In this section we come back to the question of invertibility of the operator J (which hence also implies invertibility of \({\mathcal {J}}\)). This will then lead to a globally convergent method.
Coercivity of the shifted J-operator
We first show that the operator J is – up to a shift – coercive.
Lemma 3
Given \(u\in V\) and assumption (11), the operator J(u) corresponding to the Gross–Pitaevskii operator satisfies a Gårding inequality. More precisely, for any \(\sigma \ge \frac{\kappa }{3} \Vert u \Vert _{L^4({\mathcal {D}})}^4 / \Vert u \Vert ^4\) the bilinear form
$$\begin{aligned} \langle (J(u)+\sigma {\mathcal {I}}) \cdot , \cdot \rangle :V \times V \rightarrow \mathbb {R}\end{aligned}$$
is coercive and thus, the operator \(J_\sigma (u) = J(u)+\sigma {\mathcal {I}}:V\rightarrow V^*\) is invertible.
Proof
Consider \(v\in V\). We start with considering the rotational gradient, for which we observe with Young’s inequality that
$$\begin{aligned} \int _{{\mathcal {D}}} | \nabla _{ \text{ R }}v|^2 \,\text {d}x\ge \frac{1}{2} \int _{{\mathcal {D}}} |\nabla v|^2 \,\text {d}x- \frac{3}{4} \varOmega ^2 \int _{{\mathcal {D}}} |x|^2 |v|^2 \,\text {d}x. \end{aligned}$$
Hence, with \(W_{ \text{ R }}= W(x) - \tfrac{1}{4}\, \varOmega ^2|x|^2\) and assumption (11) we have
$$\begin{aligned} \int _{{\mathcal {D}}} | \nabla _{ \text{ R }}v|^2 + W_{ \text{ R }}|v|^2 \,\text {d}x\ge \frac{1}{2} \int _{{\mathcal {D}}} |\nabla v|^2 \,\text {d}x. \end{aligned}$$
(17)
This leads to
$$\begin{aligned}&\langle J(u) v , v \rangle = \int _{{\mathcal {D}}} |\nabla _{ \text{ R }}v|^2 + W_{ \text{ R }}|v|^2 \,\text {d}x+ \frac{\kappa }{\Vert u \Vert ^2} \int _{{\mathcal {D}}} |u|^2 |v|^2 \,\text {d}x\nonumber \\&\qquad + \frac{2\kappa }{\Vert u \Vert ^2} \int _{{\mathcal {D}}} |\mathfrak {R}( u {\overline{v}})|^2 \,\text {d}x- 2\kappa \frac{ \mathfrak {R}( u , v )_{L^2({\mathcal {D}})} }{\Vert u \Vert ^4} \int _{{\mathcal {D}}} |u|^2\, \mathfrak {R}(u {\overline{v}}) \,\text {d}x\nonumber \\&\quad \overset{(17)}{\ge } \frac{1}{2}\, \Vert \nabla v \Vert ^2 + \frac{\kappa }{\Vert u \Vert ^2} \int _{{\mathcal {D}}} |u {\overline{v}}|^2 \,\text {d}x+ \frac{2\kappa }{\Vert u \Vert ^2} \int _{{\mathcal {D}}} |\mathfrak {R}( u {\overline{v}})|^2 \,\text {d}x\nonumber \\&\qquad - 2\kappa \frac{ \mathfrak {R}( u , v )_{L^2({\mathcal {D}})} }{\Vert u \Vert ^4} \int _{{\mathcal {D}}} |u|^2\, \mathfrak {R}(u {\overline{v}}) \,\text {d}x\nonumber \\&\quad \ge \frac{1}{2}\, \Vert \nabla v \Vert ^2 + 3\frac{\kappa }{\Vert u \Vert ^2} \int _{{\mathcal {D}}} |\mathfrak {R}(u {\overline{v}})|^2 \,\text {d}x- 2\kappa \frac{ \mathfrak {R}( u , v )_{L^2({\mathcal {D}})} }{\Vert u \Vert ^4} \int _{{\mathcal {D}}} |u|^2\, \mathfrak {R}(u {\overline{v}}) \,\text {d}x.\nonumber \\ \end{aligned}$$
(18)
To estimate the negative part, we apply once more Young’s inequality with some parameter \(\mu >0\) and the Cauchy Schwarz inequality to get
$$\begin{aligned} 2\, \mathfrak {R}( u , v )_{L^2({\mathcal {D}})} \int _{{\mathcal {D}}} |u|^2\, \mathfrak {R}(u {\overline{v}} )\,\text {d}x&\le \frac{1}{\mu }\, | \mathfrak {R}( u , v )_{L^2({\mathcal {D}})} |^2 + \mu \, \Big ( \int _{{\mathcal {D}}} |u|^2 \mathfrak {R}(u {\overline{v}} ) \,\text {d}x\Big )^2 \\&\le \frac{1}{\mu }\, \Vert u \Vert ^2 \Vert v \Vert ^2 + \mu \, \Vert u\Vert ^4_{L^4({\mathcal {D}})} \int _{{\mathcal {D}}} |\mathfrak {R}(u {\overline{v}} )|^2 \,\text {d}x. \end{aligned}$$
Using this estimate in (18) with the particular choice \(\mu = 3\, \Vert u\Vert ^2/ \Vert u\Vert ^4_{L^4({\mathcal {D}})}\), we obtain
