Structure and symmetry of the Gross-Pitaevskii ground-state manifold

STRUCTURE AND SYMMETR Y OF T HE GROSS–PIT AEVSKI I GR OUND-ST A TE MANIFOLD ∗ ZIXU FENG † , P A TRICK HENNING ‡ , AND QINGLIN T ANG § Abstract. The structure and degene racy of ground states of the Gross–Pitaevskii ene rgy func- tional play a cen tral role in both analysis and computation, yet a precise c haracterization of the ground-state manifold in the presence of symmetries remains a fundament al challenge. In this paper, we esta blish s harp theoretical results describing the geometric structure of l ocal minimizers and i ts implications for optimi zation algorithms. W e show that when local minimizers are non-unique, the Morse–Bott condition pr o vides a natural and suﬃcient criterion under which the ground-state set partitions int o ﬁnitely man y embedded submanifol ds, eac h c oinciding with an orbit generate d b y the int rinsic symmetries of t he energy functional, namely phase shifts and spatial rotations. This yields a structural c haracterization of the ground-state manifold purely in terms of these natural symmetries. Building on this geometric insight, we analyze the lo cal conv ergence b eha vior of the preconditioned Riemannian gradient metho d (P-RG). Under the M orse–Bott condition, we derive the optimal lo cal Q -linear conv ergence rate and prov e that the condition holds if and only if the energy sequence gen- erated b y P-RG conv erges lo cally Q -li nearly . In particular, on the ground-state set, the M orse–Bott condition is satisﬁed i f and only i f the mini mizers decompose into ﬁnit ely many symmetry orbits and the P-RG exhibits lo cal li near conv ergence in a neighborho o d of this set. When the condition fails, we e stablish a l ocal s ublinear con ve rgence rate. T ake n together, these resul ts prov ide a complete and precise picture: for the Gross–Pitaevskii minimization pr oblem, the Morse–Bott condition acts as the exact threshold separating linear fr om s ublinear conv ergence, whil e si multa neously determining the symmetry-i nduced structure of the ground-state m anifold. Our analysis th us connects geometric structure, symm etry , and algorithmic p erformance in a uniﬁed framework. Key words. Geometric structure, Gross–Pitaevskii energy functional, ground states, Bose– Einstein condensates, Riemannian optimization, Morse–Bott condition MSC co des. 35Q55, 47A75, 49J40, 49R05, 81Q05, 90C26 1. Intr oduction. The Gross–Pita evskii energy functional and th e un derlying eq ua- tions form a central mathematical mo del in quantum physics. They w ere originally in- trod uced to describ e Bose–Einstein condensates (BECs), where a large num b er of b osonic particles occupy the same quantum s tate at extremely lo w temp eratures. Du e to their ability to capture collectiv e quantum b ehavior, these mod els h a ve found applications in several ar- eas, including cold atom physics, non linear optics, astrophysica l modeling, and th e study of quantum ﬂuids an d t urbulence [4 , 10, 12, 19, 34, 37]. F or ex ample, related equations app ear in nonlinear optics to describ e ligh t propagation in nonlinear media, while in astrophysics they are u sed in mo dels where macroscopic quantum coherence is exp ected , such as ultra- ligh t dark matter or sup erﬂuid phases inside neutron stars. Moreov er, t he Gross– Pitaevskii equation provides an imp ortant framew ork for inv estigati ng vortex d ynamics and energy transfer pro cesses in q uantum tu rbulence. Accordingly , the computation and c haracterization of minimizers of the Gross–Pitaevskii energy functional are of centra l relev ance in the mathematical description of BECs and re- lated qu antum systems. F rom a mathematical stand p oin t, these minimizers arise from a constrained va riational problem u nder an L 2 normalization cond ition. F ollo wing the p resen- tation in the survey b y Bao et al. [9], the dimensionless Gross–Pitaevskii energy functional ∗ Submitted to the editors DA TE. F unding: Zixu F eng and Qinglin T ang were partiall y supp orted by the National Key R&D Program of China (Grant No. 2024YF A1012803) and the Natural Science F oundation of Sich uan Province (Grant No. 2024NSFSC0438). Patric k Henning was partially supported b y the Deutsc he F orsch ungsgemeinschaft (DFG, German Research F oundation; Grant No. 551527112). † Sc hool of M athematical Science, Chengdu Universit y of T ec hnology , Chengdu 610059, P . R. China. (zixu feng123@163.com) . ‡ Departmen t of Math ematics, Ruhr-Universit y Boc hu m, DE-44801 Bo c hu m, Germany . (patric k.henning@rub.de). § Sc hool of Mathe matics, Sic hua n Univ ersit y , Chengd u 6100064 , P . R. China. (qinglin tang@scu.edu.cn) 1 2 ZIXU FEN G, P A TRICK HENNING, AND QINGLIN T ANG in a rotating frame is deﬁned by E ( φ ) := 1 2 Z R d  1 2 |∇ φ | 2 + V ( x ) | φ | 2 − Ω φ L z φ + F ( ρ φ )  d x , (1.1) where φ denotes t he macroscopic w av e function describing t he quantum state of the con- densate. F urthermore, x ∈ R d ( d = 2 , 3) den otes the spatial v ariable , with x = ( x, y ) ⊤ in tw o dimensions and x = ( x, y , z ) ⊤ in th ree dimensions. T he trapping p otential V ( x ) (that conﬁnes the particles) is real-v alued and satisﬁes lim | x |→∞ V ( x ) = ∞ . The rotation compon ent is d escribed by th e angular momentum op erator L z = − i( x∂ y − y∂ x ) together with th e rotation frequency Ω ≥ 0. The notation φ denotes th e complex conjugate of φ . The nonlinear p article interaction term is given by F ( ρ φ ) = Z ρ φ 0 f ( s ) d s, ρ φ := | φ | 2 , (1.2) i.e., a function acting on th e particle densit y ρ φ . The function f ( s ) frequently app ears in forms such as f ( s ) = η s , η s log s , or η s + η LHY s 3 / 2 , dep ending on the p hysica l applications, cf. [21, 38, 44, 45]. The normalizati on constraint is deﬁned by N ( φ ) := k φ k 2 L 2 ( R d ) = Z R d | φ | 2 d x = 1 and represents n ormalizatio n of th e total particle n umber (mass). The corresponding ground state w a ve fun ction φ g , in a given p hysica l conﬁguration, is therefore c haracterized by the constrained minimization problem φ g := arg min φ ∈M E ( φ ) with M := n φ ∈ H 1 ( R d )   k φ k 2 L 2 ( R d ) = 1 o . (1.3) The main analytical challenges asso ciated with problem ( 1.3) arise from t he non-conv ex constrain t and th e inherent symmetry properties of the Gross–Pitaevskii functional. A ﬁrst symmetry originates from global phase in v ariance of the energy: If φ g is a lo cal minimizer, then e i α φ g is also a lo cal minimizer for every α ∈ [ − π , π ). A second sym metry is indu ced by rotational inv ariance of the trapp ing p otential: If V ( x ) is radially symmetric with resp ect to th e z - axis, i.e., V ( x ) = V ( A β x ) for all β ∈ [ − π , π ), where A β =  cos β − sin β sin β cos β  for d = 2 , A β =   cos β − sin β 0 sin β cos β 0 0 0 1   for d = 3 , then φ g ( A β x ) is lik ewise a local minimizer. These contin uous symmetry transformations imply that local minimizers are generally not isolated but instead fo rm families of symmetry- related states, which signiﬁcantly complicates b oth theoretical analysis and numerical com- putation. Over t he past tw o decades, numerous numerical approaches hav e b een developed for computing minimizers of b oth rotating and non-rotating Gross–Pitaevskii energy functionals. These approaches p rimarily include gradient-ﬂow -based energy min imization techniques [1, 5, 6, 7, 8, 14 , 15 , 16, 17, 18, 23, 25, 28, 29 , 36, 40, 46, 47 , 48, 49, 51] as well as iterativ e metho ds form ulated as nonlinear eigenv alue solve rs [2, 20, 29, 35]. Within th e class of energy minimization techniques, the present work focuses on preconditioned Riemannian gradient metho ds [23, 30], also know n as pro jected Sob olev gradien t sc hemes [15, 17 , 18, 28, 29, 36, 49, 50], whic h are particularly w ell suited for constrained minimization problems associated with th e Gross–Pitaevskii functional, in cluding rotational eﬀects. F rom the v iewpoint of local conv ergence theory , severa l rigorous results are av ailable, but they rely on sp eciﬁc structural assumptions on the set of minimizers that reﬂect t he symmetry-indu ced d egeneracy d escribed ab o ve. In p articular, lo cal linear conv ergence of a Riemannian gradient metho d tow ard the manifold of ground states g enerated b y phase inv ari- ance was recently established in [30], thereby ex ploiting th e fact that the non-uniquen ess of STRUCTURE AND SYMMETR Y OF THE GP GROUND-ST A T E MANIFOLD 3 minimizers is lo cally ind uced by a single continuous symmetry . This p ersp ective was furth er developed in [50], where sev eral Sob olev gradient v ariants were analyzed on an appropri- ate quotient space, again under th e assumption that th e ground-state manifold is locally described by symmetry orbits arising from p hase inv ariance. More recently , [23] t reated gen- eral preconditioners and obt ained exp licit linear conv ergence rates by establishing a Poly ak– Lo jasiewic z inequalit y un der a structu ral assumption on the critical set which, in the Gross– Pitaevskii setting, corresp onds to t he ground-state manifold b eing lo cally generated by the natural sy mmetries of th e mo del, namely p hase shifts and possibly spatial rotations. This condition can b e in terpreted as a Morse–Bott-t yp e assumption tied to sy mmetry-induced degeneracy . Despite th ese adva nces, the current theory do es n ot p ro vide a complete geometric char- acterization of when linear conve rgence should be exp ected. Existing results sho w that li near conv ergence holds under sy mmetry-driven structu ral assumpt ions on the ground -state m an- ifold, b ut they do not address the more general situation in whic h the critical set satisﬁes a Morse–Bott condition in an in trinsic sense, ind epen dent of a priori identiﬁcatio n with s p eciﬁc symmetry orbits. In p articular, it remains unclear whether linear conv ergence is fund amen- tally equiva lent to suc h a geometric condition, how the structure of the set of minimizers is related to it, and what converg ence b ehavior should o ccur when this structure fails. A uniﬁed picture linking the geometry of the critical set, the symmetry- induced classiﬁcation of minimizers, and the precise conv ergence regime of preconditioned Riemannian gradient metho ds is therefore still missing. A recent survey [32] highlights sev eral open prob lems related to these p henomena, in- cluding whether symmetry-related families of nearly d egenerate states can lead to arbitrarily slo w conv ergence, how the entire set of ground states can b e systematically classiﬁed, an d whic h parameters gov ern or accelerate conv ergence in concrete applications. The results developed in this p aper provide new insight into these questions by linking the geomet- ric structu re of the critical set to th e lo cal converg ence beh a vior of Riemannian gradient metho ds. In particular, th e results identify the Morse–Bott condition as the key geomet- ric mechanism that gov erns b oth the structure of th e groun d-state set and th e transition b etw een linear and sublinear local con vergence rates. The remainder of t he pap er is organized as follo ws. Section 2 introd uces notation, assumptions, and fundamental prop erties of th e minimization problem together with the relev ant asp ects of preconditioned Riemann ian optimization. In Section 3, we analyze the geometric structu re of th e energy la ndscap e a nd e stablish its connection with the con vergence b ehavio r of the P-RG metho d. Section 4 addresses implementation aspects and prop oses an eﬃcient numerical strategy guided by the theoretical results. Numerical exp eriments conﬁrming t he th eoretical ﬁnd ings are p resen ted in Section 5. Concluding remarks are given in Section 6. 2. Prelimi naries. I n this section, we introduce problem settin gs, b asic n otations, and some imp ortant prop erties of th e problem and algorithms. 