L by eigenvalues of L+ and L−. Recall pj and
λj(p) are defined in (3.4) and (3.3).
Theorem 3.8 (Interlacing of eigenvalues) Fix k ≥ 1 and p ∈ [pk+2, pk+1) where,
re-call, pj = j+1j−1. Let λj(p) = 1 −14[(p + 1) − j(p − 1)]2 be as in (3.3) and so λk+1< 1 ≤ λk+2.
For the eigenvalues µj defined by (3.35), we have
λ2j+1(p) < µj+1(p) < λ2j+2(p), (1 ≤ j < k); λ2k+1(p) < µk+1(p) ≤ 1. (3.38)
In particular, there are K simple eigenvalues µ2, . . . , µK+1in (0, 1) where K = k if µk+1 < 1
and K = k − 1 if µk+1 = 1. Moreover, K is always 1 when k = 1. Finally,
µ2≥ ( λ2λ3 (1 < p ≤ 2), λ2 (2 < p < 5), µ3 ≥ λ3λ4 (1 < p ≤ 5/3), λ3 (5/3 < p ≤ 2), 1 (2 < p < ∞), µ1 ≥ − 1 16(p − 1) 3 (p − 5) (5 ≤ p < ∞).
Remark 3.9 In view of the above lower bounds for µ2 and µ3, we conjecture that
µj+1≥ λj+1λj+2 (1 < p < pj+2); µj+1 ≥ λj+1 (pj+2 ≤ p < pj+1). (3.39)
This is further confirmed numerically for j = 3, 4, 5 (see Figure 9). Note that limp→pj+1−
λj+1
µj+1 =
1 because both λj+1 and µj+1 converge to 1. It also seems that λj+1µj+1λj+2 has a limit as
p → 1+, but it is not clear although we have (3.38) and λj = (j − 1)(p − 1) + O((p − 1)2)
−0.54 −0.3979 −0.3010 −0.1761 0 0.3010 0.92 0.94 0.96 0.98 1 3 2 1.6667 1.5 1.4 1.29 log10(p − 1) fj (p ) p p2 p3 p4 p5 p6 f1 f2 f3 f4 f5
Figure 9: p vs. fj for j = 1, . . . , 5, where fj(p) = λj+1µj+1λj+2 for 1 < p < pj+2and fj(p) = λµj+1j+1
for pj+2 ≤ p < pj+1.
Proof. We first prove the upper bound: For j < k, use the test functions Sψ2, Sψ3, . . . , Sψj+2
(we cannot use Sψ1since it is zero). Recall L−ψm= λmψm. Let a = (a2, . . . , aj+2) vary over
Cj+1− {0}. By equivalent definition (3.36), H = SL+S∗, L−= S∗S, and the orthogonality between the ψm’s, we have
µj+1≤ sup a (PmamSψm, HPℓaℓSψℓ) (PmamSψm,PℓaℓSψℓ) = sup a (Pmamψm, L−L+L−Pℓaℓψℓ) (Pmamψm, L−Pℓaℓψℓ) ≤ sup a (Pmamψm, L−L−L−Pℓaℓψℓ) (Pmamψm, L−Pℓaℓψℓ) = sup a P m|am|2λ3m P m|am|2λm ≤ max m=2,...,j+2λ 2 m= λ2j+2.
Since µj+1 ≤ λ2j+2 < 1, it is attained at some function, for which the second inequality
above cannot be replaced by an equality sign. Thus µj+1< λ2j+2.