$$\begin{aligned}&\langle J(u) v , v \rangle \\&\quad \ge \frac{1}{2}\, \Vert \nabla v \Vert ^2 + \frac{3\kappa }{\Vert u \Vert ^2} \int _{{\mathcal {D}}} |\mathfrak {R}(u {\overline{v}})|^2 \,\text {d}x- \frac{\kappa }{3} \frac{ \Vert u\Vert ^4_{L^4({\mathcal {D}})} \Vert v \Vert ^2 }{\Vert u \Vert ^4} - \frac{3\kappa }{\Vert u \Vert ^2} \int _{{\mathcal {D}}} |\mathfrak {R}(u {\overline{v}} )|^2 \,\text {d}x\\&\quad = \frac{1}{2}\, \Vert \nabla v \Vert ^2 - \frac{\kappa }{3}\frac{\Vert u \Vert _{L^4({\mathcal {D}})}^4}{\Vert u \Vert ^4} \Vert v \Vert ^2. \end{aligned}$$
Thus, we conclude
$$\begin{aligned} \langle J(u) v , v \rangle \ge \frac{1}{2}\, \Vert v \Vert _V^2 - \sigma \, \Vert v \Vert ^2 = \frac{1}{2}\, \Vert v \Vert _V^2 - \sigma \, \langle {\mathcal {I}}v , v \rangle . \end{aligned}$$
\(\square \)
Recall the definition of the energy E in Sect. 4.1. Motivated by the previous lemma, we also define the shifted energy \(E_\sigma (u) := E(u) + \frac{1}{2} \sigma \Vert u\Vert ^2\) such that \(E_0(u) = E(u)\). For \(u\in V\) with normalization constraint \(\Vert u \Vert = 1\), a sufficient shift in the sense of Lemma 3 is thus given by
$$\begin{aligned} \sigma := \frac{4}{3}\, E(u) \ge \frac{\kappa }{3}\, \Vert u \Vert _{L^4({\mathcal {D}})}^4. \end{aligned}$$
Note that for a normalized function u we can express the Rayleigh quotient in terms of the energy by
$$\begin{aligned} \lambda (u) := \langle {\mathcal {A}}(u), u\rangle = 2 E(u) + \frac{\kappa }{2}\, \Vert u \Vert _{L^4({\mathcal {D}})}^4. \end{aligned}$$
In particular, this formula relates the eigenvalues with the energies of the eigenfunctions.
Feasibility of the J-method
For the feasibility of the damped J-method, we need to guarantee a priori that \(J_\sigma (u^n)\) stays invertible throughout the iteration process. We fix the shift \(\sigma \) in the beginning of the iteration, e.g., by \(\sigma := \frac{4}{3} E(u^0)\). Now, the aim is to show that the energy of the iterates does not increase such that \(\sigma \ge \frac{4}{3} E(u^n)\) for all \(n\ge 0\).
Lemma 4
Recall the shifted energy \(E_\sigma (u) := E(u) + \frac{1}{2} \sigma \Vert u\Vert ^2\) with E(u) being defined in (12). Further recall the definition of the J-method (7) with iteration steps \({\tilde{u}}^{n+1} = (1-\tau )u^n + \tau \, \gamma _n\, J_\sigma (u^n)^{-1} {\mathcal {I}}u^n\). Then, for any step size \(\tau \le \frac{1}{2}\) we have the following guaranteed estimate for the difference of the shifted energies
$$\begin{aligned} E_{\sigma }(u^n) - E_{\sigma }({\tilde{u}}^{n+1})\ge & {} - \frac{3\kappa }{2} \int _{{\mathcal {D}}} |{\tilde{u}}^{n+1} - u^{n}|^4 \,\text {d}x\nonumber \\&+\, \left( \tfrac{1}{\tau }-\tfrac{1}{2}\right) \int _{{\mathcal {D}}} |\nabla _{ \text{ R }}({\tilde{u}}^{n+1}-u^n)|^2 \nonumber \\&+ (\sigma +W_{ \text{ R }}) |({\tilde{u}}^{n+1}-u^n)|^2 \,\text {d}x. \end{aligned}$$
(19)
Proof
We start by establishing a couple of identities for different evaluations of J. Here we exploit the \(L^2\)-orthogonality (8), i.e., \((u^n,{\tilde{u}}^{n+1}-u^n)=0\). Together with \(\Vert u^n \Vert =1\) this implies \((u^n,{\tilde{u}}^{n+1})=1\). Using these facts, we observe that