2.1. Problem settings and notations. In our analytical framework, the physical domain is truncated from the full space R d to the b ound ed d omain D and the h omogeneous Diric hlet b oundary condition is imp osed on ∂ D due to the trapping p otential. On D , w e adopt the standard notations for the Leb esgue spaces L p ( D ) = L p ( D , C ) and the S ob olev space H 1 ( D ) = H 1 ( D , C ) as well as the correspond ing norms k · k L p and k · k H 1 . F or notational simplicit y , we omit the explicit dep endence on D in these norms. Recalling th e notation F ( ρ φ ) from (1.2), w e then c onsider the Gross– Pitaevskii energy functional (1.1) and the constrained optimization problem (1.3) on D : E ( φ ) := 1 2 Z D  1 2 |∇ φ | 2 + V ( x ) | φ | 2 − Ω φ L z φ + F ( ρ φ )  d x and φ g := arg min φ ∈M E ( φ ) with M :=  φ ∈ H 1 0 ( D )   k φ k 2 L 2 = 1  . (2.1) The set M forms a Riemannian manifold, whose tangent space at φ ∈ M is given by T φ M :=  v ∈ H 1 0 ( D )    Re Z D φ v d x = 0  . (2.2) 4 ZIXU FEN G, P A TRICK HENNING, AND QINGLIN T ANG Since the Gross–Pitaevskii energy fun ctional E is real-v alued while th e w av e function φ is complex-v alued, E is not complex F r ´ ec het diﬀerentia ble in the usual sense. T o address this, we work within a real-linear space consisting of complex-v alued functions, as done in [2, 13]. In th is setting, the function space is viewed as a real Hilb ert space, meaning that all v ariatio ns are t ake n with respect to real p arameters. T o this end, we equip t he Leb esgue space L 2 ( D ) and the Sob olev space H 1 0 ( D ) with th e follow ing real inner pro du cts: ( u, v ) L 2 := Re Z D u v d x and ( u, v ) H 1 := Re  Z D u v d x + Z D ∇ u · ∇ v d x  . The corresp onding real dual space is den oted by H − 1 ( D ) :=  H 1 0 ( D )  ∗ . F or any φ ∈ H 1 0 ( D ) , let P φ : H 1 0 ( D ) → H − 1 ( D ) b e a symmetric, coercive real-linear preconditioner. It induces a bilinear form ( · , · ) P φ :=  P φ · , ·  where h· , ·i represents the canon ical duality p airing b etw een H − 1 ( D ) and H 1 0 ( D ) . This bilinear form indu ces an inner pro duct on H 1 0 ( D ) , with the associated norm given by k v k P φ := p hP φ v , v i . F urth ermore, for any closed subset W ⊂ T φ M , its orthogonal complemen t with resp ect to this inner pro duct is (2.3) W ⊥ P φ :=  u ∈ T φ M   ( u, v ) P φ = 0 ∀ v ∈ W  . Give n a subset U ⊂ M , its σ -neighborho o d is deﬁned as B σ ( U ) :=  ϕ ∈ M   ∃ φ ∈ U : k ϕ − φ k H 1 < σ  . (2.4) Throughout the pap er, w e use t w o types of constan ts: ( i ) Generic constan ts denoted by C , whic h depen d o nly on D , d , K , and V ∞ := k V k L ∞ ; ( ii ) P arameter-dep endent constants writ- ten as C v 1 ,...,v k , which increase monotonically with the H 1 -norms of the functions v 1 , . . . , v k . In p articular, if k v j k H 1 ≤ M , then C v 1 ,...,v j ,...,v k ≤ C v 1 ,...,M ,...,v k . Throughout the remainder of the pap er, we work un der t he follo wing standing assump- tions. (A1) D ⊂ R d is a b oun ded domain with C 1 , 1 b oundary , and is rotationally symmetric abou t t he z - axis for d = 2 , 3, such as a disk for d = 2 and a ball for d = 3. (A2) V ∈ L ∞ ( D ) is rotationally symmetric ab out the z -axis, i.e., V ( x ) = V ( A β x ) for all β ∈ [ − π , π ). Moreo ver, the trapping p otentia l dominates th e centrifugal con tribu- tion, i.e., there exists a constan t K > 0 suc h that V ( x ) − 1 + K 2 Ω 2 ( x 2 + y 2 ) ≥ 0 for a.e. x ∈ D . (A3) The nonlinearity f : [0 , ∞ ) → [0 , ∞ ) satisﬁes f ∈ C ([0 , ∞ )) ∩ C 1 ((0 , ∞ )) , f (0) = 0 , and the limit lim s → 0 + f ′ ( s 2 ) s 2 = 0 exists. F urthermore, there ex ists θ ∈ [0 , 3) such that f ′ ( s 2 ) s 2 is Lipschitz contin uous with p olynomial gro wth, i.e.,   f ′ ( s 2 1 ) s 2 1 − f ′ ( s 2 2 ) s 2 2   ≤ C ( s 1 + s 2 ) θ | s 1 − s 2 | , ∀ s 1 , s 2 ≥ 0 . (A4) Given φ ∈ H 1 0 ( D ) and for all u, v ∈ H 1 0 ( D ) , P φ : H 1 0 ( D ) → H − 1 ( D ) satisﬁes: ( i ) P φ is symmetric, coercive, an d con tinuous on H 1 0 ( D ) , i.e., hP φ v , v i ≥ C k v k 2 H 1 and hP φ u, v i = hP φ v , u i ≤ C φ k u k H 1 k v k H 1 . ( ii ) Given ψ ∈ H 1 0 ( D ) , the follo wing inequality holds    P φ − P ψ  u, v    ≤ C φ,ψ k u k H 1 k v k H 1 k φ − ψ k H 1 . W e brieﬂy discuss the role and interpretation of the assumptions. Assumptions (A1) and (A2) guarantee rotational inv ariance of th e Gross–P itaevskii energy functional un der rotations ab out th e z -axis. Due to th e normalization constraint, the energy functional already p ossesses a b asic inv ariance with resp ect to multiplica tion by complex phase factors, so th at minimizers are never isolated in a strict sense. The add itional STRUCTURE AND SYMMETR Y OF THE GP GROUND-ST A T E MANIFOLD 5 rotational inv ariance introduced by (A1) – (A2) has imp ortant analytical consequences: ro- tating a minimizer p rodu ces another minimizer with th e same en ergy , leading to higher- dimensional families of min imizers and additional degeneracies in th e second v ariation of the energy . Under th e assumed C 1 , 1 regularit y and rotational symmetry of D , these degeneracies admit a precis e math ematical characteri zation. In particular, for a local minimizer φ g one has L z φ g ∈ H 1 0 ( D ) , and the function i L z φ g represents an inﬁnitesimal generator of the rotatio nal symmetry . Consequently , i L z φ g b elongs to th e t angen t space T φ g M and gives rise to a zero eigenfunction of the Riemannian Hessian of E on M at φ g . A detailed justiﬁcation of the inclusion L z φ g ∈ H 1 0 ( D ) under t hese geometric assumptions is provided in App endix A. W e emphasize that the presence of rotational sy mmetry represents the most delicate setting for the analysis, since it en larges the kernel of th e second v ariation and introdu ces further degeneracies. When rotational inv ariance is absent, for instance due to th e geometry of t he domain or the ext ernal potential, these symmetry-indu ced degeneracies disapp ear, and the structure of the min imizer set simpliﬁes accordingly . Even when D lacks rotational symmetry or C 1 , 1 regularit y (e.g., rectangular domains), numerical ev idence su ggests that the symmetry-induced structu re describ ed abov e is still eﬀectivel y observe d, pro vided t he computational domain is suﬃciently large. This b eha vior can b e explained by th e exp onential decay of the ground state φ g , which renders the nu- merical solution eﬀectively insensitive to the b oundary geometry . In this sense, there exists a rotationally symmetric sub domain e D ⊂ D with C 1 , 1 -b oundary con taining the essential supp ort of φ g , such that φ g coincides with a rotationally symmetric ground state on e D up to exp onentiall y small errors. On this interior sub domain, the inclusion i L z φ g ∈ H 1 0 ( e D ) h olds rigorously , and the corresp onding zero-mode structure is therefore accurately captured in numerical compu tations on the full domain D . Regarding (A3) , we note that the condition f ≥ 0 can be relaxed to a lo w er b oun dedness condition; for clarity , we assume non-negativity . The gro wth and regularit y condition on f ′ is adapted from the classical work [13] and en sures that the energy functional E is twice conti nuously F r´ ec het d iﬀeren tiable, i.e., E ∈ C 2 ( H 1 0 ( D ) , R ). Finally , assumpt ion (A4) is not a structural p roperty of the Gross–Pi taevskii mo del itself, but rather a condition link ed to the numerical framew ork, namely to the precond itioner P φ emplo yed in t he Riemannian gradient metho d. I t ensures that P φ induces a stable local metric on M . F ollo wing [33], w e omit the compactness assumption on P φ previously i mp osed in [23, Assumption (A6)-(iii)], as it is not essential for the analysis of local conv ergence rates. Condition (A4) is satisﬁed by a wide class of p reconditioned Riemannian gradien t and pro jected Sob olev gradient metho ds in the literature. Moreov er, the Lipschitz con tinuit y of the preconditioner re quired i n (A4) -(ii) is natural, since the energy f unctional E is o f class C 2 and the associated Riemannian gradient is therefore locally Lipschitz conti nuous. Altogether, assumptions (A1) – (A4) are standard in numerical simulations and physical exp eriments, and under ( A1) – (A3) th e existence of a minimizer fo r (2.1) foll ow s from classica l var iational arguments ( see [9]). 2.2. Properties of the problem. Giv en φ ∈ H 1 0 ( D ) , we introduce a b ounded real linear op erator H φ : H 1 0 ( D ) → H − 1 ( D ) , for all u, v ∈ H 1 0 ( D ) hH φ u, v i := 1 2 ( ∇ u, ∇ v ) L 2 + (( V − Ω L z ) u, v ) L 2 +  f ( ρ φ ) u, v  , (2.5) where  f ( ρ φ ) u, v  := Re R D f ( ρ φ ) u v d x . Then the ﬁrst and second F r´ echet deriv ativ es of the energy functional E can be expressed as E ′ ( φ ) = H φ φ and E ′′ ( φ ) = H φ + f ′ ( ρ φ )  | φ | 2 + φ 2 ·  . F rom an optimizatio n p ersp ective, the local minimizer φ g satisﬁes the ﬁrst-order and second- order necessary conditions: E ′ ( φ g ) = λ φ g I φ g and  E ′′ ( φ g ) − λ φ g I  v , v  ≥ 0 for all v ∈ T φ g M , (2.6) with λ φ g b eing the Lagrange multiplier asso ciated with th e L 2 -normalization constraint and I : L 2 ( D ) → L 2 ( D ) ⊂ H − 1 ( D ) the canonical identiﬁcation I v := ( v , · ) L 2 . Equ iv alently , 6 ZIXU FEN G, P A TRICK HENNING, AND QINGLIN T ANG λ φ g =  H φ g φ g , φ g  can b e seen as an eigenv alue of the nonlinear eigenproblem E ′ ( φ ) = λ I φ with eigenfunction φ g . In the sp ecial case Ω = 0 and f ( s ) = η s, η ≥ 0, and when restricting to real-v alued functions, the lo cal minimizer is nondegenerate in the classical sense: the second-order suﬃcien t optimality cond ition holds, i.e.,  E ′′ ( φ g ) − λ φ g I  v , v  ≥ C k v k 2 H 1 for all v ∈ T φ g M . This condition implies that the lo cal minimizer is isolated. F or Ω > 0, h o w ever, this is no longer true due to symmetry , but we will see t hat a corresp onding co ercivit y prop erty still holds on a closed subspace of T φ g M . Let u s next in tro duce the set of lo cal minimizers at the same energy level as a giv en lo cal minimizer φ g : S := n φ ∈ M   φ is a local minimizer and E ( φ ) = E S := E ( φ g ) o . (2.7) T o address symmetry-ind uced degeneracy , Bott, in his seminal wo rk [11], introdu ced th e no- tion of non degenerate critical manifolds, a condition now known as the Morse– Bott cond ition [22, 42]. W e no w recall the p recise formulation. Definition 2.1 ( Morse–Bott Condition ). L et E : M → R b e a C 2 functional deﬁne d on a smo oth Riemannian submanifold M ⊂ X , wher e X is a r e al Hil b ert sp ac e. We say that E satisﬁes the lo c al Morse–Bott c ondition ne ar a l o c al mi nimizer φ g if ther e exists a suﬃciently smal l σ > 0 such that the set S σ ( φ g ) := S ∩ B σ ( φ g ) i s a ﬁni te-dimensional C 1 emb e dde d submanifold of M , and for every φ ′ g ∈ S σ ( φ g ) , we have the i dentity K φ ′ g := ker   ∇ R X  2 E ( φ ′ g )  := n v ∈ T φ ′ g M    ∇ R X  2 E ( φ ′ g ) v , u  X = 0 , ∀ u ∈ T φ ′ g M o = T φ ′ g S σ ( φ g ) . Her e, ∇ R X E ( φ ) and  ∇ R X  2 E ( φ ) denote r esp e ctively the Riemannian gr adient and the Rie- mannian Hessian of E at φ , c ompute d with r esp e ct to the R iemannian metric on M induc e d by the ambient Hil b ert sp ac e X . The local Morse–Bott condition stated ab ov e is form ulated in full generality . In particu- lar, th e form ulations used in [23, 30] (also k now n as qu asi-isolated ground states) corresp ond to sp eciﬁc symmetry -induced scenarios. In those works, the set S is assumed to b e the orbit of φ g under symmetry transformations, sp eciﬁcally phase rotations and, in t he rotationall y inv ariant setting, spatial rotations. In such cases, the local manifold S σ ( φ g ) is precisely the group orbit through φ g , whic h is automatically an embedded submanifold. Moreo ver, the tangent space at any φ ∈ S σ ( φ g ) contains the inﬁnitesimal generators of these symmetries; concretely , span { i φ, i L z φ } ⊆ T φ S σ ( φ g ) . In th is w ork, we go beyond symmetry-indu ced manifolds and consider the general se tting described by the lo cal Morse–Bott condition ab o ve. Crucially , whether this condition holds turns out to b e decisive for the conv ergence b ehavior and o verall eﬃciency of numerical optimization algori thms. W e now introduce the P φ -orthogonal complement of K φ , den oted by R φ , i.e., R φ := ( K φ ) ⊥ P φ =  u ∈ T φ M   ( u, v ) P φ = 0 , ∀ v ∈ K φ  . Let J φ : H 1 0 ( D ) → R φ denote the P φ -orthogonal p ro jection op erator on to R φ . Und er th e Morse–Bott condition, for any φ ′ g ∈ B σ ( φ g ), E ′′ ( φ ′ g ) − λ φ ′ g I is non-degenerate on R φ ′ g . Pr op erty 2.2. Let E satisfy the Morse–Bott cond ition around φ g . Then, there ex ists a suﬃciently small σ > 0 suc h th at for every φ ′ g ∈ S σ ( φ g ), th e op erator E ′′ ( φ ′ g ) − λ φ ′ g I is coercive on R φ ′ g , i.e.,  ( E ′′ ( φ ′ g ) − λ φ ′ g I ) v , v  ≥ C k v k 2 H 1 f or al l v ∈ R φ ′ g . Pr o of. The pro of is given in App endix B. STRUCTURE AND SYMMETR Y OF THE GP GROUND-ST A T E MANIFOLD 7 Finally , for an y φ ∈ H 1 0 ( D ) , th e imp ortant prop erties of E ( φ ) and E ′′ ( φ ) are su mmarized b elo w (cf. [23, Propsotion 2.3]). Pr op erty 2.3. Given φ ∈ H 1 0 ( D ) and for all u, v ∈ H 1 0 ( D ) , the f ollo wing conclusions hold: ( i ) E ′′ ( φ ) is a continuous op erator on H 1 0 ( D ) , i.e.,    E ′′ ( φ ) u, v    ≤ C φ k u k H 1 k v k H 1 . ( ii ) Given ψ ∈ H 1 0 ( D ) , the follo wing inequality holds    E ′′ ( φ ) − E ′′ ( ψ )  u, v    ≤ C φ,ψ k u k H 1 k v k H 1 k φ − ψ k H 1 . ( iii ) The follo wing Lipsc hitz-type inequalit y h olds E ( φ + v ) − E ( φ ) ≤  E ′ ( φ ) , v  + 1 2  E ′′ ( φ ) v , v  + C φ,v k v k 3 H 1 . 