For the lower bound of eigenvalues, we use only the special case j = 1 of (3.27):
H = SL+S∗= SL0S∗ = S∗L3S. (3.40)
In particular, we have for 1 < p < 3,
H ≥ S∗L2S = S∗SS∗S = L21= L2−, (3.41)
which implies that
λ2j+1 ≤ µj+1 (1 < p < 3) (3.42)
For the second eigenvalue µ2, we can get a more precise estimate by using (3.21) for
L3 ≥ λ′3 together with
L1|Q⊥≥ λ′2, (3.43)
which follows from spec(L1). Combining these estimates, we have for any f ⊥ Q and p < 5,
(Hf, f ) ≥ λ′3(Sf, Sf ) ≥ λ′3λ′2(f, f ), (3.44)
which implies that µ2≥ λ′3λ′2, i.e.,
µ2 ≥
(
λ2λ3 (1 < p ≤ 2),
λ2 (2 < p < 5)
(3.45) For p > 3, we have L3≥ λ2 = −(p − 1)(p − 5)/4 and
L3− L2 ≥ −(p − 1)(p − 3)/2 =: −a. (3.46)
Hence for any t ∈ [0, 1], we have
L3≥ tL2− at + (1 − t)λ2. (3.47)
and so for b > 0, we have
(Hf, f ) + b(f, f ) ≥ (S∗(tL2− at + (1 − t)λ2)Sf, f ) + b(f, f )
= tkL1f k2− (at − (1 − t)λ2)(L1f, f ) + bkfk2,
(3.48) which is nonnegative if
b ≥ (at − (1 − t)λ2)2/(4t), (3.49)
whose infimum is attained at t = −λ2/(a + λ2) = (p − 5)/(p − 1) ∈ (0, 1) for p > 5. Plugging
this back in, we obtain the lower bound µ1 ≥ λ2(a + λ2) = −
1
16(p − 1)
3(p − 5) (p > 5). (3.50)
We have a similar bound on µ3 by using the even-odd decomposition L2(R) = L2ev(R) ⊕
L2od(R). Let ψj, ξj be the eigenfunction of L1 and L3 such that
L1ψj = λjψj, L3ξj = λjξj. (3.51)
ψj starts from j = 1 and ξj starts with j = 3. They are even for odd j and odd for even j.
For any even function f ⊥ Q = ψ1, Sf is odd and so we have f ⊥ ψ1 = Q, ψ2 and Sf ⊥ ξ3.
Hence by spec(L3) and spec(L1), we have
(Hf, f ) = (L3Sf, Sf ) ≥ ˜λ4(Sf, Sf ) = ˜λ4(L1f, f ) ≥ ˜λ4˜λ3(f, f ), (3.52) where we denote ˜ λj := ( λj (1 < p < pj), 1 (pj < p). (3.53)
Thus the second eigenvalue of H on L2ev is ≥ ˜λ4˜λ3. Next for any odd function f ⊥ ψ2, we
have f ⊥ ψ1, ψ2, ψ3. Hence we have
(Hf, f ) = (L3Sf, Sf ) ≥ λ′3(Sf, Sf ) = λ′3(L1f, f ) ≥ λ′3˜λ4(f, f ). (3.54)
Similarly, every odd function f ⊥ S∗ξ3 satisfies f ⊥ ψ1 and Sf ⊥ ξ3, ξ4, so
(Hf, f ) ≥ ˜λ5λ˜2(f, f ). (3.55)
Hence the second eigenfunction on L2
od is ≥ max(˜λ4λ′3, ˜λ5λ˜2) ≥ ˜λ4λ˜3. Therefore we have
µ3 ≥ ˜λ3λ˜4, i.e., µ3≥ λ3λ4 (1 < p < 5/3), λ3 (5/3 < p < 2), 1 (2 < p). (3.56)
This argument, however, does not yield any useful estimates for the higher µj.
3.7 Resonance for p= 3
In the theory of dispersive estimates for the linear Schr¨odinger evolution, it is important to know whether or not the endpoints of the continuous spectrum of the linear operator are eigenvalues or resonances. For our L, the endpoints are λ = ±i. Resonance here refers to a function φ which satisfies the eigenvalue problem locally in space with eigenvalue i or −i, but which does not belong to L2(Rn). For dimension n = 1, one requires φ ∈ L∞(R). (Note for comparison’s sake that in one dimension, the operator −d2/dx2 has a resonance
– corresponding to the constant function – at the endpoint 0 of its continuous spectrum.) Before we made the numerical calculation, we did not expect to see any resonance. However, from Figure 1, one sees that κ = √µ2 converges to 1 as p → 3. What does the
point κ = 1 at p = 3 correspond to? A natural conjecture is that it is a resonance or an eigenvalue, since the p = 3 case is well-known to be completely integrable and special phenomena may occur.
This is indeed the case since we have the following solution to the eigenvalue problem (2.12) when p = 3, φ = 1 − Q2 i , λ = i. (3.57)
It is clear that φ ∈ L∞(R) but φ 6∈ Lq(R) for any q < ∞.