$$\begin{aligned}&\langle J(u^n) {\tilde{u}}^{n+1} , {\tilde{u}}^{n+1} \rangle \nonumber \\&\quad = \int _{{\mathcal {D}}} |\nabla _{ \text{ R }}{\tilde{u}}^{n+1}|^2 + W_{ \text{ R }}|{\tilde{u}}^{n+1}|^2 \,\text {d}x+ \kappa \int _{{\mathcal {D}}} |{\tilde{u}}^{n+1}|^2 \, |u^n|^2 \,\text {d}x\nonumber \\&\qquad + 2 \kappa \int _{{\mathcal {D}}} \left( \mathfrak {R}( u^n \, \overline{{\tilde{u}}^{n+1}} ) - |u^n|^2 \right) \mathfrak {R}(u^n \, \overline{{\tilde{u}}^{n+1}}) \,\text {d}x\nonumber \\&\quad = 2E({\tilde{u}}^{n+1}) + \kappa \int _{{\mathcal {D}}} |{\tilde{u}}^{n+1}|^2 \, \big ( |u^n|^2- |{\tilde{u}}^{n+1}|^2 \big ) \,\text {d}x\nonumber \\&\qquad + \frac{\kappa }{2} \int _{{\mathcal {D}}} |{\tilde{u}}^{n+1}|^4 \,\text {d}x\nonumber \\&\qquad + 2 \kappa \int _{{\mathcal {D}}} \big ( \mathfrak {R}( u^n \, \overline{{\tilde{u}}^{n+1}}) - |u^n|^2 \big )\, \mathfrak {R}( u^n \, \overline{{\tilde{u}}^{n+1}}) \,\text {d}x. \end{aligned}$$
(20)
Next, using again the \(L^2\)-orthogonality together with the definition of \({\tilde{u}}^{n+1}\) in (7), we have
$$\begin{aligned}&\tfrac{1}{\tau } \langle J_{\sigma }(u^n) ({\tilde{u}}^{n+1} - u^n) , {\tilde{u}}^{n+1} - u^n \rangle \nonumber \\&\quad = - \langle J_{\sigma }(u^n) u^n , {\tilde{u}}^{n+1} - u^n \rangle + \gamma ^n \langle J_{\sigma }(u^n) J_{\sigma }(u^n)^{-1}{\mathcal {I}}u^n , {\tilde{u}}^{n+1} - u^n \rangle \nonumber \\&\quad = - \langle J_{\sigma }(u^n) u^n , {\tilde{u}}^{n+1} - u^n \rangle + \gamma ^n ( u^n , {\tilde{u}}^{n+1} - u^n ) \nonumber \\&\quad = - \langle J_{\sigma }(u^n) u^n , {\tilde{u}}^{n+1} - u^n \rangle . \end{aligned}$$
(21)
With this equality, we conclude
$$\begin{aligned} \langle J_{\sigma }(u^n) u^n , u^n \rangle&= \langle J_{\sigma }(u^n) u^n , {\tilde{u}}^{n+1} \rangle - \langle J_{\sigma }(u^n) u^n , {\tilde{u}}^{n+1} - u^n \rangle \nonumber \\&\overset{(21)}{=} \langle J_{\sigma }(u^n) u^n , {{\tilde{u}}}^{n+1} \rangle + \tfrac{1}{\tau } \langle J_{\sigma }(u^n) ({\tilde{u}}^{n+1} - u^n) , {\tilde{u}}^{n+1} - u^n \rangle \nonumber \\&= \langle J_{\sigma }(u^n) {\tilde{u}}^{n+1} , {\tilde{u}}^{n+1} \rangle - \langle J_{\sigma }(u^n) ( {\tilde{u}}^{n+1} - u^n) , u^{n} \rangle \nonumber \\&\qquad + (\tfrac{1}{\tau }-1) \langle J_{\sigma }(u^n) ({\tilde{u}}^{n+1} - u^n) , {\tilde{u}}^{n+1} - u^n \rangle . \end{aligned}$$
(22)
Here we need to have a closer look at the term \(\langle J_{\sigma }(u^n) ( {\tilde{u}}^{n+1} - u^n) , u^{n} \rangle \). By the definition of \(J(u^n)\) it is easily seen that
$$\begin{aligned}&\langle J(u^n) ( {\tilde{u}}^{n+1} - u^n), u^{n} \rangle - \langle J(u^n) u^n , {\tilde{u}}^{n+1} - u^n \rangle \nonumber \\&\quad = 2 \kappa \int _{{\mathcal {D}}} |u^n|^2 \mathfrak {R}\left( ( {\tilde{u}}^{n+1} - u^n) \overline{u^n} \right) \,\text {d}x. \end{aligned}$$
(23)
Plugging this into (22) yields for the shifted energy
$$\begin{aligned}&2E_{\sigma }(u^n) \\&\quad = \langle J_{\sigma }(u^n) u^n, u^{n} \rangle - \frac{\kappa }{2}\int _{{\mathcal {D}}}|u^n|^4 \,\text {d}x\\&\quad \overset{(22)}{=} \langle J_{\sigma }(u^n) {\tilde{u}}^{n+1} , {\tilde{u}}^{n+1} \rangle - \langle J_{\sigma }(u^n) ( {\tilde{u}}^{n+1} - u^n) , u^{n} \rangle \\&\ \qquad + (\tfrac{1}{\tau }-1) \langle J_{\sigma }(u^n) ({\tilde{u}}^{n+1} - u^n), {\tilde{u}}^{n+1} - u^n \rangle - \frac{\kappa }{2}\int _{{\mathcal {D}}}|u^n|^4 \,\text {d}x\\&\quad \overset{(21), (23)}{=} \langle J_{\sigma }(u^n) {\tilde{u}}^{n+1} , {\tilde{u}}^{n+1} \rangle +\tfrac{1}{\tau } \langle J_{\sigma }(u^n) ({\tilde{u}}^{n+1} - u^n) , {\tilde{u}}^{n+1} - u^n \rangle \\&\ \qquad - \frac{\kappa }{2}\int _{{\mathcal {D}}}|u^n|^4 \,\text {d}x+ (\tfrac{1}{\tau }-1) \langle J_{\sigma }(u^n) ({\tilde{u}}^{n+1} - u^n), {\tilde{u}}^{n+1} - u^n \rangle \\&\ \qquad - 2 \kappa \int _{{\mathcal {D}}} |u^n|^2 \mathfrak {R}\left( ( {\tilde{u}}^{n+1} - u^n) \overline{u^n} \right) \,\text {d}x\\&\quad \overset{(20)}{=} 2E_{\sigma }({\tilde{u}}^{n+1}) - \kappa \int _{{\mathcal {D}}} |{\tilde{u}}^{n+1}|^2 \, ( |{\tilde{u}}^{n+1}|^2 - |u^n|^2 ) \,\text {d}x+ \frac{\kappa }{2}\int _{{\mathcal {D}}}|{\tilde{u}}^{n+1}|^4 \,\text {d}x\\&\ \qquad - \frac{\kappa }{2}\int _{{\mathcal {D}}}|u^n|^4 \,\text {d}x-2 \kappa \int _{{\mathcal {D}}} |u^n|^2 \mathfrak {R}\left( ( {\tilde{u}}^{n+1} - u^n) \overline{u^n} \right) \,\text {d}x\\&\ \qquad +\,2 \kappa \int _{{\mathcal {D}}} \big ( \mathfrak {R}\big ( u^n \, \overline{{\tilde{u}}^{n+1}} \big ) - |u^n|^2 \big )\, \mathfrak {R}( u^n \, \overline{{\tilde{u}}^{n+1}}) \,\text {d}x\\&\ \qquad +\, (\tfrac{2}{\tau }-1) \langle J_{\sigma }(u^n) ({\tilde{u}}^{n+1} - u^n) , {\tilde{u}}^{n+1} - u^n \rangle . \end{aligned}$$
Since \(\mathfrak {R}\, z = \mathfrak {R}\, {\overline{z}}\) for all \(z\in \mathbb {C}\), we conclude that
$$\begin{aligned}&2E_{\sigma }(u^n) - 2E_{\sigma }({\tilde{u}}^{n+1}) \nonumber \\&\quad = - \kappa \int _{{\mathcal {D}}} |{\tilde{u}}^{n+1}|^2 \, ( |{\tilde{u}}^{n+1}|^2 - |u^n|^2 ) \,\text {d}x+ \frac{\kappa }{2}\int _{{\mathcal {D}}}|{\tilde{u}}^{n+1}|^4 \,\text {d}x- \frac{\kappa }{2}\int _{{\mathcal {D}}}|u^n|^4 \,\text {d}x\nonumber \\&\qquad + 2 \kappa \int _{{\mathcal {D}}} \left( |u^n|^2 - \mathfrak {R}( {\tilde{u}}^{n+1} \overline{u^n} )\right) ^2 \,\text {d}x\ +\, (\tfrac{2}{\tau }-1) \langle J_{\sigma }(u^n) ({\tilde{u}}^{n+1} - u^n) , {\tilde{u}}^{n+1} - u^n \rangle . \end{aligned}$$
(24)
Using that
$$\begin{aligned} \langle J(u^n) ({\tilde{u}}^{n+1} - u^n) , ({\tilde{u}}^{n+1} - u^n) \rangle&= \int _{{\mathcal {D}}} |\nabla _{ \text{ R }}({\tilde{u}}^{n+1} - u^n)|^2 + W_{ \text{ R }}\, |{\tilde{u}}^{n+1} - u^n|^2 \,\text {d}x\nonumber \\&\quad + \kappa \int _{{\mathcal {D}}} |\mathfrak {R}( u^n (\overline{{\tilde{u}}^{n+1} - u^n}) )|^2 + |u^n|^2 \, |{\tilde{u}}^{n+1} - u^n|^2 \,\text {d}x, \end{aligned}$$
(25)
and \(\mathfrak {R}(u^n (\overline{{\tilde{u}}^{n+1} - u^n})) = \mathfrak {R}({\tilde{u}}^{n+1}\overline{u^n} ) - |u^n|^2\) we see that
$$\begin{aligned}&2E_{\sigma }(u^n) - 2E_{\sigma }({\tilde{u}}^{n+1}) \\&\quad \overset{(24),(25)}{\ge } - \kappa \int _{{\mathcal {D}}} |{\tilde{u}}^{n+1}|^2 ( |{\tilde{u}}^{n+1}|^2 - |u^n|^2 ) \,\text {d}x+ \frac{\kappa }{2}\int _{{\mathcal {D}}}|{\tilde{u}}^{n+1}|^4 \,\text {d}x-\, \frac{\kappa }{2}\int _{{\mathcal {D}}}|u^n|^4 \,\text {d}x\\&\qquad +\, (\tfrac{2}{\tau }-1) \int _{{\mathcal {D}}} |\nabla _{ \text{ R }}({\tilde{u}}^{n+1}-u^n)|^2 + (W_{ \text{ R }}+ \sigma )\, |({\tilde{u}}^{n+1}-u^n)|^2 \,\text {d}x\\&\qquad + \kappa \, (\tfrac{2}{\tau }-1) \int _{{\mathcal {D}}} |u^n|^2 | {\tilde{u}}^{n+1} - u^n|^2 \,\text {d}x. \end{aligned}$$
Finally, an application of the triangle and Young’s inequality yields the estimate
$$\begin{aligned}&- 2 \int _{{\mathcal {D}}} |{\tilde{u}}^{n+1}|^2 (|{\tilde{u}}^{n+1}|^2 - |u^n|^2) \,\text {d}x- \int _{{\mathcal {D}}}|u^n|^4 \,\text {d}x+ \int _{{\mathcal {D}}}|{\tilde{u}}^{n+1}|^4 \,\text {d}x\\&\quad = - \int _{{\mathcal {D}}} |{\tilde{u}}^{n+1}+ u^{n}|^2 |{\tilde{u}}^{n+1} - u^{n}|^2 \,\text {d}x\\&\quad \ge - \int _{{\mathcal {D}}} 3\, |{\tilde{u}}^{n+1} - u^{n}|^4 + 6\, |u^n|^2 |{\tilde{u}}^{n+1} - u^{n}|^2 \,\text {d}x, \end{aligned}$$
which then implies the desired inequality if \(\tau \le \frac{1}{2}\).\(\square \)
With this result we are now in the position to prove uniform boundedness of the energy of the iterates and even a reduction of the energy.