2.3. Properties of the al gorithms. F or a nondegenerate sequence of step size pa- rameters τ n > 0, the precond itioned R iemannian gradient metho d takes th e form φ n +1 = R φ n ( τ n d n ) := φ n + τ n d n   φ n + τ n d n   L 2 (2.8) with th e descent direction given by t he negative Riemannian gradien t in the P φ -metric: d n := −∇ R P E ( φ n ) = −P − 1 φ n H φ n φ n + λ φ n P − 1 φ n I φ n , λ φ :=  φ, P − 1 φ H φ φ  L 2  φ, P − 1 φ I φ  L 2 . Note that the formula ex ploits that the Riemannian gradient ∇ R P E ( φ ) is give n by ∇ R P E ( φ ) = Pro j P φ φ P − 1 φ E ′ ( φ ) , where Pro j P φ φ ( v ) = v − ( φ, v ) L 2 ( φ, P − 1 φ I φ ) L 2 P − 1 φ I φ. Although the assumptions on the preconditioner diﬀer from those in [23], the proof of the follo wing basic prop erties is la rgely analogous (cf. [23, Prop osition 3.1]); see also Remark 2.5. W e brieﬂy recall them here. Pr op erty 2.4. Given φ ∈ H 1 0 ( D ) and f or al l u, v ∈ H 1 0 ( D ) and w ∈ H − 1 ( D ) , the follo wing conclusions hold: ( i ) If E satisﬁes th e Morse–Bott condition around φ g , then there exists σ > 0 such t hat for all φ ∈ S σ ( φ g ), th e op erators P φ and E ′′ ( φ ) − λ φ I are spectrally equ iv alent on R φ , i.e., inf v ∈ R φ \{ 0 }  E ′′ ( φ ) − λ φ I  v , v   P φ v , v  = µ > 0 , sup v ∈ R φ \{ 0 }  E ′′ ( φ ) − λ φ I  v , v   P φ v , v  = L < ∞ . (2.9) ( ii ) F or any φ ∈ M , th ere exists σ > 0 suc h that for all ψ ∈ B σ ( φ ), the op erator ∇ R P E ( · ) : H 1 0 ( D ) → H 1 0 ( D ) and the fun ctional λ ( · ) : H 1 0 ( D ) → R are Lipschitz contin uous at φ , i.e.,   ∇ R P E ( φ ) − ∇ R P E ( ψ )   H 1 ≤ C φ k φ − ψ k H 1 and   λ φ − λ ψ   ≤ C φ k φ − ψ k H 1 . ( iii ) Let φ ∈ M , then for all v ∈ T φ M , it holds   R φ ( tv ) − ( φ + tv )   ≤ 1 2 t 2 k v k 2 L 2 | φ + tv | p oint wise a.e. in D . R em ark 2.5. I n con trast to p revious w ork (e.g ., [23, 30]), w e do not a ssume in v ariance of the preconditioner under t he u nderlying symmetry group (su c h as rotations or phase shifts), nor do w e imp ose additional symmetry-based structural assumptions on the critical set. A s a consequence, the lo cal constants L and µ in (2.9) may dep end on the base p oint φ along the em b edded submanifold. How eve r, our analysis is local in n ature. Since the energy fun ctional 8 ZIXU FEN G, P A TRICK HENNING, AND QINGLIN T ANG E is of class C 2 and the pro jection op erator J φ dep ends continuously on φ ∈ S σ ( φ g ), this v ariatio n is w ell controlle d. The contin uity of J φ follo ws from the fact that S σ ( φ g ) is a ﬁnite-dimensional C 1 em b edded submanifold, so that t he orthogonal pro jection onto th e tangent space T φ S σ ( φ g ) v aries continuously with φ . Combined with the contin uity of the preconditioner P φ , this implies th at t he p ro jection on to t he P φ -orthogonal complement also dep ends continuously on φ . Consequently , th e constants app earing in our estimates dep end only on the size of the neighborho o d around the submanifold. Any such dep enden ce can b e absorb ed into the small parameter ε , which go verns b oth the stepsize restriction and the asymptotic con vergence rate. F or notational simplicity , we therefore d enote all such lo cal b ounds un iformly by L and µ . 3. Sharp res ults on conv ergence and structure. In this section, we establish a set of sh arp theoretical c haracterizations describing the lo cal behavior and g eometric structure of minimizers for the Gross–Pitaevskii energy functional. First, in the case of non-isolated min- imizers, we show th at th e Morse–Bott cond ition p ro vides a su ﬃcien t condition for the set of ground states to b e p artitioned into ﬁnitely many embedded submanifolds, on each of which the energy functional is constant. Second, for t he P-RG metho d, w e derive its opt imal local conv ergence rate and prov e that lo cal Q -linear conve rgence of the energy sequence occurs if and only if the Morse– Bott condition holds, thereby characterizing precisely when such fast rates are attainable. F urthermore, when restricted to th e ground state set, the Morse–Bott condition holds if and only if t his set decomp oses in to ﬁ nitely many sy mmetry orbits gener- ated by the phase and rotational inv ariances. In th is case, the P- RG algorithm exhibits lo cal linear conv ergence in a neighborho o d of the ground state set, thereby connecting the geo- metric regularit y of the critical manifold, the top ological classiﬁcation of minimizers, and the algorithm’s lo cal converge nce b ehavior. Finally , when E is real analyt ic and the Morse–Bott condition fails, the P-RG algorithm conv erges lo cally at a sublinear rate. 3.1. Main results. 3.1.1. The Morse–Bott condition for the ﬁnite classiﬁcation of global mi n- imizers. In the p resence of contin uous symmetries, such as phase inv ariance and spatial rotations, the Gross–Pitaevskii energy functional typical ly admits non-unique global mini- mizers t hat are related by symmetry transformations. These states organize into contin uous families, or orbits, each forming a compact C 1 em b edded sub manifold. A fund amenta l q ues- tion arises: under what conditions can all physically distinct global minimizers b e cleanly separated into such orbits, so th at n o tw o orbits interse ct or accumulate arbitrarily close t o one anoth er? Such a classiﬁcation is not only essentia l for a rigorous understanding of t he solution landscape, bu t also crucial for the design of optimization algorithms that aim to recov er all relev ant physical conﬁ gurations. I n what follo ws, we introduce a notion of w ell-deﬁned clas- siﬁcation, and show that the Morse–Bott condition provides a suﬃcient geometric criterion for this prop erty . Definition 3.1 ( W el l-deﬁned clas siﬁcation ). We say that the set of glob al mi nimiz- ers S g admits a wel l-deﬁne d classiﬁc ation if it c an b e written as a ﬁnite disjoint union of distinct c onne cte d symmetry orbits: S g = N G ℓ =1 S φ g,ℓ = N G ℓ =1 n ψ ∈ M   ψ = e i α φ g,ℓ ( A β x ) , α, β ∈ [ − π , π ) o , for some ﬁnite c ol le ction of gr ound state r epr esentatives φ g,ℓ ∈ S g , with 1 ≤ ℓ ≤ N . This n otion of classiﬁcatio n captu res the idea that global minimizers can b e group ed into ﬁnitely many geometricall y distinct families, each closed under the inherent symmetries of the system. F rom a computational p erspective, it guarantees th at th e solution landscap e is enumerable and resolv able: with a ﬁnite num ber of app ropriately chosen initial guesses, one can in p rinciple recov er every inequiv alent global minimizer using stand ard optimization al- gorithms. The follo wing t heorem pro vides a ge ometric criterion for when such a c lassiﬁcation holds. Theorem 3.2. Supp ose the ener gy functional E satisﬁes the Morse–Bott c ondi tion in a STRUCTURE AND SYMMETR Y OF THE GP GROUND-ST A T E MANIFOLD 9 neighb orho o d of e ach symmetry orbit S φ g := n ψ ∈ M | ψ = e i α φ g ( A β ( x )) , α, β ∈ [ − π , π ) o , φ g ∈ S g . Then the set of glob al minim izers S g admits a wel l-deﬁne d classiﬁc ation. In other words, the M orse–Bott condition preven ts the app earance of i nﬁnitely many distinct families of minimizers or accumulation of symmetry orbits, and enforces a ﬁ nite decomp osi- tion of the ground state set into well-separa ted comp onents. The proof of Theorem 3.2 is giv en in Section 3.3. 3.1.2. Morse–Bott condition and optimal lo cal line ar con vergence. The foll ow - ing theorem establishes the optimal local con vergence rate for the p reconditioned Riemannian gradien t (P-RG) metho d. The proof is giv en in Section 3.3. Theorem 3.3. L et E satisfy the Morse–Bott c ondition ar ound a lo c al m inimizer φ g ∈ S . Then, for every suﬃciently smal l ε > 0 , ther e exists σ > 0 such that for al l φ 0 ∈ B σ ( φ g ) , the se quenc e { φ n } n ∈ N gener ate d by the P-R G c onver ges Q -line arly to a lo c al minimi zer φ ∗ g (dep ending on φ 0 ) and satisﬁes k φ ∗ g − φ g k H 1 ≤ C σ , i.e., k φ n − φ ∗ g k P φ ∗ g ≤ k φ n − 1 − φ ∗ g k P φ ∗ g (max {| 1 − τ µ | , | 1 − τ L |} + ε ) , for al l τ ∈ (0 , 2 / ( L + ε )) , n ≥ 1 . In p articular, with the asymptotic al ly optimal choic e τ = 2 / ( L + µ ) , the P-RG yields the optimal Q-line ar c onver genc e r ate k φ n − φ ∗ g k P φ ∗ g ≤ k φ n − 1 − φ ∗ g k P φ ∗ g  L − µ L + µ + ε  , ∀ n ≥ 1 . (3.1) This result p ro vides a natu ral generalizatio n of the classical optimal linear conv ergence theory for gradient d escen t on strongly conv ex qu adratic p roblems to a nonconv ex setting gov erned by the Morse–Bott condition. In particular, the contraction factor ( L − µ ) / ( L + µ ) coincides with t he b est p ossible rate k now n for gradient d escen t on strongly conv ex q uadratics, which w as shown to b e opt imal in [41, Theorem 3 in Chapter 1, Section 4]. The th eorem therefore shows that, despite the presence of continuous symmetries and the resulting degeneracy of the Hessian along symmetry directions, the P-RG method with general preconditioners retains the same optimal local Q -linear conv ergence b ehavior as in the classical strongly convex setting where second-order suﬃcient conditions hold. The follo wing th eorem, again p ro ved in Section 3.3, provides a sharp characterization of the Morse–Bo tt condition in terms of Q -linear converg ence of the energy for the P-RG metho d. Theorem 3.4. The ener gy functional E satisﬁes the Morse–Bott c ondition ar ound a lo c al minim izer φ g ∈ S if and only if for every suﬃciently smal l ε > 0 , ther e exist c onstan ts ρ ∈ (0 , 1) and σ > 0 such that for al l φ 0 ∈ B σ ( φ g ) , the ener gy se quenc e { E ( φ n ) } n ∈ N gener ate d by the P-RG c onver ges Q-line arly to E ( φ g ) , i.e., E ( φ n +1 ) − E ( φ g ) ≤ ( ρ + ε )  E ( φ n ) − E ( φ g )  , ∀ τ ∈ (0 , 2 / ( L + ε )) , n ≥ 0 . This naturally raises the qu estion of whether the symmetry- generated critical manifold it- self satisﬁes the Morse–Bott condition. This question is motiv ated by extensive numerical evidence in important models suc h as the BEC model, which consisten tly indicate that the non-uniqu eness of ground states arises from th e action of the sy mmetry group U (1) × S O (2), i.e., global phase shifts and spatial rotations, with n o further degeneracies observ ed. The follow ing result pro vides a theoretical explanation for th is phen omenon. In partic- ular, it sho ws that if the Morse–Bott condition holds along the symmet ry-generated critical manifolds, then every contin uous family of ground states is induced by these symmetries. Under assumpt ions (A1) – (A4) , this rules out additional bifurcation-typ e or accidental de- generacies. The follo wing theorem mak es th is connection precise. Theorem 3.5. The ener gy functional E satisﬁes the Morse–Bott c ondition along every symmetry orbit S φ g := n ψ ∈ M | ψ = e i α φ g ( A β ( x )) , α, β ∈ [ − π , π ) o , φ g ∈ S g . 