Let up(x) denote the real-valued (and suitably normalized) solution of (2.14)
corre-sponding to µ = µ2. It is the first component of the eigenfunction of (2.12). A natural
question is: does up(x) converge in some sense to u3(x) := 1 − Q2(x) as p → 3? Since
up − u3 is not in Lq(R) for all q ∈ [1, ∞), it seems natural to measure the convergence in
the following weighted norm,
kfkw:=
Z
R
where a weighting operator w is defined by w(f )(x) := f (x)√ 1
1+x2. This de-emphasizes the
value of up− u3 for x large, and so it should converge to 0 as p goes to 3. This is confirmed
numerically as follows.
Let u3 := 1 − Q2 and δ := ku3kw. In Section 4 we will propose a numerical method to
solve for the eigenpair {λ, [up(x), wp(x)]T} of (2.12) corresponding to µ2 = −λ2.
Renor-malize up(x) for p 6= 3 so that it is real-valued, up(0) < 0, and kupkw = δ. In Figure 10(c)
we plot u3 in a large interval |x| < 130 with δ = 1.3588. According to the numerical
method in Section 4, we get u2.8, u2.9, u3.1 and u3.2plotted in Figure 10(a), (b), (d) and (e),
respectively. The vertical range is roughly [−1, 1]. In Figure 10(f)–(j) we plot w(up) for
p = 2.8, 2.9, 3, 3.1 and 3.2, for |x| < 130 and vertical range [−1, 0.5].
−100 −50 0 50 100 −1 −0.5 0 0.5 1 p = 2.8 u2.8 (x ) (a) −100 −50 0 50 100 −1 −0.5 0 0.5 1 p = 2.9 u2.9 (x ) (b) −100 −50 0 50 100 −1 −0.5 0 0.5 1 p = 3.0 u3 (x ) (c) −100 −50 0 50 100 −1 −0.5 0 0.5 1 p = 3.1 u3.1 (x ) (d) −100 −50 0 50 100 −1 −0.5 0 0.5 1 p = 3.2 u3.2 (x ) (e) −100 −50 0 50 100 −1 −0.5 0 0.5 replacemen x w (u 2 .8 )( x ) (f) −100 −50 0 50 100 −1 −0.5 0 0.5 x w (u 2 .9 )( x ) (g) −100 −50 0 50 100 −1 −0.5 0 0.5 x w (u 3 )( x ) (h) −100 −50 0 50 100 −1 −0.5 0 0.5 x w (u 3 .1 )( x ) (i) −100 −50 0 50 100 −1 −0.5 0 0.5 x w (u 3 .2 )( x ) (j) Figure 10: up(x) & w(up)(x) for p = 2.8, 2.9, 3, 3.1 and 3.2.
In Figure 11 we plot p vs. kup− u3kw and observe that up(x) converge to u3(x) in the
weighted norm k·kw as p → 3. In the numerical calculation for Figure 11, our increment for
p is 0.01. 2.8 2.85 2.9 2.95 3 3.05 3.1 3.15 3.2 0 0.01 0.03 0.06 0.09 0.12 0.15 p k up − u3 kw Figure 11: p vs. kup− u3kw.
Remark 3.10 For the operators L+L−and L−L+, and in general 4-th order operators, it
seems difficult to exclude the possibility that µ = 1 is an eigenvalue. Consider the following example. Let ˜H := (L+)2with p =√8−1. Note −1 is an eigenvalue of L+when p =√8−1.
Hence 1 is an eigenvalue of ˜H, at the endpoint of its continuous spectrum.
It would be interesting to prove the above convergence analytically and characterize the leading order behavior near p = 3 as we did in Theorem 2.6.
4
Numerical method for the spectra of the ground states
In this section we propose a numerical method to compute the spectrum of the linear operator L defined by (1.8) for p > 1 and space dimension n ≥ 1. There are two main steps in this method. First, we will solve the nonlinear problem (1.3) for Q: we will discretize it into a nonlinear algebraic equation, and then solve it by an iterative method. Second, we will compute the spectrum of L: we will discretize the operator L into a large-scale linear algebraic eigenvalue problem and then use implicitly restarted Arnoldi methods to deal with this problem.