Theorem 4
(energy reduction) Given \(u^0 \in V\) with \(\Vert u^0\Vert =1\), we let \(\sigma \ge \frac{4}{3} E(u^0)\) and \(W(x) \ge \varOmega ^2 |x|^2\). Then, there exists a \(\tau ^{*}>0\) (that only depends on \(u^0\) and its energy) such that for all \(0 < \tau _n \le \tau ^{*}\) the sequence obtained by the damped J-method (7) is well-posed and strictly energy diminishing for all n, i.e., it holds
$$\begin{aligned} E(u^{n+1}) \le E(u^{n}) \le E(u^0), \end{aligned}$$
where \(E(u^{n+1}) < E(u^{n})\) if \(u^{n}\) is not already a critical point of E and thus an eigenstate.
Proof
We proceed inductively. From estimate (17) in the proof of Lemma 3 we know that \(J_{\sigma }(u^0)\) is coercive with constant 1/2, i.e., \(\Vert \nabla v\Vert ^2 = \Vert v \Vert _{V}^2 \le 2\, \langle J_{\sigma }(u^0)v, v \rangle \). Together with (21) and the \(L^2\)-orthogonality (8), this implies
$$\begin{aligned}&\frac{1}{2\tau _0} \int _{{\mathcal {D}}} |\nabla {\tilde{u}}^1 - \nabla u^{0} |^2 \,\text {d}x\\&\quad \le -\, \langle J_{\sigma }(u^0) u^0 , {\tilde{u}}^{1} - u^0 \rangle \\&\quad = \mathfrak {R}\left( \int _{{\mathcal {D}}} \nabla _{ \text{ R }}u^0 \cdot \overline{\nabla _{ \text{ R }}(u^0 - {\tilde{u}}^{1})} + W_{ \text{ R }}\, u^0 \overline{(u^0 - {\tilde{u}}^{1})} \,\text {d}x+ \kappa \int _{{\mathcal {D}}} |u^0|^2 u^0 \, \overline{( u^0 - {\tilde{u}}^{1})} \,\text {d}x\right) \\&\quad \le \Vert \nabla _{ \text{ R }}u^0 \Vert \Vert \nabla _{ \text{ R }}({\tilde{u}}^{1}-u^0) \Vert + \Vert \sqrt{W_{ \text{ R }}} u^0 \Vert \Vert \sqrt{W_{ \text{ R }}} ({\tilde{u}}^{1}-u^0) \Vert + \kappa \Vert u^0 \Vert _{L^6({\mathcal {D}})}^3 \Vert {\tilde{u}}^{1}-u^0 \Vert \\&\quad \le C \sqrt{E(u^0)} \Vert \nabla ({\tilde{u}}^{1}-u^0) \Vert . \end{aligned}$$
Hence, we get
$$\begin{aligned} \Vert \nabla {\tilde{u}}^{1} - \nabla u^0\Vert ^2 = \int _{{\mathcal {D}}} | \nabla {\tilde{u}}^{1} - \nabla u^0|^2 \,\text {d}x\le 4\, \tau _0^2 C^2 E(u^0) =: \tau _0^2\, C_0. \end{aligned}$$
Assume that \(\tau _0\le \tau ^{*}\le 2\), where \(\tau ^{*}\) is selected sufficiently small compared to \(C_0\). Then we have that \(\Vert \nabla {\tilde{u}}^{1} - \nabla u^0\Vert <1\) and the Sobolev embedding of \(L^4({\mathcal {D}}) \hookrightarrow H^1_0({\mathcal {D}})\) with constant \(C_S\) implies that
$$\begin{aligned} \int _{{\mathcal {D}}} | {\tilde{u}}^{1} - u^0|^4 \,\text {d}x\le C_S \int _{{\mathcal {D}}} | \nabla {\tilde{u}}^{1} - \nabla u^0|^2 \,\text {d}x. \end{aligned}$$
Consequently, we can use (19) and (17) to observe that the energy difference fulfills
$$\begin{aligned}&E_{\sigma }(u^0) - E_{\sigma }({\tilde{u}}^{1})\nonumber \\&\quad \ge - \frac{3\kappa }{2} \int _{{\mathcal {D}}} |{\tilde{u}}^{1}-u^0|^4 \,\text {d}x+ \tfrac{1}{2}\left( \tfrac{1}{\tau _0}-\tfrac{1}{2}\right) \int _{{\mathcal {D}}} |\nabla ({\tilde{u}}^{1}-u^0)|^2 \,\text {d}x\nonumber \\&\qquad + \sigma \left( \tfrac{1}{\tau _0}-\tfrac{1}{2}\right) \int _{{\mathcal {D}}} |{\tilde{u}}^{1}-u^0|^2 \,\text {d}x\nonumber \\&\quad \ge \tfrac{1}{2}\left( \tfrac{1}{\tau _0}-\tfrac{1}{2}-3\kappa \, C_S\right) \int _{{\mathcal {D}}} |\nabla ({\tilde{u}}^{1}-u^0)|^2 \,\text {d}x\nonumber \\&\qquad + \sigma \left( \tfrac{1}{\tau _0}-\tfrac{1}{2}\right) \int _{{\mathcal {D}}} |{\tilde{u}}^{1}-u^0|^2 \,\text {d}x. \end{aligned}$$
(26)
Hence, if