10 ZIXU FEN G, P A TRICK HENNING, AND QINGLIN T ANG if and only if ( i ) The set of gl ob al minimi zers S g admits a wel l-deﬁne d classiﬁc ation. ( ii ) F or every suﬃciently smal l ε > 0 , ther e exist c onstants ρ ∈ (0 , 1) and σ > 0 such that for al l φ 0 ∈ B σ ( S g ) , the se quenc e { φ n } n ∈ N gener ate d by the P-RG c onver ges line arly to a gr ound state φ ∗ g , which dep ends on φ 0 and satisﬁes k φ ∗ g − φ g k H 1 ≤ C σ , i.e., k φ n − φ ∗ g k H 1 ≤ C ε k φ 0 − φ ∗ g k H 1 ( ρ + ε ) n , ∀ τ ∈ (0 , 2 / ( L + ε )) , n ≥ 0 . Again, th e pro of is p ostp oned to S ection 3.3. 3.1.3. Be yond the Morse–Bott condition. The p revious results rely crucially on the Morse–B ott condition, which ensu res that the critical set of the energy functional E is a sub manifold and th at th e H essian of E is non- degenerate in the normal directions. When the Morse–Bott condition fails, the lo cal geometry of t he energy functional E near local minimizers may ex hibit higher-order d egeneracies, leading to more intricate d ynamical b ehavio r of the P-RG. I n particular, by Theorem 3.5 , local linear conv ergence is n o longer p ossible in this setting. Nevertheless , if the nonlinearity f in E is real-analytic, which is indeed the case for the Bose–Einstein condensation model since f ( s ) = η s is linear and hence analytic, then the P-RG iteration still enjo ys a wea ker but meaningful conv ergence guaran tee: global conv ergence to a critical p oint with a sub linear asymptotic rate. This is made p recise in the follo wing result, to b e prov ed in Section 3.3. Theorem 3.6. Supp ose that the Morse–Bott c ondition fails for E , and that the nonlin- e arity f i s r e al-analytic. Then, for every suﬃciently smal l ε > 0 , ther e exists σ > 0 such that for any step size τ ∈ (0 , 2 / ( L + ε )) and any i nitial iter ate φ 0 ∈ B σ ( Crit ( E )) , the se quenc e { φ n } n ∈ N gener ate d by the P-R G c onver ges to a critic al p oint φ s ∈ Cri t ( E ) . Mor e over, the c onver genc e r ate is subline ar: k φ n − φ s k H 1 ≤ C n − ν 1 − 2 ν , ∀ n ≥ 0 , wher e ν ∈ (0 , 1 / 2) i s the Loj asiewicz exp onent asso ciate d with E at the l imiting critic al p oint φ s , indep endent of P φ . Consequently , an imp ortant implication is that preconditioning cannot impro ve the asymp- totic converg ence order of th e P-RG, since t he L o jasiewicz exp onent (and thus the sub linear rate) does not depend on the p reconditioner P φ . Moreo v er, this result implies that any P-RG metho d that is kno wn to con v erge globally m ust in fact con v erge strongly along the en tire se- quence, rather than only along subsequences. In particular, this applies to the Bose–Einstein condensation mo del. 3.2. T echnical lemmas. Before presen ting the proofs of the main results, we introduce sever al key lemmas that will b e instrumental in establishing v arious aspects of our results. Lemma 3.7. Given a lo c al minimi zer φ g of E , the fol lowing e quality holds f or any pr e- c onditioner P φ g : sup v ∈ R φ g \{ 0 } h ( E ′′ ( φ g ) − λ φ g I ) v , v i hP φ g v , v i = sup v ∈ T φ g M\{ 0 } h ( E ′′ ( φ g ) − λ φ g I ) v , v i hP φ g v , v i . Pr o of. F or any v ∈ T φ g M , we ha ve the decomposition v = v 1 + v 2 with v 1 ∈ R φ g and v 2 ∈ K φ g . Noting th at, by th e self-adjointness of E ′′ ( φ g ) − λ φ g I with resp ect to th e d ualit y pairing, h ( E ′′ ( φ g ) − λ φ g I ) v , v i hP φ g v , v i = h ( E ′′ ( φ g ) − λ φ g I ) v 1 , v 1 i hP φ g v 1 , v 1 i + hP φ g v 2 , v 2 i , it follow s that sup v ∈ T φ g M\{ 0 } h ( E ′′ ( φ g ) − λ φ g I ) v , v i hP φ g v , v i ≤ sup v ∈ R φ g \{ 0 } h ( E ′′ ( φ g ) − λ φ g I ) v , v i hP φ g v , v i . Consequently , we obtain sup v ∈ R φ g \{ 0 } h ( E ′′ ( φ g ) − λ φ g I ) v , v i hP φ g v , v i = sup v ∈ T φ g M\{ 0 } h ( E ′′ ( φ g ) − λ φ g I ) v , v i hP φ g v , v i . STRUCTURE AND SYMMETR Y OF THE GP GROUND-ST A T E MANIFOLD 11 Lemma 3.8. L et E satisfy the Morse–Bott c ondition ar ound φ g ∈ S . Then, for every suﬃciently smal l ε > 0 , th er e exists σ > 0 such that for any φ ∈ B σ ( φ g ) , the Pol yak– Lojasiewicz ine quality holds E ( φ ) − E ( φ g ) ≤ 1 2( µ − ε )    ∇ R P E ( φ )    2 P φ . Pr o of. F or some suﬃciently small σ > 0 , consider the pro jection of φ ∈ B σ ( φ g ) onto S , deﬁned by φ ′ g := arg min u ∈S 1 2 k φ − u k 2 P φ g . W e ﬁ rst establish the ex istence of φ ′ g for suﬃciently small σ > 0. T o this end, in tro duce the P φ g -metric n eigh b orho od B P σ ( φ g ) := n ψ ∈ M   k ψ − φ g k P φ g < σ o and S P σ ( φ g ) := S ∩ B P σ ( φ g ) . Since t he en ergy functional E satisﬁes th e Morse–Bott cond ition around φ g , th ere exists σ 0 > 0 such that S P σ 0 ( φ g ) is a smooth embedded submanifold of M , and E ( φ ) ≥ E ( φ g ) for all φ ∈ B P σ 0 ( φ g ). Consequently , for any φ ∈ B P σ 0 ( φ g ), w e hav e E ( φ ) = E ( φ g ) if and only if φ ∈ S . Now deﬁne th e lo cal set S P σ 0 / 2 ( φ g ) := S ∩ B P σ 0 / 2 ( φ g ) = E − 1 ( { E ( φ g ) } ) ∩ B P σ 0 / 2 ( φ g ) , this set is closed as the intersectio n of tw o closed sets. Because S P σ 0 / 2 ( φ g ) is a ﬁnite- dimensional embedd ed C 1 submanifold, S P σ 0 / 2 ( φ g ) is compact. F or an y φ ∈ B P σ 0 / 4 ( φ g ) and any u ∈ S \ S P σ 0 / 2 ( φ g ) , w e estimate k φ − u k P φ g ≥ k u − φ g k P φ g − k φ − φ g k P φ g > σ 0 2 − σ 0 4 = σ 0 4 > k φ − φ g k P φ g . Hence, the local minimizer of k φ − u k P φ g o ver u ∈ S must lie in S P σ 0 / 2 ( φ g ). Since the norm is contin uous and S P σ 0 / 2 ( φ g ) is compact, the minimum is attained. Therefore, by (A4) -( i ), there exists suﬃ cien tly small σ > 0 such that for all φ ∈ B σ ( φ g ) , th e pro jection φ ′ g exists. Moreo ve r, since φ ′ g ∈ S P σ 0 ( φ g ), it satisﬁes th e ﬁ rst-order optimality condition: ( φ − φ ′ g , v ) P φ g = 0 , ∀ v ∈ T φ ′ g S P σ 0 ( φ g ) . By assumption (A4) -( ii ) and the follo wing inequalit y k φ ′ g − φ g k P φ g ≤ k φ − φ g k P φ g + k φ − φ ′ g k P φ g ≤ 2 k φ − φ g k P φ g , w e deduce that as σ → 0 + , ( φ − φ ′ g , v ) P φ ′ g = o ( k φ − φ ′ g k H 1 ) , ∀ v ∈ T φ ′ g S P σ 0 ( φ g ) . Combining the follo wing decomp osition φ − φ ′ g = Pro j L 2 φ ′ g ( φ − φ ′ g ) − 1 2 k φ − φ ′ g k 2 L 2 φ ′ g , w e h a ve φ − φ ′ g = J φ ′ g ( φ − φ ′ g ) + o ( k φ − φ ′ g k H 1 ) as σ → 0 + , (3.2) where we recall J φ ′ g as th e P φ ′ g -orthogonal pro jection onto the normal space R φ ′ g . The remainder of th e pro of follo ws verbatim from [23, Lemma 4.2]. Sp eciﬁcally , according to E ( φ ′ g ) = E ( φ g ), T aylor’s formula at φ , and (3.2), w e hav e 12 ZIXU FEN G, P A TRICK HENNING, AND QINGLIN T ANG E ( φ ) − E ( φ g ) =  E ′ ( φ ) , φ − φ ′ g  − 1 2  E ′′ ( φ )( φ − φ ′ g ) , φ − φ ′ g  + o  k φ − φ ′ g k 2 H 1  =  ∇ R P E ( φ ) , φ − φ ′ g  P φ − 1 2  E ′′ ( φ ) − λ φ I  ( φ − φ ′ g ) , φ − φ ′ g  + o  k φ − φ ′ g k 2 H 1  =  ∇ R P E ( φ ) , J φ ′ g ( φ − φ ′ g )  P φ − 1 2  E ′′ ( φ ) − λ φ I  J φ ′ g ( φ − φ ′ g ) , J φ ′ g ( φ − φ ′ g )  + o  k φ − φ ′ g k 2 H 1  . Based on Property 2.3 -( ii ), Property 2.4 -( ii ), an d (A4) -( ii ), the follo wing estima- tions hold D  E ′′ ( φ ) − E ′′ ( φ ′ g )  J φ ′ g ( φ − φ ′ g ) , J φ ′ g ( φ − φ ′ g ) E = o  k φ − φ ′ g k 2 H 1  , D  λ φ ′ g I − λ φ I  J φ ′ g ( φ − φ ′ g ) , J φ ′ g ( φ − φ ′ g ) E = o  k φ − φ ′ g k 2 H 1  , D  P φ − P φ ′ g  J φ ′ g ( φ − φ ′ g ) , J φ ′ g ( φ − φ ′ g ) E = o  k φ − φ ′ g k 2 H 1  . According to Prope rt y 2.4 - ( i ), the follow ing lo w er b oun d estimate holds D  E ′′ ( φ ′ g ) − λ φ ′ g I  J φ ′ g ( φ − φ ′ g ) , J φ ′ g ( φ − φ ′ g ) E D P φ ′ g J φ ′ g ( φ − φ ′ g ) , J φ ′ g ( φ − φ ′ g ) E ≥ µ. In summary , the estimate w e w an t is derived − 1 2 D  E ′′ ( φ ) − λ φ I  J φ ′ g ( φ − φ ′ g ) , J φ ′ g ( φ − φ ′ g ) E ≤ − µ 2 D P φ J φ ′ g ( φ − φ ′ g ) , J φ ′ g ( φ − φ ′ g ) E + o  k φ − φ ′ g k 2 H 1  . Then, combined with (3.2), w e futher get E ( φ ) − E ( φ g ) ≤  ∇ R P E ( φ ) , J φ ′ g ( φ − φ ′ g )  P φ − µ 2   J φ ′ g ( φ − φ ′ g )   P φ + o  kJ φ ′ g ( φ − φ ′ g ) k 2 H 1  . By the minimization prop ert y of φ ′ g , we obtain k φ − φ ′ g k H 1 ≤ C k φ − φ g k H 1 . Consequently , for all suﬃcien tly sma ll ε > 0, th ere exists σ > 0 such t hat for any φ ∈ B σ ( φ g ), the Poly ak– L o jasiewicz inequality is d educed as follo ws E ( φ ) − E ( φ g ) ≤  ∇ R P E ( φ ) , J φ ′ g ( φ − φ ′ g )  P φ − µ − ε 2  J φ ′ g ( φ − φ ′ g ) , J φ ′ g ( φ − φ ′ g )  P φ ≤ sup v ∈ H 1 0 ( D )  ∇ R P E ( φ ) , v  P φ − µ − ε 2 ( v , v ) P φ ! = 1 2( µ − ε )    ∇ R P E ( φ )    2 P φ . Lemma 3.9. L et E satisfy the Polyak– Lojasiewicz ine quality ar ound φ g ∈ S , i.e. , ther e exists σ > 0 and a c onstant µ P L > 0 such that for any φ ∈ B σ ( φ g ) , the fol lowing ine quality holds E ( φ ) − E ( φ g ) ≤ 1 2 µ P L    ∇ R P E ( φ )    2 P φ . Then, for every φ ′ g ∈ S σ ( φ g ) , the R iemannian Hessian ( ∇ R P ) 2 E ( φ ′ g ) i s uniformly c o er cive on R φ ′ g , i.e.,  ( ∇ R P ) 2 E ( φ ′ g ) v , v  P φ ′ g ≥ µ PL k v k 2 P φ ′ g for al l v ∈ R φ ′ g . STRUCTURE AND SYMMETR Y OF THE GP GROUND-ST A T E MANIFOLD 13 Pr o of. The Riemannian Hessian at φ ′ g ∈ B σ ( φ g ) is given by ( ∇ R P ) 2 E ( φ ′ g ) = Pro j P φ ′ g φ ′ g P − 1 φ ′ g  E ′′ ( φ ′ g ) − λ φ ′ g I    T φ ′ g M . Observe that the kernel K φ ′ g is indep en dent of the choice of the Riemannian m etric. Indeed , for any v , u ∈ T φ ′ g M ,  ( ∇ R P ) 2 E ( φ ′ g ) v , u  P φ ′ g =  ( E ′′ ( φ ′ g ) − λ φ ′ g I ) v , u  . Hence, to study the kernel K φ ′ g , we may c hoose the con venien t H 1 0 -metric, i.e., P φ ′ g = − ∆. Under this choi ce, the Riemannian Hessian becomes ( ∇ R H 1 0 ) 2 E ( φ ′ g ) = Pro j H 1 0 φ ′ g ( − ∆) − 1  E ′′ ( φ ′ g ) − λ φ ′ g I    T φ ′ g M = Pro j H 1 0 φ ′ g  1 2 I + ( − ∆) − 1  V − Ω L z + f ( ρ φ ′ g ) + f ′ ( ρ φ ′ g )  | φ ′ g | 2 + ( φ ′ g ) 2 ·  − λ φ ′ g I    T φ ′ g M =: 1 2 I   T φ ′ g M + A φ ′ g   T φ ′ g M . The op erator A φ ′ g is compact due to kA φ ′ g v k 2 H 1 0 = D V − Ω L z + f ( ρ φ ) + f ′ ( ρ φ )  | φ | 2 + φ 2 ·  − λ φ ′ g I  v , A φ ′ g v E ≤ C ( k v k L 2 + k v k L 6 / (4 − θ ) ) kA φ ′ g v k H 1 0 , ∀ v ∈ T φ ′ g M . Therefore, th e H 1 0 -Riemannian Hessian is a compact p erturbation of t he identi ty op erator and hen ce F redholm of index zero. Its kernel K φ ′ g is ﬁn ite-dimensional. As in the pro of of [23, Prop osition 2.2], we now establish pointw ise co ercivity on R φ ′ g µ φ ′ g := inf v ∈ R φ ′ g \{ 0 }  ( ∇ R P ) 2 E ( φ ′ g ) v , v  P φ ′ g k v k 2 P φ ′ g = inf v ∈ R φ ′ g \{ 0 }  ( E ′′ ( φ ′ g ) − λ φ ′ g I ) v , v  k v k 2 P φ ′ g > 0 . F or any v ∈ R φ ′ g with k v k P φ ′ g = 1 and suﬃciently small t ∈ R such that φ = R φ ′ g ( tv ) ∈ B σ ( φ g ) , the T aylor expansion yields E ( φ ) − E ( φ ′ g ) = t 2 2  ( ∇ R P ) 2 E ( φ ′ g ) v , v  P φ ′ g + o ( t 2 ) . Since the P olya k– L o jasiewicz ineq ualit y holds by assumption, we hav e E ( φ ) − E ( φ ′ g ) = E ( φ ) − E ( φ g ) ≤ 1 2 µ P L   ∇ R P E ( φ )   2 P φ . Noting that ∇ R P E ( φ ) = t ( ∇ R P ) 2 E ( φ ′ g ) v + o ( t ), we obtain t 2 2  ( ∇ R P ) 2 E ( φ ′ g ) v , v  P φ ′ g + o ( t 2 ) ≤ t 2 2 µ P L    ( ∇ R P ) 2 E ( φ ′ g ) v    2 P φ ′ g + o ( t 2 ) . Dividing b oth sides by t 2 / 2, letting t → 0, and noting that ( ∇ R P ) 2 E ( φ ′ g ) is a b ounded self- adjoin t operator, w e obtain µ P L ≤ 1 µ φ ′ g inf v ∈ R φ ′ g \{ 0 }  ( ∇ R P ) 2 E ( φ ′ g ) v , ( ∇ R P ) 2 E ( φ ′ g ) v  P φ ′ g ( v , v ) P φ ′ g = 1 µ φ ′ g inf Sp ec   ( ∇ R P ) 2 E ( φ ′ g )  2   R φ ′ g  = µ φ ′ g . This completes the proof. 