Hereafter, we use the bold face letters or symbols to denote a matrix or a vector. For A ∈ RM ×N, q = (q1, . . . , qN)⊤ ∈ RN, q p = q ◦ · · · ◦ q denotes the p-time Hadamard
product of q, and [[q]] := diag(q) the diagonal matrix of q.
Step I. We first discretize equation (1.3) into a nonlinear algebraic equation and consider it on an n-dimensional ball Ω = {x ∈ Rn: |x| ≤ R, R ∈ R}. We rewrite the Laplace operator −∆ in the polar coordinate system with a Dirichlet boundary condition. Based on the recently proposed discretization scheme [15], the standard central finite difference method, we discretize −∆q(x) into
Aq = A[q1, . . . , qN]⊤, A ∈ RN ×N, (4.1)
where q is an approximation of the function Q(x). The matrix A is irreducible and diagonally-dominant with positive diagonal entries. The discretization of the nonlinear equation (1.3) can now be formulated as the following nonlinear algebraic equation,
Aq + q − q p
= 0. (4.2)
We introduce an iterative algorithm [12] to solve (4.2):
Aeqj+1+ eqj+1= q jp, (4.3)
where eqj+1 and qj are the unknown and known discrete values of the function Q(x),
re-spectively. The iterative algorithm is shown in Algorithm 1. Algorithm 1. Iterative Algorithm for Solving Q(x). Step 0 Let j = 0.
Choose an initial solution eq0> 0 and let q0=
e
q0
keq0k2
Step 1 Solve the equation (4.3), then obtain eqj+1.
Step 2 Let αj+1= k 1
e
qj+1k2 and normalize eqj+1 to obtain qj+1 = αj+1eqj+1.
Step 3 If (convergent) then
Output the scaled solution (αj+1)
1 p−1q j+1. Stop. else Let j := j + 1. Goto Step 1. end
If the components of q0 are nonnegative, this property is preserved by each iteration qj,
and hence also by the limit vector if it exists (see [12, Theorem 3.1]). The convergence of a subsequence of this iteration method to a nonzero vector is proved in [12, Theorem 2.1]. Although the convergence of the entire sequence is not proved, it is observed numerically to be very robust. See Chen-Zhou-Ni [5] for a survey on numerically solving nonlinear elliptic equations.
Step II. Next we discretize the operator L of (1.10) into a linear algebraic eigenvalue problem: L u w = λ u w , (4.4) where L = 0 A + I − [[q γ ]] −A − I + [[p q γ]] 0 ,
γ = p − 1, u = (u1, . . . , uN)⊤ ∈ RN, w = (w1, . . . , wN)⊤ ∈ RN, and q is the output of
the previous step, and satisfies the equation in (4.2). We use ARPACK [17] in MATLAB version 6.5 to deal with the linear algebraic eigenvalue problem (4.4) and obtain eigenvalues λ of L near the origin for p > 1 and space dimension n ≥ 1. Furthermore, the eigenvectors of L can be also produced.
The Step II above can in principle be used to compute all eigenfunctions in L2(Rn).
However, in producing Figures 2–7, we look for eigenfunctions of the form φ(r)eimθ. These problems can be reformulated as 1-D eigenvalue problems for φ(r), which can be computed using the same algorithm and MATLAB code. This dimensional reduction saves a lot of computation time and memory. Even with this dimensional reduction, and applying an algorithm for sparse matrices, the computation is still very heavy, and we cannot compute all eigenvalues in one step. We can only compute a portion of them each time.
5
Excited states with angular momenta
In this section we consider excited states with angular momenta in Rn, n ≥ 2. Let k = [n/2], the largest integer no larger than n/2. For x = (x1, . . . , xn) ∈ Rn, use polar coordinates rj
![Figure 12: Spectra of L in R 2 for m = 1, k = 0 and various p ∈ [1.5, 2.1].](https://thumb-ap.123doks.com/thumbv2/9libinfo/7491733.115237/41.918.137.789.147.843/figure-spectra-l-r-m-k-various-p.webp)
![Figure 13: Spectra of L in R 2 for m = 1, k = 0 and various p ∈ [2.2, 3.25].](https://thumb-ap.123doks.com/thumbv2/9libinfo/7491733.115237/42.918.136.789.149.839/figure-spectra-l-r-m-k-various-p.webp)