$$\begin{aligned} \tau _0 \le \tau ^{*} < \min \Big \{ \frac{2}{1 + 6 \kappa C_S},\, C_0^{-1/2},\, \frac{1}{2} \Big \}, \end{aligned}$$
then we have
$$\begin{aligned} E_{\sigma }(u^{1} ) \le E_{\sigma }({\tilde{u}}^{1} ) \le E_{\sigma }(u^0), \end{aligned}$$
where we have used that the iterations increase the mass intermediately, i.e., \(\Vert {\tilde{u}}^{1} \Vert \ge 1\) as shown in (9). Note that since \(u^0\) and \(u^1\) are normalized in \(L^2({\mathcal {D}})\), we can drop the shift, leading to \(E(u^{1} ) \le E(u^0)\). Hence, we have \(\sigma \ge \frac{4}{3}E(u_0) \ge \frac{4}{3}E(u_1)\) and Lemma 3 guarantees that \(J_{\sigma }(u^1)\) is still coercive. Inductively, we can repeat the arguments for \(u^n\) with the same generic constant \(C_0\) to show that for \(\tau ^n \le \tau ^{*}\) we have
$$\begin{aligned} E_{\sigma }(u^{n+1}) \le E_{\sigma }(u^{n}). \end{aligned}$$
Since the energy is diminished in every iteration, the coercivity of \(\langle J_\sigma (u^n) \cdot , \cdot \rangle \) is maintained and all the iterations are well-defined. Finally, we note that because of (26) we have \(E_{\sigma }(u^n) = E_{\sigma }({\tilde{u}}^{n+1} )\) if and only if \(u^n={\tilde{u}}^{n+1}\). However, this can only happen if
$$\begin{aligned} J_\sigma (u^n) u^{n} = \, \gamma _n\, {\mathcal {I}}u^n, \end{aligned}$$
i.e., if \(u^n\) is already an eigenfunction with eigenvalue \(\lambda =\gamma _n\).\(\square \)
It is interesting to note that the \(L^2\)-norm of \({\tilde{u}}^{n}\) cannot diverge. We see this in the following conclusion.
Conclusion 5
In the setting of Theorem 4 it holds that \(\Vert {\tilde{u}}^{n} \Vert \rightarrow 1\) for \(n\rightarrow \infty \).
Proof
In the proof of Theorem 4 we have seen that
$$\begin{aligned} E_{\sigma }(u^n) - E_{\sigma }(u^{n+1} ) \ge \tfrac{1}{2}\left( \tfrac{1}{\tau ^{*}}-\tfrac{1}{2} - 3\, \kappa \, C_S\right) \int _{{\mathcal {D}}} |\nabla ({\tilde{u}}^{n+1}-u^n)|^2 \,\text {d}x. \end{aligned}$$
Since \(E_{\sigma }(u^n)\) is monotonically decreasing and bounded from below, we have \(E_{\sigma }(u^n) - E_{\sigma }(u^{n+1} ) \rightarrow 0\). This together with the Poincaré-Friedrichs inequality implies that
$$\begin{aligned} \Vert {\tilde{u}}^{n+1} \Vert \le \Vert {\tilde{u}}^{n+1} - u^{n} \Vert + \Vert u^n \Vert \rightarrow 1 \end{aligned}$$
for \(n\rightarrow \infty \).\(\square \)
Convergence and optimal damping
In this subsection we prove the convergence of the J-method for a suitable choice of damping parameters. We can make practical use of this result by selecting \(\tau _n\) in each iteration step by the minimizer of a simple one-dimensional minimization problem.
Theorem 6
(global convergence) Suppose that the assumptions of Theorem 4 are fulfilled. Additionally assume that \(\tau _n\) is selected such that it does not degenerate, i.e., is uniformly bounded away from zero. Then there exists a limit energy \(E^{*}:=\lim _{n\rightarrow \infty } E(u^{n})\) and, up to a subsequence, we have that the iterates \(u^n\) of the damped J-method converge strongly in \(H^1({\mathcal {D}})\) to a limit \(u^{*}\in V\). The limit is an \(L^2\)-normalized eigenfunction with some eigenvalue \(\lambda ^{*}>0\), i.e.,
$$\begin{aligned} {\mathcal {A}}(u^{*}) = \lambda ^{*} {\mathcal {I}}u^{*} \end{aligned}$$
and we have \(E(u^{*})=E^{*}\). If \(u^{*}\) is the only eigenfunction on the energy level \(E^{*}\), then we have convergence of the full sequence \(u^{n}\).