14 ZIXU FEN G, P A TRICK HENNING, AND QINGLIN T ANG Lemma 3.10. L et E satisfy the Polyak– Lojasiewicz ine quali ty ar ound φ g ∈ S , i .e., ther e exists σ > 0 and a c onstant µ P L > 0 such that for any φ ∈ B σ ( φ g ) , the fol lowing ine quality holds E ( φ ) − E ( φ g ) ≤ 1 2 µ P L    ∇ R P E ( φ )    2 P φ . Then, E satisﬁes the Morse–Bott c ondition ar ound φ g . Pr o of. W e ﬁ rst observ e that, for σ > 0 suﬃciently small, the Poly ak– Lo jasiewicz inequ al- it y implies that any critical p oint in B σ ( φ g ) has the same energy as φ g . Indeed, i f φ ′ g ∈ B σ ( φ g ) satisﬁes ∇ R P E ( φ ′ g ) = 0, th en the Poly ak– L o jasiewicz in equalit y giv es E ( φ ′ g ) − E ( φ g ) ≤ 0. Since φ g is a local minimizer, we als o hav e E ( φ ′ g ) ≥ E ( φ g ), hence E ( φ ′ g ) = E ( φ g ). Therefore, for σ > 0 suﬃcien tly small, the set of critical p oints in B σ ( φ g ) coincides with S σ ( φ g ) = n φ ∈ B σ ( φ g )   ∇ R P E ( φ ) = 0 o . Moreo ve r, for any φ ′ g ∈ S σ ( φ g ), since B σ ( φ g ) is op en, there exists σ φ ′ g > 0 su c h th at B σ φ ′ g ( φ ′ g ) ⊂ B σ ( φ g ) . Consequently , for all φ ∈ B σ φ ′ g ( φ ′ g ), the Po lyak– L o jasiewi cz inequ al- it y h olds around φ ′ g with th e same constant µ P L . W e now prov e th at, for σ small enough , S σ ( φ g ) is a ﬁnite-dimensional C 1 em b ed- ded submanifold. F or any φ ′ g ∈ S σ ( φ g ), t he kernel K φ ′ g is ﬁnite-dimensional by Lemma 3.9. Let R φ ′ g ( v ), v ∈ T φ ′ g M , b e a smooth local retraction satisfying R φ ′ g (0) = φ ′ g and R ′ φ ′ g (0) = I | T φ ′ g M . Then , by the inv erse function t heorem, there exists a suﬃcien tly small neighborho od B σ φ ′ g ( φ ′ g ) su c h that ev ery φ ∈ B σ φ ′ g ( φ ′ g ) can b e uniquely written a s φ = R φ ′ g ( v ) for some small v ∈ T φ ′ g M . Deﬁne t he pulled-back gradient map G ( v ) := ∇ R P E ( R φ ′ g ( v )) . Then G : T φ ′ g M → T R φ ′ g ( v ) M is a C 1 map with G ( 0) = 0. Decomp ose T φ ′ g M = K φ ′ g ⊕ R φ ′ g . W rite v = v 1 + v 2 with v 1 ∈ K φ ′ g , v 2 ∈ R φ ′ g , and deﬁn e G ( v 1 , v 2 ) := G ( v 1 + v 2 ). Then ∂ v 2 G (0 , 0) = ( ∇ R P ) 2 E ( φ ′ g )   R φ ′ g . By Lemma 3.9, t he Poly ak– L o jasiewicz inequ alit y implies that this restriction is co ercive  ( ∇ R P ) 2 E ( φ ′ g ) v 2 , v 2  P φ ′ g ≥ µ P L k v 2 k 2 P φ ′ g , ∀ v 2 ∈ R φ ′ g , hence ∂ v 2 G (0 , 0) is an isomorphism from R φ ′ g onto R φ ′ g . By th e implicit function th eorem, there exists a C 1 map g from a neighb orhoo d of 0 in K φ ′ g into a neighborho o d of 0 in R φ ′ g such th at ∀ v 1 ∈ U σ (0) := { v ∈ K φ ′ g | k v k H 1 < σ } , G ( v 1 + g ( v 1 )) = 0 and g ′ (0) = ( ∂ v 2 G (0 , 0)) − 1 ∂ v 1 G (0 , 0) = 0 , and every solution of G ( v ) = 0 in a neighborho od of 0 is of this form. Thus, we have constructed a local C 1 chart for the set S σ ( φ g ) around any p oint φ ′ g , given by Φ( v 1 ) = R φ ′ g ( v 1 + g ( v 1 )) with Φ ′ (0) = I | K φ ′ g . By the in verse function theorem, this is a lo cal d iﬀeomorphism from a neighborho od of 0 in K φ ′ g onto a neighborho o d of φ ′ g in S σ ( φ g ). It remains to sho w that dim K φ ′ g ≡ d im K φ g , ∀ φ ′ g ∈ S σ ( φ g ) . Using (B.3) in the app en- dix together with (A4) -( i ), w e can d educe that for an y φ ′ g ∈ B σ ( φ g ) with σ > 0 suﬃ cien tly small, the R iemannian H essian ( ∇ R P ) 2 E ( φ ′ g ) is co ercive on the subspace T φ ′ g φ g ( R φ g ) ⊂ T φ ′ g M , i.e.,  ( ∇ R P ) 2 E ( φ ′ g ) T φ ′ g φ g v , T φ ′ g φ g v  P φ ′ g = D ( E ′′ ( φ ′ g ) − λ φ ′ g I ) T φ ′ g φ g v , T φ ′ g φ g v E ≥ C k T φ ′ g φ g v k 2 H 1 ≥ C k T φ ′ g φ g v k 2 P φ ′ g , ∀ v ∈ R φ g . STRUCTURE AND SYMMETR Y OF THE GP GROUND-ST A T E MANIFOLD 15 Consequently , K φ ′ g ∩ T φ ′ g φ g ( R φ g ) = { 0 } . This implies that if J φ ′ g v 1 = J φ ′ g v 2 for v 1 , v 2 ∈ T φ ′ g φ g ( R φ g ), then v 1 − v 2 ∈ K φ ′ g ∩ T φ ′ g φ g ( R φ g ) = { 0 } , and hence v 1 = v 2 . In other words, J φ ′ g is injective on T φ ′ g φ g ( R φ g ). W e consider the restriction of the pro jection J φ ′ g to the tangent space T φ ′ g M , denoted by J φ ′ g | T φ ′ g M : T φ ′ g M → R φ ′ g ⊂ T φ ′ g M . Then J φ ′ g | T φ ′ g M is the P φ ′ g -orthogonal pro jection onto R φ ′ g with kernel K φ ′ g , hen ce J φ ′ g | T φ ′ g M = I − Π K φ ′ g : T φ ′ g M → T φ ′ g M , where Π K φ ′ g denotes the P φ ′ g -orthogonal pro jection onto K φ ′ g . S ince dim K φ ′ g < ∞ , Π K φ ′ g is a ﬁ nite-rank operator, and therefore J φ ′ g | T φ ′ g M is a F redh olm op erator of index zero. Consider the inclusion map ι φ ′ g : T φ ′ g φ g ( R φ g ) ֒ → T φ ′ g M . S ince T φ ′ g φ g ( R φ g ) is a closed sub - space of ﬁnite co dimension in T φ ′ g M , the map ι φ ′ g is a F redholm operator with ind ι φ ′ g = − codim T φ ′ g M T φ ′ g φ g ( R φ g ) . Therefore, the restricted op erator J φ ′ g | T φ ′ g φ g ( R φ g ) = J φ ′ g | T φ ′ g M ◦ ι φ ′ g is a composition of F redholm operators and h ence itself F redh olm. By the index formula for compositions (see [39, Theorem 2.8 in Chapter XVI I, Section 2]), ind J φ ′ g | T φ ′ g φ g ( R φ g ) = ind ι φ ′ g + ind J φ ′ g | T φ ′ g M = − codim T φ ′ g M T φ ′ g φ g ( R φ g ) + 0 . On th e other hand, by deﬁn ition of th e F redh olm index, ind J φ ′ g | T φ ′ g φ g ( R φ g ) = dim ker J φ ′ g | T φ ′ g φ g ( R φ g ) − codim T φ ′ g M J φ ′ g  T φ ′ g φ g ( R φ g )  . Since J φ ′ g is injective on T φ ′ g φ g ( R φ g ), th e kernel v anishes. Hence, ind J φ ′ g | T φ ′ g φ g ( R φ g ) = − codim T φ ′ g M J φ ′ g  T φ ′ g φ g ( R φ g )  . Comparing t he tw o expressions for th e index yields codim T φ ′ g M J φ ′ g  T φ ′ g φ g ( R φ g )  = codim T φ ′ g M T φ ′ g φ g ( R φ g ) . Moreo ve r, since th e map T φ ′ g φ g is an isomorphism, w e ha ve codim T φ ′ g M T φ ′ g φ g ( R φ g ) = codim T φ g M R φ g = dim K φ g . Finally , b ecause R φ ′ g ⊃ J φ ′ g  T φ ′ g φ g ( R φ g )  , it follo ws t hat dim K φ ′ g = codim T φ ′ g M R φ ′ g ≤ co dim T φ ′ g M J φ ′ g  T φ ′ g φ g ( R φ g )  = dim K φ g . Reversing the roles of φ g and φ ′ g yields th e opp osite inequality dim K φ g ≤ d im K φ ′ g . There- fore, dim K φ ′ g = d im K φ g , ∀ φ ′ g ∈ S σ ( φ g ) . This show s that the set S σ ( φ g ) is a C 1 em b edded submanifold, and that th e k ernel of the Riemannian Hessi an coincides w ith the tangen t s pace to this submanifold at each p oint, i.e., the Morse–Bo tt condition holds. 16 ZIXU FEN G, P A TRICK HENNING, AND QINGLIN T ANG T o sum u p, for the Gross–Pi taevskii energy funcational, we establish that the Morse–Bott condition and the Poly ak– Lo jasiewicz inequ alit y are eq uiv alen t on Hilbert manifold M . T o the b est of our knowle dge,[42] is the ﬁrst wor k to prov e this equiv alence in the setting of ﬁnite-dimensional manifolds. T o prov e Theorem 3.3 , we introd uce the op erator G τ ( φ ∗ g ) : R φ ∗ g → R φ ∗ g deﬁned b y G τ ( φ ∗ g ) := J φ ∗ g  I − τ P − 1 φ ∗ g  E ′′ ( φ ∗ g ) − λ φ ∗ g I     R φ ∗ g . The op erator norm characterization of G τ ( φ ∗ g ) is given as follow s. Lemma 3.11. L et E satisfy the Morse–Bott c ondition ar ound φ g . Endow the normal sp ac e R φ ∗ g with the inner pr o duct ( · , · ) P φ ∗ g . Then, ther e exist σ > 0 such that for al l φ ∗ g ∈ S σ ( φ g ) ,the op er ator norm of G τ ( φ ∗ g ) satisﬁes kG τ ( φ ∗ g ) k = max { | 1 − τ µ | , | 1 − τ L | } . Pr o of. The op erator G τ ( φ ∗ g ) is clearly b ounded and linear on th e Hilb ert space R φ ∗ g endow ed with th e inner produ ct ( · , · ) P φ ∗ g . Moreov er, it is self-adjoin t. In deed, for any u, v ∈ R φ ∗ g , we hav e ( G τ ( φ ∗ g ) u, v ) P φ ∗ g =  J φ ∗ g  I − τ P − 1 φ ∗ g ( E ′′ ( φ ∗ g ) − λ φ ∗ g I ) u, v  P φ ∗ g =  I − τ P − 1 φ ∗ g ( E ′′ ( φ ∗ g ) − λ φ ∗ g I )  u, v  P φ ∗ g = hP φ ∗ g u, v i − τ h ( E ′′ ( φ ∗ g ) − λ φ ∗ g I ) u, v i = hP φ ∗ g v , u i − τ h ( E ′′ ( φ ∗ g ) − λ φ ∗ g I ) v , u i = ( G τ ( φ ∗ g ) v , u ) P φ ∗ g . Since G τ ( φ ∗ g ) is self-adjoin t, it follo ws from the p olarization identit y that its op erator norm admits th e v ariational characterizatio n kG τ ( φ ∗ g ) k = sup v ∈ R φ ∗ g \{ 0 } | ( G τ ( φ ∗ g ) v , v ) P φ ∗ g | ( v , v ) P φ ∗ g . Using t he deﬁnition of G τ ( φ ∗ g ) and the fact that J φ ∗ g acts as the iden tity on R φ ∗ g , we obtain ( G τ ( φ ∗ g ) v , v ) P φ ∗ g = ( v , v ) P φ ∗ g − τ h ( E ′′ ( φ ∗ g ) − λ φ ∗ g I ) v , v i = ( v , v ) P φ ∗ g  1 − τ R ( v )  , where R ( v ) = h ( E ′′ ( φ ∗ g ) − λ φ ∗ g I ) v , v i hP φ ∗ g v , v i . Hence, kG τ ( φ ∗ g ) k = sup v ∈ R φ ∗ g \{ 0 } | 1 − τ R ( v ) | . Then, by Prop erty 2.4-( i ), kG τ ( φ ∗ g ) k = max {| 1 − τ µ | , | 1 − τ L |} , as claimed. Lemma 3.12. If the nonline arity f i s r e al-analytic and φ s ∈ M is a critic al p oi nt of E (i.e., H φ s φ s = λ s φ s , λ s = h H φ s I φ s , φ s ), then the ener gy functional E satisﬁes a r eﬁne d Lojasiewicz–Simon R iemannian gr adient ine quality in a neighb orho o d of φ s , i.e., ther e exist c onstants ν ∈ (0 , 1 2 ] and σ > 0 , wi th ν indep endent of P φ , such that for any φ ∈ B σ ( φ s ) , we have | E ( φ ) − E ( φ s ) | 1 − ν ≤ C    ∇ R P E ( φ )    P φ . Pr o of. W e ﬁrst establish th e L o jasiewicz–Simon gradient ineq ualit y for the R iemannian gradien t associated with th e H 1 0 -metric. Sin ce th e n onlinearit y f is real-analytic, the energy functional E is real-analytic on H 1 0 ( D ) , and the constraint manifold M is also real-analytic. Therefore, w e ma y app ly [43, Corolla ry 5.2] with V = H = H 1 0 ( D ) , equipp ed with the H 1 0 inner pro duct, provided that the Hessian operator ∇ 2 H 1 0 E ( φ ) : H 1 0 ( D ) → H 1 0 ( D ) is F redholm of index zero. A direct computation yields ∇ 2 H 1 0 E ( φ ) =  − ∆  − 1 E ′′ ( φ ) =  − ∆  − 1  − 1 2 ∆ + V − Ω L z + f ( ρ φ ) + f ′ ( ρ φ )  | φ | 2 + φ 2 ·   = 1 2 I +  − ∆  − 1  V − Ω L z + f ( ρ φ ) + f ′ ( ρ φ )  | φ | 2 + φ 2 ·   =: 1 2 I + A φ . STRUCTURE AND SYMMETR Y OF THE GP GROUND-ST A T E MANIFOLD 17 The op erator A φ is compact due to k A φ v k 2 H 1 0 = D V − Ω L z + f ( ρ φ ) + f ′ ( ρ φ )  | φ | 2 + φ 2 ·   v , A φ v E ≤ C φ ( k v k L 2 + k v k L 6 / (4 − θ ) ) k A φ v k H 1 0 , ∀ v ∈ H 1 0 ( D ) . Consequently , ∇ 2 H 1 0 E ( φ ) is a compact p ertu rbation of the (scaled) id entit y op erator and hence F redholm of index zero. By [43, Corollary 5.2], there exist constants ν ∈ (0 , 1 2 ] and σ > 0 such that for all φ ∈ B σ ( φ s ), | E ( φ ) − E ( φ s ) | 1 − ν ≤ C   ∇ R H 1 0 E ( φ )   H 1 0 . (3.3) Next, w e sho w that the Riemannian gradient with resp ect to an a rbitrary preconditioner P φ is equiva lent to the one induced by the scaled H 1 0 -metric (cf. (A4) -( i )). U sing the relation b etw een the tw o m etrics, we compute   ∇ R P φ E ( φ )   2 P φ =  H φ φ − λ φ φ, ∇ R P φ E ( φ )  =  ( − ∆) ∇ R H 1 0 E ( φ ) , ∇ R P φ E ( φ )  ≤ C φ   ∇ R H 1 0 E ( φ )   H 1 0   ∇ R P φ E ( φ )   P φ , whic h implies   ∇ R P φ E ( φ )   P φ ≤ C φ   ∇ R H 1 0 E ( φ )   H 1 0 . Conv ersely , this follow s from the equiv- alence b etw een t he preconditioned metric P φ and the H 1 0 -metric. Hence, the tw o gradien ts are norm-equ iv alen t:   ∇ R H 1 0 E ( φ )   H 1 0 ≃   ∇ R P φ E ( φ )   P φ , with constants indep end ent of φ in a neighborho od of φ s . Com bining t his equ iv alence with (3.3), w e obt ain | E ( φ ) − E ( φ s ) | 1 − ν ≤ C   ∇ R P φ E ( φ )   P φ . Moreo ve r, since the abov e eq uiv alence holds for an y p reconditioner P φ , and the ex p onent ν arises solely from th e analytic structure of E and M (v ia the F redholm prop erty of the Hessian), it follo ws that ν is indep end ent of the choice of preconditioner P φ . With this, w e are ready to prove th e theorems. 