Proof
The proof is similar to the arguments presented in [36, Th. 4.9]. First, Theorem 4 guarantees the existence of the limit \(E^{*}:=\lim _{n\rightarrow \infty } E(u^n)\). Hence, \(u^n\) is uniformly bounded in V and we can extract a subsequence (still denoted by \(u^n\)) that converges weakly in \(H^1({\mathcal {D}})\) and strongly in \(L^p({\mathcal {D}})\) (for \(p<6\)) to a limit \(u^{*} \in V\) with \(\Vert u^{*} \Vert =1\). Since \(J_{\sigma }(u^{*})\) is a real-linear operator that depends continuously on the data and which induces the coercive bilinear form \(\langle (J(u)+\sigma {\mathcal {I}}) \cdot , \cdot \rangle \), we have that
$$\begin{aligned} J_{\sigma }(u^{*})^{-1} {\mathcal {I}}u^n \rightarrow J_{\sigma }(u^{*})^{-1} {\mathcal {I}}u^{*} \quad \text{ strongly } \text{ in } H^1({\mathcal {D}}). \end{aligned}$$
Together with the strong convergence \(u^{n} \rightarrow u^{*}\) in \(L^4({\mathcal {D}})\), we conclude that
$$\begin{aligned} J_{\sigma }(u^{n})^{-1} {\mathcal {I}}u^n \rightarrow J_{\sigma }(u^{*})^{-1} {\mathcal {I}}u^{*} \quad \text{ strongly } \text{ in } H^1({\mathcal {D}}). \end{aligned}$$
This shows that
$$\begin{aligned} (\gamma _n)^{-1} = ( J_{\sigma }(u^{n})^{-1} {\mathcal {I}}u^n , u^n) \overset{n \rightarrow \infty }{\longrightarrow } ( J_{\sigma }(u^{*})^{-1} {\mathcal {I}}u^{*} , u^{*}) =: (\gamma ^{*})^{-1}. \end{aligned}$$
Furthermore, we have seen in the proof of Theorem 4 (respectively Conclusion 5) that the strong energy reduction implies that for \(n\rightarrow 0\)
$$\begin{aligned} \Vert {\tilde{u}}^{n+1} - u^{n} \Vert _{H^1({\mathcal {D}})} \rightarrow 0 \end{aligned}$$
and consequently we have with \({\tilde{u}}^{n+1} = (1-\tau _n)u^n + \tau _n \gamma ^n J_{\sigma }(u^{n})^{-1} {\mathcal {I}}u^n\) and the boundedness of \(\tau _n\) that
$$\begin{aligned} u^n = \gamma ^n J_{\sigma }(u^{n})^{-1} {\mathcal {I}}u^n - \tau _n^{-1}({\tilde{u}}^{n+1} - u^{n}) \ \rightarrow \ \gamma ^{*} J_{\sigma }(u^{*} )^{-1} {\mathcal {I}}u^{*} \end{aligned}$$
strongly in \(H^1({\mathcal {D}})\). Since we already know that \(u^n\) converges weakly in \(H^1({\mathcal {D}})\) to \(u^{*}\), we can now conclude that this is even a strong convergence and we have
$$\begin{aligned} J_{\sigma }(u^{*} ) u^{*} = \gamma ^{*} {\mathcal {I}}u^{*}. \end{aligned}$$
This shows that \(u^{*}\) is an eigenfunction with eigenvalue \(\gamma ^{*}\). The strong convergence in \(H^1({\mathcal {D}})\) also implies convergence of the energies, i.e., it holds \(E^{*} =\lim _{n\rightarrow \infty } E(u^n) = E(u^{*})\).\(\square \)
For all sufficiently small \(\tau _n\), Theorem 4 proves the energy reduction and Theorem 6 global convergence. However, since we do not know a priori what a sufficiently small value for \(\tau _n\) is, we can combine the damped J-method with a line search algorithm that optimizes \(\tau _n\) in each iteration step such that the energy reduction is (quasi) optimal. Theorems 4 and 6 show that such an optimal \(\tau _n\) exists and that it does not degenerate to zero. We stress that finding such a \(\tau _n\) does not require any additional inversions, which makes the procedure very cheap, cf. “Appendix A” for details.
Conclusion 7
(J-method with optimal damping) Consider a shift \(\sigma \) such that the assumptions of Theorem 4 are fulfilled. Given \(u^n\in V\) with \(\Vert u^n\Vert =1\) the next iteration is obtained by selecting the optimal damping parameter with
$$\begin{aligned} \tau _{n} := \underset{0<\tau \le 2}{\text{ arg } \text{ min }} E\left( \frac{(1- \tau )u^n + \tau \, \gamma _n\, J_\sigma (u^n)^{-1} {\mathcal {I}}u^n}{\Vert (1-\tau )u^n + \tau \, \gamma _n\, J_\sigma (u^n)^{-1} {\mathcal {I}}u^n \Vert } \right) \end{aligned}$$
and defining \(u^{n+1}\) as in (7). The approximations are energy diminishing and converge (up to a subsequence) strongly in V to an \(L^2\)-normalized eigenfunction of the GPEVP.
Finally, if there is no rotation, we can even achieve guaranteed global convergence to the ground state provided that the selected initial value is non-negative.
Proposition 3
(global convergence to ground state) Assume the setting of Theorem 6. Furthermore, we consider that there is no rotation, i.e., \(\varOmega =0\), and a non-negative and \(L^2\)-normalized starting value \(u^0 \in V\), i.e., \(u^0 \ge 0\). If \(\tau _n \le 1\) and if the shift parameter \(\sigma >0\) is selected sufficiently large, then the iterates \(u^n\) of the damped J-method converge strongly in \(H^1({\mathcal {D}})\) to the unique (positive) ground state \(u^{*} \ge 0\).