3.3. Proof of main results. Pr o of of T heorem 3.2 . W e ﬁrst show that under the Morse–Bott cond ition, each sym- metry orbit S φ g ⊂ S g is separable. That is, there exists σ > 0 such th at for an y φ ∈ B σ ( S φ g ), E ( φ ) = E S g , implies φ ∈ S φ g , or equiv alen tly B σ ( S φ g ) ∩ S g = S φ g . By applying th e lo cal pro jection construction from [23, Lemma 4.1], there exists σ > 0 such th at for every φ ∈ B σ ( φ g ) ⊂ M , there ex ists φ ′ g ∈ S σ ( φ g ) satisf ying ( φ − φ ′ g , i φ ′ g ) L 2 = 0 , ( φ − φ ′ g , i L z φ ′ g ) L 2 = 0 , and k φ − φ ′ g k H 1 ≤ C k φ − φ g k H 1 . Moreo ve r, we have t he decomp osition φ − φ ′ g = Pro j L 2 φ ′ g ( φ − φ ′ g ) − 1 2 k φ − φ ′ g k 2 L 2 φ ′ g . Now , choose σ > 0 suﬃciently small. F or any φ ∈ B σ ( S φ g ) and all suﬃciently small ε > 0, a T aylor expansion of E at φ ′ g yields E ( φ ) − E ( φ g ) = E ( φ ) − E ( φ ′ g ) = h E ′ ( φ ′ g ) , φ − φ ′ g i + 1 2 h E ′′ ( φ ′ g )( φ − φ ′ g ) , φ − φ ′ g i + o  k φ − φ ′ g k 2 H 1  = 1 2  E ′′ ( φ ′ g ) − λ φ ′ g I  ( φ − φ ′ g ) , φ − φ ′ g  + o  k φ − φ ′ g k 2 H 1  ≥ µ − ε 2 ( φ − φ ′ g , φ − φ ′ g ) E ′′ ( φ ′ g ) , 18 ZIXU FEN G, P A TRICK HENNING, AND QINGLIN T ANG where E ′′ ( φ ′ g ) is co ercive by Prope rt y 2.2 together with the fact that λ φ ′ g > 0. Conse- quently , if E ( φ ) = E ( φ g ), the ab ov e inequality forces φ = φ ′ g ∈ S φ g . This prov es that each orbit S φ g is separable in S g . It remains to sh o w that S g is compact in H 1 0 ( D ) . Clearl y , S g is b ound ed in H 1 0 ( D ) . T o prove compactness, it suﬃces to verify sequential compactn ess. Let { v n } n ∈ N ⊂ S g b e an arbitrary sequence. By the b oun dedness of S g and th e Rellic h–Kondracho v compact em b edding H 1 0 ( D ) ⊂⊂ L p ( D ) for 1 ≤ p < 6, there exist a subsequen ce ( still denoted b y { v n } ) and some v ∗ ∈ H 1 0 ( D ) such that v n ⇀ v ∗ w eakly in H 1 0 ( D ) , v n → v ∗ strongly in L p ( D ) for 1 ≤ p < 6 . Since the nonlinearity F ( ρ v ) satisﬁes suitable gro wth conditions (A3) (i.e., | F ( | z | 2 ) | ≤ C (1 + | z | θ +3 ) with θ < 3), the sequence { F ( ρ v n ) } n ∈ N is uniformly integra ble in L 1 , and hence w e obtain lim n →∞ Z D F ( ρ v n ) d x = Z D F ( ρ v ∗ ) d x . T ogether with the fact that each v n ∈ S g , it follo ws that k v n k H 0 → k v ∗ k H 0 with H 0 = H φ − f ( ρ φ ) , whic h implies strong conv ergence v n → v ∗ in H 1 0 ( D ) . Moreo ver, v ∗ ∈ M and E ( v ∗ ) = E S g . Hence, S g is sequentially compact and th erefore compact in H 1 0 ( D ) . By th e separable prop erty of S φ g , th ere ex ists σ φ g > 0 such th at B σ φ g ( S φ g ) ∩ S g = S φ g . The collection  B σ φ g ( S φ g )  φ g ∈S g forms an op en co ver of S g . Since S g is compact, there exist ﬁ nitely many orbits S φ g, 1 , . . . , S φ g,N such th at S g = N G ℓ =1 S φ g,ℓ , whic h establishes a w ell-deﬁned classi ﬁcation of the global minimizers. Pr o of of T heorem 3.3 . By Le mma 3. 7 , Le m ma 3.8 , and [23, Theorem 4.2], w e hav e that for arbitrary ε > 0, there exists σ > 0, such that the s equence { φ n } n ∈ N generated by the P-RG con verg es linearly for any initial p oint φ 0 ∈ B σ ( φ g ) and an y step s ize τ ∈  0 , 2 / ( L + ε )  . Moreo ve r, it holds φ ∗ g := lim n →∞ φ n ∈ S φ g . The optimal local Q - linear conv ergence rate is established b elow. F or all u ∈ T φ ∗ g M and v ∈ K φ ∗ g , th e follo wing orthogonalit y relations h old:  Pro j P φ ∗ g φ ∗ g P − 1 φ ∗ g ( E ′′ ( φ ∗ g ) − λ φ ∗ g I ) u, v  P φ ∗ g =  Pro j P φ ∗ g φ ∗ g P − 1 φ ∗ g ( E ′′ ( φ ∗ g ) − λ φ ∗ g I ) v , u  P φ ∗ g = 0 . A lo cal linearization of the P-RG iteration at φ ∗ g yields φ n +1 − φ ∗ g = φ n − φ ∗ g − τ Pro j P φ ∗ g φ ∗ g P − 1 φ ∗ g ( E ′′ ( φ ∗ g ) − λ φ ∗ g I )( φ n − φ ∗ g ) + o ( k φ n − φ ∗ g k H 1 ) = ( I − J φ ∗ g )( φ n − φ ∗ g ) + G τ ( φ ∗ g ) J φ ∗ g ( φ n − φ ∗ g ) + o ( k φ n − φ ∗ g k H 1 ) . F rom th is expansion, the follo wing decoupled asymptotic relations follo w: ( ( I − J φ ∗ g )( φ n +1 − φ n ) = o ( k φ n − φ ∗ g k H 1 ) J φ ∗ g ( φ n +1 − φ ∗ g ) = G τ ( φ ∗ g ) J φ ∗ g ( φ n − φ ∗ g ) + o ( k φ n − φ ∗ g k H 1 ) . T elescopic summation of the ﬁrst relation gives ( I − J φ ∗ g )( φ n − φ ∗ g ) = ∞ X k = n ( I − J φ ∗ g )( φ k − φ k +1 ) , (3.4) STRUCTURE AND SYMMETR Y OF THE GP GROUND-ST A T E MANIFOLD 19 with k ( I − J φ ∗ g )( φ k − φ k +1 ) k P φ ∗ g → 0 as k φ k − φ ∗ g k H 1 → 0. In particular, for every σ > 0 there exists δ σ → 0 as σ → 0 such th at k ( I − J φ ∗ g )( φ k − φ k +1 ) k P φ ∗ g ≤ δ σ k φ k − φ ∗ g k P φ ∗ g whenever k φ k − φ ∗ g k H 1 ≤ σ. Hence, from the telescoping identit y (3.4) and for n large enough so that k φ k − φ ∗ g k H 1 ≤ σ for all k ≥ n , it follo ws that   ( I − J φ ∗ g )( φ n − φ ∗ g )   P φ ∗ g ≤ δ σ ∞ X k = n k φ k − φ ∗ g k P φ ∗ g . (3.5) Give n th e linear conv ergence of { φ n } n ∈ N , th ere exists ρ ∈ (0 , 1) suc h that k φ k − φ ∗ g k P φ ∗ g ≤ C ρ k − n k φ n − φ ∗ g k P φ ∗ g for all k ≥ n. Consequently ,   ( I − J φ ∗ g )( φ n − φ ∗ g )   P φ ∗ g (3.5) ≤ δ σ ∞ X k = n k φ k − φ ∗ g k P φ ∗ g ≤ C δ σ k φ n − φ ∗ g k P φ ∗ g , where δ σ → 0 + as σ → 0 + . It follo ws that ( I − J φ ∗ g )( φ n − φ ∗ g ) = o ( k φ n − φ ∗ g k H 1 ) , and hen ce φ n − φ ∗ g = J φ ∗ g ( φ n − φ ∗ g ) + o ( k φ n − φ ∗ g k H 1 ) J φ ∗ g ( φ n +1 − φ ∗ g ) = G τ ( φ ∗ g ) J φ ∗ g ( φ n − φ ∗ g ) + o  kJ φ ∗ g ( φ n − φ ∗ g ) k H 1  Therefore, the asymptotic conv ergence beh a vior of φ n − φ ∗ g is en tirely determined by its pro jected component J φ ∗ g ( φ n − φ ∗ g ). By Lem ma 3.11 , the op erator norm of G τ ( φ ∗ g ) restricted to R φ ∗ g is equal to max {| 1 − τ µ | , | 1 − τ L |} . F or all φ 0 ∈ B σ ( φ g ) and τ ∈ (0 , 2 / ( L + ε )) , the local Q -linear estimate holds   J φ ∗ g ( φ n − φ ∗ g )   P φ ∗ g ≤   J φ ∗ g ( φ n − 1 − φ ∗ g )   P φ ∗ g  max {| 1 − τ µ | , | 1 − τ L |} + ε  , ∀ n ≥ 1 . Com bined with the d ecomposition φ n − φ ∗ g = J φ ∗ g ( φ n − φ ∗ g ) + o ( k φ n − φ ∗ g k H 1 ), this implies that k φ n − φ ∗ g k P φ ∗ g ≤ k φ n − 1 − φ ∗ g k P φ ∗ g (max {| 1 − τ µ | , | 1 − τ L |} + ε ) , ∀ n ≥ 1 , for σ suﬃciently small (so the kernel comp onent is negligible uniformly), p ossibly enlarging ε . In particular, t he optimal contraction factor is attained when τ = 2 / ( L + µ ). In this case, max {| 1 − τ µ | , | 1 − τ L |} = L − µ L + µ , and the optimal local Q -linear converge nce rate is giv en by k φ n − φ ∗ g k P φ ∗ g ≤ k φ n − 1 − φ ∗ g k P φ ∗ g  L − µ L + µ + ε  , ∀ n ≥ 1 . 20 ZIXU FEN G, P A TRICK HENNING, AND QINGLIN T ANG Pr o of of T heorem 3.4 . By Lemma 3. 7 , Lemma 3.8 , and [23, Lemma 4.3], t he suf- ﬁciency is immediate. W e now prov e the necessit y . By T a ylor expansion, we hav e E ( φ 1 ) = E ( φ 0 ) − τ k d 0 k 2 P φ 0 + τ 2 2  ( E ′′ ( φ 0 ) − λ φ 0 I ) d 0 , d 0  + o ( τ 2 k d 0 k 2 H 1 ) . Note th at since d 0 = ∇ R P E ( φ 0 ), t here exists a constant C d > 0 such that k d 0 k H 1 ≤ C d k φ 0 − φ g k H 1 . In particular, for φ 0 ∈ B σ ( φ g ), we hav e k d 0 k H 1 ≤ C σ . W e obtain E ( φ 0 ) = E ( φ 1 ) + τ k d 0 k 2 P φ 0 − τ 2 2  ( E ′′ ( φ 0 ) − λ φ 0 I ) d 0 , d 0  + o ( k d 0 k 2 H 1 ) . By the contin uit y of E ′′ ( φ ) − λ φ I , P φ , and Pro j P φ φ with respect to φ , together with Lemma 3.7 , we get for all d ∈ T φ M (with small k d k H 1 ) that    ( E ′′ ( φ ) − λ φ I ) d, d    ≤ L k d k 2 P φ + o ( k d k 2 H 1 ) . Consequently , for ever y suﬃciently small ε > 0, there exists suﬃciently small σ > 0 such that for all φ ∈ B σ ( φ g ), E ( φ 0 ) − E ( φ 1 ) ≤  τ + τ 2 2 ( L + ε )  k d 0 k 2 P φ 0 . On th e other hand, by the assumed Q - linear conv ergence of the energy sequence, w e h a ve E ( φ 0 ) − E ( φ 1 ) =  E ( φ 0 ) − E ( φ g )  −  E ( φ 1 ) − E ( φ g )  ≥  1 − ρ − ε  E ( φ 0 ) − E ( φ g )  . Com bining the tw o estimates yields E ( φ 0 ) − E ( φ g ) ≤ τ + τ 2 2 ( L + ε ) 1 − ρ − ε k d 0 k 2 P φ 0 , whic h establishes the Pol yak– L o jasiewicz in equalit y . This, together with Lemma 3. 10 , completes the pro of of necessity . Pr o of of T heorem 3.5 . The suﬃciency follo ws immediately from Theorem 3.2 and Theorem 3.3 . W e now p ro ve th e n ecessit y . Assume that th e well-deﬁned classiﬁcation h olds and that the iterates generated by the P-RG conv erge linearly to the gro und state manifo ld S g . Then, for an y ground state φ g ∈ S g , the sequence converge s linearly in a neighborho o d of φ g . As in th e b eginning of the pro of of Theorem 3.3 , one obtains φ 0 − φ ∗ g = J φ ∗ g ( φ 0 − φ ∗ g ) + o ( k φ 0 − φ ∗ g k H 1 ) as σ → 0 + , here φ 0 ∈ B σ ( φ g ) and th e limit p oint φ ∗ g ∈ S φ g satisﬁes k φ 0 − φ ′ g k H 1 ≤ C σ . W e remark that this expansion is identical to the one in (3.2). Let I β α denote the linear group action correspondin g to phase shifts and rotations. By the inv ariance of the en ergy fun ctional E under th e action of the linear group, for all φ ∈ S φ g , we hav e the eq uiv ariance of the t angen t bundle and the second v ariation I β α  T φ M  = T I β α φ M and  ( E ′′ ( I β α φ ) − λ I β α φ I ) I β α v , I β α v  =  ( E ′′ ( φ ) − λ φ I ) v , v  . Moreo ve r, by (A4) and the fact that S φ g is b ound ed in H 1 , the n orms k · k P φ and k · k H 1 are uniformly equiv alent for φ ∈ S φ g (and hence also in a suﬃcien tly small neighborho od of the orbit). Therefo re, coercivity of the quadratic form in H 1 transfers to a uniform co ercivity b ound in th e P φ -norm. Consequently , the R iemannian Hessian of E is uniformly co ercive on t he subspace R φ o ver th e entire orbit S φ g . By rep eating the T a ylor-expansion argument used in Lemma 3.8 , together with th e assumed linear conv ergence of { φ n } n ∈ N , we obtain the Poly ak– L o jasiewicz inequality in a neighborho od of φ g . Sin ce the Po lyak– L o jasiewi cz inequality is equiv alen t to the Morse–Bott condition, the n ecessit y is established. STRUCTURE AND SYMMETR Y OF THE GP GROUND-ST A T E MANIFOLD 21 Pr o of of T heorem 3.6 . By [2 3, Lemma 4.3] , which pro vides the upp er bound E ′′ ( φ g ) − λ φ g I in the precond itioner-norm and do es not require the Morse–Bott assumption for this estimate, together with Lemma 3.7 , w e obtain for an y critical p oint φ ′ s and ev ery suﬃciently small ε > 0, there exists σ > 0 suc h that for all φ ∈ B σ ( φ ′ s ), the lo cal energy dissipation satisﬁes (3.6) E ( φ n +1 ) − E ( φ n ) ≤ − C τ    ∇ R P E ( φ n )    2 P φ n , where C τ = τ − τ 2 2 ( L + ε ) > 0 for all τ ∈  0 , 2 / ( L + ε )  . Moreo ve r, by Proposition 2.4 -( iv ), w e h a ve k φ n +1 − φ n k P φ n =    ∇ R P E ( φ n )    P φ n + o     ∇ R P E ( φ n )    P φ n  . Com bining this with the lo cal energy d issipation estimate (3.6), we obtain E ( φ n ) − E ( φ n +1 ) ≥ C τ    ∇ R P E ( φ n )    P φ n k φ n +1 − φ n k P φ n . Hence, all the conditions of [49, Theorem 2.1] are satisﬁed, and in t he absence of th e Morse– Bott condition on th e energy fun ctional E , the P-RG admits th e lo cal sub linear conv ergence rate k φ n − φ s k H 1 ≤ C n − ν 1 − 2 ν , ν ∈ (0 , 1 / 2) , ∀ n ≥ 0 , for some φ s ∈ Crit ( E ). 4. Practical preconditioning unde r the Morse–Bott condition. In Chapter 3, w e established a sharp corresp ondence b etw een the Morse–Bott condition for the en ergy functional E and the lo cal conv ergence b ehavior of th e P-RG. I n particular, [23] prop osed an op timal p reconditioner, and similar constructions ha ve also app eared in [3]. I t takes t he form P opt = E ′′ ( φ ) −  e λ φ − σ 0  I , where e λ φ = hH φ φ, φ i and σ 0 > 0 is a tun able regularization parameter. By decreasing σ 0 , one can achieve a lo cal conve rgence rate for P-RG t hat surp asses any prescrib ed linear rate. How ever, th is theoretical adv antag e comes at a signiﬁcan t computational cost in practice. A stronger preconditioning eﬀect often renders the linear system P opt x = b increasingly ill- conditioned: although the sp ectral gap is artiﬁcially widened, the smalles t eigen v alues ma y approac h zero, causing iterative solvers such as CG or MINRES to stagnate or converg e extremely slowl y . I n other words, while P opt is ideal in the continuous setting, its d iscrete realization ma y incur prohibitive inner-solve costs th at oﬀset the gains from faster outer optimization iterations. This tension highligh ts the essen tial insigh t in preconditioner design: the intrinsic trade- oﬀ b etw een accelera tion capability and computational eﬃciency . F or large-scale symmetric linear systems arising from the discretization o f inﬁ nite-dimensional mathematical or ph ysical problems, a wide range of classical techniques, such as incomplete Cholesky factorization, al- gebraic multig rid, an d subspace pro jection metho ds, h a ve been developed to strike a balance b etw een approximation accuracy and algorithmic cost. Notably , the problem und er consid- eration originates from the high-accuracy discretizatio n of an inﬁnite-d imensional nonlinear mathematical or p hysica l mo del. Nevertheless, as our theoretical analysis demonstrates, eﬃcien t preconditioning strategies d evel op ed for the associated linearized systems remain highly b eneﬁcial for solving th e original n onlinear problem. In particular, constructing a sparse, symmetric positive d eﬁnite op erator as a low-accuracy approximation of the high- accuracy discrete system [24, 26, 27, 48], though sacriﬁcing some accuracy relative to the optimal preconditioner P opt , eﬀectively captures the d ominan t sp ectral characteristics of th e underlying continuous op erator. Consequently , such approximations signiﬁcantly accelerate conv ergence within nonlinear iterative framew orks by enhancing the eﬃciency of inn er linear solv es. Sp eciﬁcally , we seek a sparse lo w er-triangular matrix L such that P opt ≈ LL ⊤ . The preconditioning step then red uces to solving tw o triangular systems: L y = b, L ⊤ x = y , 22 ZIXU FEN G, P A TRICK HENNING, AND QINGLIN T ANG whic h can b e carried out eﬃcien tly via forw ard and backw ard substitution. Crucially , th e computational cost of this pro cess dep ends only on t he sparsity p attern of L , and is entirely indep endent of the condition number of the original coeﬃcient matrix. Thus, even if P opt is highly ill-conditioned, as long as its sparse surrogate LL ⊤ captures the dominant sp ectral features, it can dramatically accelerate P-RG at minimal linear-solv e ex p ense. This is precisely the “art of preconditioning”: not t o seek an o verly ac curate approxima- tion of an ideal op erator at all costs, but to construct a s urrogate that, near local minimizers, induces eﬀective clustering of th e eigenv alues of the p reconditioned Hessian while remaining inexp ensive to apply . 5. Numerical exp erime nt . In this section, to numerically val idate the theoretical conv ergence results and assess the practical guidance pro vided by t he optimal preconditioning theory , we consider the Riemann ian optimization problem (2.1) on the disk D :=  ( x, y ) = ( r co s Θ , r sin Θ) | r ∈ [0 , 12] , Θ ∈ [0 , 2 π ]  . The trapp ing p otential, nonlinear in teraction, and angular velocity are set as V ( x ) = | x | 2 2 , f ( s ) = η s, η = 500 , and Ω = 0 . 9 . Under this conﬁguration, it has been numerically veri ﬁed in [23] that the energy fun ctional E satisﬁes th e Morse–Bott condition at its ground state. T o numerically solve problem (2.1), we discretize all related deriv ativ es in the P- RG metho d using an eighth-order centra l ﬁnite diﬀerence scheme in the angular v ariable Θ and a second-order cen tral ﬁnite diﬀerence sc heme in the radial vari able r on an equ ally- spacing grids e D =:  ( r i +1 / 2 , Θ j ) | i = 0 , · · · , N r − 1 , j = 0 , · · · , N Θ − 1  . Here, r i +1 / 2 = ( i + 1 / 2) h r , Θ j = j h Θ with h r = 12 / 2 8 and h Θ = 2 π / 2 10 the mes h siz es in r - and Θ-direction. Homogeneous Dirichlet b ound ary cond itions are imposed. The P-RG is stopped when m eet the criterion r n := kH φ n φ n − hH φ n φ n , φ n i φ n k ∞ ≤ 10 − 12 , and the resulted iterate φ n is regarded as th e ground state φ g . In addition, for the study of the asymptotic Q -linear con vergence rate, we deﬁne tw o error ratios Q E n and Q φ n as: Q E n := s E ( φ n +1 ) − E ( φ g ) E ( φ n ) − E ( φ g ) and Q φ n := k φ n +1 − φ g k P φ g k φ n − φ g k P φ g . W e ﬁrst compute φ g via the P-RG metho d in tw o stages, employing diﬀerent precon- ditioners. In the ﬁrst stage, we u se the preconditioner P φ = LL ⊤ ≈ E ′′ ( φ ), where φ is ﬁxed th roughout each block of 100 iterations. Sp eciﬁcally , the preconditioner is held con- stant for 100 consecutive iterations and up dated only at th e b eginning of eac h new block using the current iterate φ n . This pro cedure is rep eated for 10 4 iterations with time step τ = 1. In the second stage, we switc h to an appro ximately lo cally optimal precond itioner P φ = L L ⊤ ≈ P opt with σ 0 = 10 − 1 , obtained via an incomplete Cholesky factorization with Approximate Minim um Degree (AMD) reordering and a drop t olerance of 10 − 5 . This pre- conditioner is also up dated every 100 iterations. The time step is kept constant at τ = 1, and the iterations con tinue until th e prescrib ed termination criteria are met. Fig. 1 s hows the contour plots of the density | φ g | 2 . The a symptotic Q -linear conv ergence rates measured by Q E n and Q φ n are sho wn in Fig. 2-( i ) and 2- ( ii ), resp ectively . F urther- more, w e compute the parameters ( µ, L ) th at characterize t he lo cal conv ergence rate as p re- dicted by Theorem 3.3. These are obtained under th ree settings for t he preconditioner: ( i ) P φ = LL ⊤ ≈ P opt with AMD reordering and a tolerance of 10 − 5 ; and ( ii ) P no φ = LL ⊤ ≈ P opt without AMD reordering, also with tolerance 10 − 5 . All numerical results indicate that: ( i ) AMD reordering signiﬁcantly improv es the eﬀectiveness of th e incomplete Cholesky factor- ization as a preconditioner in reducing the condition num ber (cf. T ab. 1), demonstrating its necessit y; ( ii ) th e observed con vergence rate of the P-RG closely matches the th eoretical prediction (cf. Fig. 2). STRUCTURE AND SYMMETR Y OF THE GP GROUND-ST A T E MANIFOLD 23 Fig. 1 . Contour plots of the density of the gr ound state | φ g ( x ) | 2 . T able 1 The values of µ , L , κ = L/µ , and the or etica l c onver genc e r ate ρ τ = max {| 1 − τ µ | , | 1 − τ L | } with τ = 1 w.r.t diﬀe r ent pr ec onditioners. µ L κ ρ 1 P φ g 6 . 41 × 10 − 4 1 . 25622 795 1960 . 58 0 . 999359 P no φ g 4 . 49 × 10 − 5 1 . 64263 642 36584 . 3 3 0 . 999955 6. Conclusion. In t his pap er, our analysis oﬀers partial bu t sharp insigh ts into sev eral open chal lenges recently highligh ted in the literature regarding the interpla y of symmetry , degeneracy , and conv ergence in Gross–Pita evskii energy functional minimization. F or the rotating Gross–Pitaevskii en ergy functional, a canonical model for Bose–Einstein conden- sates, sy mmetry-induced degeneracy do es not lead to arbitrarily slow conv ergence of the preconditioned Riemannian gradient meth od ( P-RG). On the contrary , w e prove that P-RG alw a ys con verges locally t o a well-deﬁned minimizer at aprecise and q uantiﬁable rate, de- termined entirel y by the geometric structure of the m inimizer set. Sp eciﬁcally , when the Morse–Bott condition holds, the set of non- isolated global minimizers is p artitioned in to ﬁ- nitely many equiv alence classes, eac h corresp onding to an orbit of t he underlyin g contin uous symmetries, such as phase shifts and spatial rotations. In this case, the P- RG exhibits local linear conv ergence, and the con verge nce rate is gov erned almost exclusive ly by the choice of preconditioner, allo wing for systematic acceleration via op timal design. I n contrast, when the Morse–Bott condition fails, we are only able t o establish a local sub linear conv ergence rate; moreov er, the preconditioner no longer app ears to control the asymptotic con vergence sp eed in our analysis. Th us, for th e Gross–Pitaevskii model, the Morse–Bott condition provides a sharp th reshold that clariﬁes—under the cu rrent framework—the relationship among sy mmetry , geometric structure, an d algorithmic p erformance. W e b eliev e th at the analytical framew ork and con vergence characterizatio ns develo p ed here ex tend naturally to multi-component Gross–Pitaevskii systems, where similar symmetry struct ures and non- isolated minimizers arise. Imp ortantly , th ese theoretical ﬁndings provide concrete guidance for designing eﬃcient n umerical algorithms in practice: preconditioning strategies developed for symmetric linear systems, such as t he incomplete Cholesky factorization with AMD re- ordering employ ed in our exp erimen ts, are n ot only compatible with, bu t indeed essential to, ac hieving fast converge nce in Gross–Pitaevskii optimization. Moreo v er, data-d rive n n eural netw ork metho ds show promise as a nov el class of preconditioning techniques and represen t a comp elling d irection for fut ure researc h. Coupling such preconditioners with Riemann- ian conjugate gradient metho ds, whic h exhibit w eake r dep end ence on the condition number compared to stand ard gradient descent, h as the p otential to further enhance comput ational eﬃciency . 24 ZIXU FEN G, P A TRICK HENNING, AND QINGLIN T ANG ( i ) 1 12000 0.998 1 5500 6500 7500 8500 0.99934 0.99937 ( ii ) 1 24000 0.998 1 1 1.1 1.2 1.3 10 4 0.99934 0.99937 Fig. 2 . (i) Err or r atio Q E n = p ( E ( φ n +1 ) − E ( φ g )) / ( E ( φ n ) − E ( φ g )) versus n (r e d solid line). The black dashe d line r epr esents the the or etic al r ate ρ 1 pr e dicte d in Theorem 3 .3 . (ii) Err or r atio Q φ n = k φ n +1 − φ g k P φ g / k φ n − φ g k P φ g versus n (r ed solid line). The black dashe d line r e pr esent s the the or et i c al r ate ρ 1 pr e dicte d in Theorem 3.3 . REFERENCES [1] Y. Ai, P . Henning, M . Y adav, and S. Y uan, Ri emannian conjugate Sobol ev gradients and their application to compute ground stat es of BECs, J. Comput. Appl. Math., 473 (2026), article 116866. [2] R. Altmann, P . Henning, and D. Peterseim, The J -method for the Gross-Pi taevskii eigenv alue problem, Numer. Math., 148 (2021), pp. 575–610. [3] R. Altmann, M. Herm ann, D . Pet erseim, and T. Styk el, Riemannian optimisation metho ds for ground states of multicomponen t Bose-Einstein condensates, [4] M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Co rnell, Observ ation of Bose-Einstein condensation in a di lute atomic v apor, Sci., 269 (1995), pp. 198–201. [5] X. An toine and R. Duboscq, Robust and eﬃcient preconditioned Krylov spectral s olv ers f or computing the ground states of fast rotating and strongly i n teracting Bose-Einstein con- densates, J. Comput. Phys., 258 (2014), pp. 509–523. [6] X. Anto ine, A. Levitt, and Q. T ang, Eﬃcient sp ectral computation of the stationary states of rotating Bose-Einstein condensates b y the preconditioned nonlinear conjugate gradien t method, J. Comput. Phys., 343 (2017), pp. 92–109. [7] W. Bao and Q. Du, Computing the gr ound s tate solution of Bose-Einstein condensate s b y a normalized gr adien t ﬂow, SIAM J. Sci. Comput., 25 (2004), pp. 1674–1697. [8] W. Bao, I. Chern, and F. Lim, Eﬃcient and sp ectrally accurate n umerical methods for com- puting ground and ﬁrst excited states in Bose-Einstein condensates, J. Comput. Phys., 219 (2006), pp. 836–854. [9] W. Bao and Y. Cai, Mathemat ical theory and n umerical methods f or Bose-Einstein condensa- tion, Kinet. Relat. Mo dels, 6 (2013), pp. 1–135. [10] C. F. Barenghi, L. Sk rb ek, and K. R. Sreeniv asan, Introduction to quan tum turbulence , PNAS, 111 (2014), pp. 4647–4652. [11] R. Bott, Nondegenerate critical m anifolds, A nn. of Math., 60 (1954), pp. 248–261. [12] I. Carusotto and C. Ci uti, Quan tum ﬂuids of li gh t, Rev. Mo d. Phys., 85 (2013), pp. 299–366. [13] T. Cazenav e, Semilinear Schr¨ odinger Equations, Courant Lect. Notes Math., 10, Amer. Math. Soc., Providence, R.I., 2003. [14] H. Chen, G. Dong, W. Liu, and Z. Xie, Second-order ﬂows for computing the ground states of rotating Bose-Einstein condensates, J. Comput. Phys., 475 (2023), article 111872. [15] Z. Chen, J. Lu, Y. Lu, and X. Zhang, On the con v ergence of Sobolev gradient ﬂo w for the Gross-Pitaevskii eigenv alue problem, SIAM J. Numer. A nal., 62 (2024), pp. 667–691. [16] M. Chiofalo, S. Succi, and M. T osi, Ground s tate of trapped interact ing Bose-Einstein conden- sates by an explicit imaginary-time al gorithm, Phys. Rev. E, 62 (2000), pp. 7438–7444. [17] I. Danaila and P . Kazemi, A new Sob olev gradien t method for direct minimization of the Gross-Pitaevskii energy with rotation, SIAM J. Sci. Comput., 32 (2010), pp. 2447–2467. [18] I. Danaila and B. Protas, Computation of ground states of the Gross-Pi taevskii f unctional via Riemannian optimization, SIAM J. Sci. Comput., 39 (2017), pp. B1102–B1129. [19] K. B. Da vis, M. Mew es, and M. R. Andrews, Bose-Einstein condensation in a gas of sodium atoms, Phys. Rev. Lett., 75 (1995), pp. 