Proof
If \(\varOmega =0\), then the problem can be fully formulated over \(\mathbb {R}\) and admits a unique positive ground state \(u^{*} \in V\), cf. [21]. The only other ground state is \(- u^{*}\). Furthermore, all excited states (i.e., all other eigenfunctions) must necessarily change their sign on \({\mathcal {D}}\), cf. [36, Lem. 5.4]. Hence, if we can verify that the iterates of the damped J-method (7) preserve positivity, then Theorem 6 guarantees that the global \(H^1\)-limit must be the desired ground state \(u^{*}\).
Recall the damped J-iteration from (7) together with (14), which (over \(\mathbb {R}\)) reduces to
$$\begin{aligned} \langle J_{\sigma }(u^n) v , w \rangle = \langle S_{\sigma } v , w \rangle + \langle G v , w \rangle , \end{aligned}$$
where \(S_{\sigma }\) is the linear self-adjoint operator given by
$$\begin{aligned} \langle S_{\sigma } v , w \rangle := ( \nabla v , \nabla w )_{L^2({\mathcal {D}})} + ( (W+\sigma + 3 \kappa |u^n|^2 ) v, w )_{L^2({\mathcal {D}})} \end{aligned}$$
and G characterizes the non-symmetric rank-1 remainder, i.e.,
$$\begin{aligned} \langle G v , w \rangle := - (u^n , v )_{L^2({\mathcal {D}})} (f , w )_{L^2({\mathcal {D}})},\qquad \text{ where } f := 2 \kappa \, |u^n|^2 u^n . \end{aligned}$$
Analogously to the Sherman–Morrison formula for matrices [49], we can see that
$$\begin{aligned} (S_{\sigma } + G)^{-1} = S_{\sigma }^{-1} -\frac{S_{\sigma }^{-1} \circ G \circ S_{\sigma }^{-1} }{1 - (u^n , S_{\sigma }^{-1} {\mathcal {I}}f )_{L^2({\mathcal {D}})} }. \end{aligned}$$
Consequently we can write the effect of the inverse as
$$\begin{aligned} J_{\sigma }(u^n)^{-1} {\mathcal {I}}v = (S_{\sigma } + G)^{-1} {\mathcal {I}}v = S_{\sigma }^{-1} ({\mathcal {I}}v) - \frac{S_{\sigma }^{-1} \circ G \circ S_{\sigma }^{-1} ({\mathcal {I}}v) }{1 - (u^n , S_{\sigma }^{-1} {\mathcal {I}}f )_{L^2({\mathcal {D}})} }. \end{aligned}$$
Since \(\sigma >0\) and \(W\ge 0\), \(S_{\sigma }\) is a self-adjoint elliptic operator and hence preserves positivity, i.e., we have \(S_{\sigma }^{-1} {\mathcal {I}}v \ge 0\) if \(v\ge 0\). This immediately follows by writing the action of \(S_{\sigma }^{-1}\) as an energy minimization problem. Starting (inductively) from a function \(u^n\ge 0\) we conclude that
$$\begin{aligned} S_{\sigma }^{-1} ({\mathcal {I}}u^n) \ge 0 \qquad \text{ and } \qquad - S_{\sigma }^{-1} \circ G \circ S_{\sigma }^{-1} ({\mathcal {I}}u^n) \ge 0. \end{aligned}$$
If we can ensure that \(1 - (u^n , S_{\sigma }^{-1} {\mathcal {I}}f )_{L^2({\mathcal {D}})} >0\), then we have \(J_{\sigma }(u^n)^{-1} {\mathcal {I}}u^n \ge 0\). Let us hence consider \((u^n , S_{\sigma }^{-1} {\mathcal {I}}f )_{L^2({\mathcal {D}})}\), for which we obtain
$$\begin{aligned} | (u^n , S_{\sigma }^{-1} {\mathcal {I}}f )_{L^2({\mathcal {D}})} | \le \Vert u^n\Vert \, \Vert S_\sigma ^{-1}\Vert _{{\mathcal {L}}(V^*,V)} \Vert {\mathcal {I}}f\Vert _{V^*} \le 2 \kappa \, \Vert u^n\Vert ^3_{L^6({\mathcal {D}})} \, \Vert S_\sigma ^{-1}\Vert _{{\mathcal {L}}(V^*,V)}. \end{aligned}$$
Since \(S_{\sigma }\) is self-adjoint, we have
$$\begin{aligned} \Vert S_\sigma ^{-1}\Vert _{{\mathcal {L}}(V^*,V)} = 1/\lambda _\text {min}(S_\sigma ) = 1/(\sigma + \lambda _\text {min}(S_0))\le 1/\sigma , \end{aligned}$$
where \( \lambda _\text {min}(S_0)>0\) is the minimal eigenvalue of \(S_0\). Consequently, if the shift is such that
$$\begin{aligned} 2 \kappa \, \Vert u^n\Vert ^3_{L^6({\mathcal {D}})} < \sigma , \end{aligned}$$
then we have positivity of \(J_{\sigma }(u^n)^{-1} {\mathcal {I}}u^n\). Note that by the energy reduction property, we can bound \(\Vert u^n\Vert ^3_{L^6({\mathcal {D}})}\) uniformly for all n by a constant that only depends on the initial energy \(E(u^0)\). Together with the obvious positivity of \((1-\tau _n)u^n\) for \(\tau _n\le 1\), we conclude the existence of a sufficiently large shift \(\sigma \) so that \(u^{n+1}\ge 0\) for all \(n\ge 0\) and hence global convergence to the ground state.\(\square \)