3969–3973. [20] C. M. Dion and E. Canc ´ es, Ground state of the time-independent Gross- Pitaevskii equation, STRUCTURE AND SYMMETR Y OF THE GP GROUND-ST A T E MANIFOLD 25 Comput. Phys. Commun., 177 (2007), pp. 787–798. [21] L. D ong and Y. V. K artasho v, Rotating multidimensional quan tum droplets, Phys. Rev. Lett., 126 (2021), article 244101. [22] P . M. F eehan and M. Maridakis, Lo jasiewicz-Simon gradien t inequalities for analytic and Morse-Bott f unctions on Banac h spaces, J. Reine Angew. Math., 765 (2020), pp. 35–67 [23] Z. F eng and Q. T ang, O n pr econditioned Riemannian gradient methods for m inimizing the Gross-Pitaevskii energy functional: algorithms, global conv ergence and optimal local con- v ergence rate, ar Xiv:2510.13516. [24] Z. F eng, W. Qu, and Q. T ang, A new preconditioned L 2 -Riemannian con jugate gradien t method for computing the ground states of Bose-Ei nstein condensates with rotation, Preprint. [25] J. J. Garc ´ ıa. Rip oll and V. M. P´ erez-Garc ´ ıa, Optimi zing Sc hr¨ odinger f unctionals using Sobolev gradien ts: Applications t o quan tum me cha nics and nonlinear o ptics, SIAM J. Sci. Comput., 23 (2001), pp. 1316–1334. [26] S. D. Kim and S. V. Parter, Preconditioning Cheb yshev s pectral collo cation b y ﬁnite-diﬀerence operators, SIAM J. Numer. Anal., 34 (1997), pp. 939–958. [27] S. V. P arter, Preconditioning Legendre sp ectral collo cation methods for elliptic problems I: Finite di ﬀerence op erators, SIAM J. Numer. Anal., 39 (2001), pp. 330–347. [28] P . Henning and D. Pete rseim, Sobolev gradient ﬂow for the Gross-Pi taevskii eigen v alue prob- lem: global conv ergenc e and computational eﬃciency , SIAM J. Numer. Anal., 58 (2020), pp. 1744–177 2. [29] P . Henning, The dep endency of sp ectral gaps on the conv ergence of the inv erse iteration for a nonlinear eigenv ector problem, Math. Mo d. Meth. Appl. S., 33 (2023), pp. 1517–1544 . [30] P . Henning and M . Y ada v, Conv ergence of a Riemannian gr adien t metho d for the Gr oss- Pitaevskii energy f unctional in a r otating f rame, ESAIM Math. Model. Numer. Anal., 59 (2025), pp 1145–1175. [31] P . Henning and M. Y ada v, On di screte ground states of rotating Bose–Einstein condensates, Math. C omp., 94 (2025), pp 1–32. [32] P . H enning and E. Jarlebring, The Gr oss-Pitaevskii equation and eigenv ector nonlinearities: n umerical methods and algorithms, SIAM Rev., 67 (2025), pp. 256–317. [33] M. Herm ann, T. Styk el , and M. Y adav, Qualitativ e and Quan titativ e Analysis of Riemann- ian Optimization Metho ds for Ground States of Rotating M ulticomponent Bose-Einstein Condensates, [34] W. Hu, R. Bark ana, and A. Gruzino v, F uzzy cold dark matter: the wa ve properties of ultralight particles, Phys. Rev. Lett., 85 (2000), pp. 1158–1161 . [35] E. Jarlebring, S. Kv aal, and W. Michiels, An inv erse iteration method for eigen v alue problems with eigenv ector nonlinearities, SIAM J. Sci. Comput., 36 (2014), pp. A1978–A2001. [36] P . K azemi and M. Eck art, M inimizing the Gross- Pitaevskii energy functional with the Sob olev gradien t-analytical and numerical results, Int. J. Comput. Meth., 7 (2010), pp. 453–475. [37] J. Klaers , J. Sc hmitt, F. V ewinger, and M. W eitz, Bose-Einstein condensation of photons in an optical m icrocavit y, Nat., 468 (2010), pp. 545-548. [38] E. H. Lieb and R. Seiringer, Deri v ation of the Gr oss-Pitaevskii equation for rotating Bose gases, Commun. Math. Phys., 264 (2006), pp. 505–537 . [39] S. Lang, Real and F unctional Analysis, 3rd ed., Graduate T exts in Mathematics, vol. 142, Springer-V erlag, New Y ork, 1993. [40] W. Liu and Y. Cai, Normali zed gradient ﬂow with Lagrange multiplier for computing ground states of Bose-Einstein condensates, SIAM J. Sci. Comput., 43 (2021), pp. B219–B242. [41] B. T. Po lyak, Int ro duction to Optimization. New Y ork: Optimization Softw are, Inc., 1987. [42] Q. Rebj ock and N . Boumal, F ast con v ergence to non-isolated minima: four equ iv alen t co nditions for C 2 functions, M ath. Pr ogram., 213 (2025), pp 151–199. [43] F. Rupp, On the Lo jasi ewicz-Simon gradient i nequalit y on s ubmanifolds, J. F unct. Anal., 279 (2020), article 108708. [44] E. Shamri z, Z. Chen, and B. A . Malomed, Suppression of the quasi-tw o-dimensional quant um collapse in the attraction ﬁeld b y the Lee-Huang-Y ang eﬀect, Phys. Rev. A., 101 (2020), article 063628. [45] M. N. T engstrand, P . St¨ urmer, E. ¨ O. Karabulut, and S. M . Reimann, Rotating binary Bose- Einstein conden sates and v ortex clusters in quan tum droplets, Ph ys. Rev. Lett., 123 ( 2019), article 160405. [46] Y. W u, C. Liu, and Y. Cai, Nor malized ﬂo ws based on Sob olev gr adien ts f or computing ground states of spinor Bose-Einstein condensates, J. Comput. Phys., 538 (2025), article 114153. [47] X. W u, Z. W en, and W. Bao, A r egularized newton metho d f or computing ground states of Bose-Einstein condensates, J. Sci. Comput., 73 (2017), pp. 303–329. [48] T. Zhang and F. Xue, A new preconditioned nonlinear conjugate gradien t metho d i n real 26 ZIXU FEN G, P A TRICK HENNING, AND QINGLIN T ANG arithmetic for computing the ground state s of r otational Bose-Ei nstein condensate, SIAM J. Sci. Comput., 46 (2024), pp. A1764–A1792. [49] Z. Zhang, Exp onen tial conv ergence of Sob olev gradient descent for a class of nonli near eigen- problems. Commun. Math. Sci., 20 (2022), pp. 377–403. [50] C. Zhang, P . Henning, M. Y adav, and W. Chen, Con v ergence analysis of Sobolev Gradient ﬂo ws for the rotating Gross-Pitaevskii energy f unctional, arXiv: 2510.15604. [51] Q. Zhuan g and J. Shen, Eﬃcient SA V approac h for imaginary time gradient ﬂows wi th appli- cations to one- and multi-component Bose-Einstein Condensates, J. Comput. Ph ys., 396 (2019), pp. 72–88. App endix A. Pro of of L z φ ∈ H 1 0 ( D ) . In this app endix, for the local minimizer φ g , w e p ro ve th at L z φ g ∈ H 1 0 ( D ) . T o this end, we establish the follo wing lemma. Lemma A.1. L et D ⊂ R d ( d = 2 , 3 ) b e a b ounde d C 1 , 1 -domain that is r otat ional ly symmetric ab out the z -axis, i.e., A β D = D for al l β ∈ R . Then for every u ∈ H 2 ( D ) ∩ H 1 0 ( D ) we have L z u ∈ H 1 0 ( D ) . In p art icular, assumption (A3) holds whenever φ g ∈ H 2 ( D ) ∩ H 1 0 ( D ) . Pr o of. S ince u ∈ H 2 ( D ) and L z u = − i( x∂ y u − y ∂ x u ), we immediately hav e L z u ∈ H 1 ( D ) . Because A β D = D for all β and A β is a homeomorphism of R d , we h a ve A β ( ∂ D ) = ∂ D . Hence, for every x ∈ ∂ D the curve β 7→ A β x lies in ∂ D . D iﬀeren tiating with respect to β at β = 0 yields 0 = d dβ A β x     β =0 · n ( x ) = ( − y , x, 0) · n ( x ) for a.e. x = ( x, y , z ) ∈ ∂ D . Consequently , t he vector ﬁeld ( − y , x, 0) is tangential to ∂ D . Since ( x∂ y − y ∂ x ) u = ( − y , x, 0) · ∇ u, this op erator represents diﬀerentiation in a tangen tial direction along ∂ D . Because D is a C 1 , 1 -domain and u ∈ H 2 ( D ) , the trace u | ∂ D b elongs to H 3 / 2 ( ∂ D ), and t angen tial deriv ativ es admit traces in H 1 / 2 ( ∂ D ). S ince u ∈ H 1 0 ( D ) , we h a ve u | ∂ D = 0, and therefore all t angen tial deriv ativ es v anish on the b oundary . In particular,  ( x∂ y − y ∂ x ) u    ∂ D = 0 in the sense of traces. H ence L z u ∈ H 1 ( D ) has v anishing trace on ∂ D , whic h shows that L z u ∈ H 1 0 ( D ) . Finally , since the local minimizer φ g satisﬁes t he Euler–Lagrange equation H φ g φ g = λ φ g I φ g , standard elliptic regularity theory implies that φ g ∈ H 2 ( D ) ∩ H 1 0 ( D ) , provided D is C 1 , 1 and the nonlinearity satisﬁes the regularity assumptions in (A3) . The precise regularity argument is e.g. ellobrated in [31, Lem. 2.5]. Hence, by th e lemma ab o ve, w e conclude that L z φ g ∈ H 1 0 ( D ) . App endix B. Proof of Property 2.2. Pr o of. The co ercivity at φ g is iden tical to that established in [23, Proposition 2.2]. W e now p ro ve u niform co ercivity in a neighb orhoo d of φ g . Consider the restriction of t he L 2 -orthogonal pro jection operator (also know n as vector transp ort): T φ ′ g φ g := Pro j L 2 φ ′ g   T φ g M . (B.1) Since φ ′ g ∈ B σ ( φ g ), we hav e | ( φ ′ g , φ g ) L 2 | =     1 − 1 2 k φ ′ g − φ g k 2 L 2     ≥ C > 0 STRUCTURE AND SYMMETR Y OF THE GP GROUND-ST A T E MANIFOLD 27 for some constan t C indep endent of φ ′ g , provided σ > 0 is suﬃciently small. T o show that T φ ′ g φ g is bijectiv e, consider the equation T φ ′ g φ g ( v ) = w , v ∈ T φ g M , w ∈ T φ ′ g M . This is eq uiv alen t to ﬁnd ing a scalar x ∈ R such that v = w + xφ ′ g ∈ T φ g M . Imp osing the condition ( v , φ g ) L 2 = 0 y ields ( w , φ g ) L 2 + x ( φ ′ g , φ g ) L 2 = 0 , which has a solution x = − ( w , φ g ) L 2 ( φ ′ g , φ g ) L 2 . T o prov e uniqueness, sup p ose v = w + xφ ′ g and v 1 = w + x 1 φ ′ g b oth b elong to T φ g M . Then v − v 1 = ( x − x 1 ) φ ′ g ∈ T φ g M , so ( x − x 1 )( φ ′ g , φ g ) L 2 = 0 . Since ( φ ′ g , φ g ) L 2 6 = 0, it follo ws that x = x 1 , and thus v = v 1 . Therefore, T φ ′ g φ g is bijective, and its inverse is given exp licitly by ( T φ ′ g φ g ) − 1 ( w ) = w − ( w , φ g ) L 2 ( φ ′ g , φ g ) L 2 φ ′ g , ∀ w ∈ T φ ′ g M . (B.2) Com bined with the b oundedn ess of T φ ′ g φ g (its inverse is also b ounded), it is a linear homeo- morphism b etw een T φ g M and T φ ′ g M for all φ ′ g ∈ B σ ( φ g ) with σ > 0 su ﬃcien tly small. Using the contin uit y of E ′′ ( φ ) − λ φ I and the L 2 -pro jection Pro j L 2 φ with resp ect to φ , w e d educe th at E ′′ ( φ ) − λ φ I is co ercive on the closed subspace T φ ′ g φ g ( R φ g ) ⊂ T φ ′ g M . More precisely , for some constant δ σ with δ σ → 0 + for σ → 0 + , we hav e D ( E ′′ ( φ ′ g ) − λ φ ′ g I ) T φ ′ g φ g v , T φ ′ g φ g v E ≥ D ( E ′′ ( φ g ) − λ φ g I ) v , v E − δ σ   T φ ′ g φ g v   2 H 1 ≥ C k v k 2 H 1 − δ σ   T φ ′ g φ g v   2 H 1 ≥ C   T φ ′ g φ g v   2 H 1 , ∀ v ∈ R φ g , (B.3) for σ suﬃcien tly small. This implies that K φ ′ g ∩ T φ ′ g φ g ( R φ g ) = { 0 } . W e n o w prov e that the pro jection operator J φ ′ g is a bijection from T φ ′ g φ g ( R φ g ) to R φ ′ g . Since T φ ′ g φ g is a linear homeomorphism and R φ g ⊂ T φ g M is closed, it follo ws that T φ ′ g φ g ( R φ g ) is closed in T φ ′ g M and dim K φ g = codim T φ g M R φ g = codim T φ ′ g M T φ ′ g φ g ( R φ g ) = dim( T φ ′ g φ g ( R φ g )) ⊥ P φ ′ g . Because K φ ′ g is ﬁn ite-dimensional and T φ ′ g φ g ( R φ g ) is closed, T φ ′ g φ g ( R φ g ) + K φ ′ g is a closed sub- space of T φ ′ g M . It s orthogonal complement is ( T φ ′ g φ g ( R φ g ) + K φ ′ g ) ⊥ P φ ′ g = ( T φ ′ g φ g ( R φ g )) ⊥ P φ ′ g ∩ R φ ′ g . W e compute dimensions dim( T φ ′ g φ g ( R φ g )) ⊥ P φ ′ g = dim(( T φ ′ g φ g ( R φ g )) ⊥ P φ ′ g ∩ R φ ′ g ) + dim(( T φ ′ g φ g ( R φ g )) ⊥ P φ ′ g ∩ K φ ′ g ) , dim K φ ′ g = dim( K φ ′ g ∩ T φ ′ g φ g ( R φ g )) + dim( K φ ′ g ∩ ( T φ ′ g φ g ( R φ g )) ⊥ P φ ′ g ) . By K φ ′ g ∩ T φ ′ g φ g ( R φ g ) = { 0 } , we hav e dim K φ ′ g = dim  K φ ′ g ∩ ( T φ ′ g φ g ( R φ g )) ⊥ P φ ′ g  . Substituting into th e ﬁ rst equation and using dim K φ ′ g = dim K φ g = dim( T φ ′ g φ g ( R φ g )) ⊥ P φ ′ g , we obtain dim(( T φ ′ g φ g ( R φ g )) ⊥ P φ ′ g ∩ R φ ′ g ) = 0 . 28 ZIXU FEN G, P A TRICK HENNING, AND QINGLIN T ANG Therefore, T φ ′ g φ g ( R φ g ) + K φ ′ g = T φ ′ g φ g ( R φ g ) + K φ ′ g = T φ ′ g M . Then, for all u ∈ R φ ′ g , there exist unique v ∈ T φ ′ g φ g ( R φ g ) and w ∈ K φ ′ g such that u = v + w . Applying t he orthogonal p ro jection J φ ′ g yields u = J φ ′ g u = J φ ′ g v + J φ ′ g w = J φ ′ g v . Thus, J φ ′ g is a linear isomorphism from T φ ′ g φ g ( R φ g ) to R φ ′ g , and consequently the comp osition op erator J φ ′ g T φ ′ g φ g is a linear isomorphism from R φ g to R φ ′ g . Finally , t he co ercivity estimate transfers to R φ ′ g : there exists a constant C > 0, indep end ent of φ ′ g ∈ B σ ( φ g ), such that D ( E ′′ ( φ ′ g ) − λ φ ′ g I ) J φ ′ g T φ ′ g φ g v , J φ ′ g T φ ′ g φ g v E = D ( E ′′ ( φ ′ g ) − λ φ ′ g I ) T φ ′ g φ g v , T φ ′ g φ g v E ≥ C   T φ ′ g φ g v   2 P φ ′ g ≥ C   J φ ′ g T φ ′ g φ g v   2 H 1 , ∀ v ∈ R φ g .

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