Let G > 0 be Newton’s gravitational constant, c > 0 the speed of light, and t time. A particle position x and momentum per unit mass p belong to Euclidean space ℝ 3 , with x ˙ = p ; a dot denotes a time derivative. We write | x | for Euclidean length and x ⋅ p for the Euclidean inner product. A phase point is z = ( x , p ) ∈ ℝ 6 , and d z = d x d p is Lebesgue phase volume. The non-negative phase-space mass density F ( t , x , p ) has spatial density ρ F and total mass M [1][21], defined by

Define the cosmological frequency Ω Λ > 0 by Ω Λ 2 = c 2 Λ / 3 . The total particle potential Φ F Λ satisfies the Vlasov–Poisson–Λ equations

Here ∇ x and ∇ p are gradients in position and momentum, and Δ x = ∇ x ⋅ ∇ x is the spatial Laplacian. The matter potential U F g is specified by

Thus the matter acceleration is − ∇ U F g , whereas the cosmological acceleration is + Ω Λ 2 x . For differentiable phase-space functions f , g , the canonical Poisson bracket is [22]

For a spherical state let r = | x | . Finite spherical mass gives U F g ( r ) = − G M / r + o ( r − 1 ) as r → ∞ , where o ( r − 1 ) denotes a remainder whose product with r tends to zero. Define L = | x × p | , the magnitude of angular momentum per unit mass. Outside the spherical matter support the radial acceleration is

No normalization at infinity is imposed on the total potential Φ F Λ .

Let Φ ext ( t , x ) be a smooth external gravitational potential near the centre of mass. Its constant term has no effect on motion, and its gradient at the centre is removed by adopting the freely falling frame. Let ∂ i denote differentiation with respect to the i th position coordinate, and let δ i j equal one for i = j and zero otherwise. The symmetric trace-free tidal matrix T ^ 3 ( t ) is

For a real matrix A , A T denotes its transpose and tr   A the sum of its diagonal entries. The tidal matrices form the five-dimensional space

The retained external potential and acceleration are

Consequently the forced kinetic equation is

The cosmological contribution is included in Φ F Λ , not in T ^ 3 .

For a spherical potential Φ ( r ) , define the particle energy E = | p | 2 / 2 + Φ ( r ) . Energy and L 2 are invariants, that is, constants along a particle trajectory. A global Jeans distribution is F = Ψ ( E , L 2 ) for one non-negative function Ψ, with the same value on every connected component carrying the same invariants. Here “global” requires one function of the numerical invariant values on the entire phase space. This restriction does not exclude stationary distributions assigning different occupations to disconnected components of the same invariant level. We call the potential anti-confining when Φ ( r ) → − ∞ . Measurability and almost-everywhere statements below refer to Lebesgue measure; “almost everywhere” excludes sets of measure zero.

Theorem3.1(Global Jeans states in anti-confining spherical potentials).Let a > 0 , 0 < β ≤ 2 ,andleta spherical potential satisfy

where o ( r β ) / r β → 0 . Let Ψ : ℝ × [ 0 , ∞ ) → [ 0 , ∞ ] bemeasurable, non-negative, andnotalmosteverywherezero. Thenthesphericalphase-spacedensity

satisfies, forsome C > 0 andallsufficientlylarge r ,

Consequentlyitsmassisinfinite. Moreprecisely, themassinsideradius R growsatleastas R 1 − β / 2 when 0 < β < 2 andatleastas log R when β = 2 .

Proof. Since Ψ ≥ 0 , all integrals below are understood in the extended sense and Tonelli’s theorem allows their order to be interchanged for non-negative integrands. Because Ψ is not almost everywhere zero, its superlevel sets in bounded rectangles give m > 0 and a bounded measurable set A ⊂ ℝ × [ 0 , ∞ ) with positive area | A | such that Ψ ( E , s ) ≥ m almost everywhere on A .

At a fixed radius r = | x | > 0 , let e r : = x / r be the radial unit vector, write p = p r e r + p ⊥ , where p r = p ⋅ e r and p ⊥ ⋅ e r = 0 , and set s = L 2 = r 2 | p ⊥ | 2 . Then

Changing from p r to E on the two branches p r = ± 2 [ E − Φ ( r ) − s / ( 2 r 2 ) ] gives

Choose E − < E + and s + < ∞ such that A ⊂ [ E − , E + ] × [ 0 , s + ] . The asymptotic hypothesis implies that there exist constants r 0 > 0 and C A > 0 such that, for every r ≥ r 0 and every ( E , s ) ∈ A ,

Thus A lies in the integration domain and restriction of Equation (5) to A yields

which is Equation (4). Finally,

and the last integral has the stated power or logarithmic divergence. □

Corollary3.2 (Global Jeans obstruction for Vlasov-Poisson-Λ).Let Λ > 0 . No nontrivial spherically symmetric self-consistent Vlasov–Poisson–Λstationary state of finite total mass can have the global form F ( x , p ) = Ψ ( E , L 2 ) with a continuous non-negativeΨ. Within this ansatz,a finite-mass equilibrium must distinguish the connected invariant components.

Proof. Finite spherical mass and the Newtonian normalization give Φ ( r ) = − Ω Λ 2 r 2 / 2 + O ( r − 1 ) . Here and below O ( g ) denotes a remainder bounded in magnitude by a constant multiple of | g | in the stated limit. Continuity and nontriviality of Ψ provide a bounded positive-measure rectangle on which Ψ has a positive lower bound; if the positive point has s = 0 , take a one-sided rectangle in s ≥ 0 . Theorem 3.1 with β = 2 gives logarithmic mass divergence. □

Remark3.3(Restriction to the inner connected component). The smooth states below occupy the inner bounded component of the admissible phase set. Replacing Ψ ( E , L 2 ) by a smooth global function does not remove the exterior component with the same invariant values and therefore does not restore finite mass.

Corollary 3.2 excludes a single global Ψ ( E , L 2 ) . We therefore make the connected component part of the stationary-state data.

Definition (Component-selected stationary distribution). Let H ( x , p ) = | p | 2 / 2 + Φ ( | x | ) be the spherical particle Hamiltonian, the energy as a function on phase space. For invariant values ( E , s ) , with s = L 2 ≥ 0 , suppose the joint level { H = E , L 2 = s } has a bounded inner connected component C in ( E , s ) and an unbounded exterior component C out ( E , s ) . A non-negative stationary distribution is component-selected on the inner component when it equals Ψ ( E , s ) on C i n ( E , s ) and zero on C out ( E , s ) , for the same invariant values and a non-negative function Ψ . Thus the component, not only the numerical invariants, determines occupation.

The construction below makes this selection without a material boundary. We write C m for functions with m continuous derivatives, C ∞ for smooth functions, and a subscript c for compact support. The support supp   f is the closure of the set where f ≠ 0 ; compact phase-space support bounds both position and momentum. Choose a nondecreasing smooth function ϑ : ℝ → [ 0 , 1 ] equal to zero on ( − ∞ , 1 ] and one on [ 2 , ∞ ) . Specifically, throughout the construction and numerical illustration, for a real argument u we take

Its transition is flat: every derivative of positive order vanishes at both endpoints. For a cutoff parameter ε > 0 , define the microscopic weight φ ε and its isotropic momentum integral g ε by

with g ε ( y ) = 0 for y ≤ 0 . Here η is the scalar argument of the energy cutoff, and y is the relative potential, the cutoff energy minus the particle potential. For any real u , write u + = max ( u , 0 ) .

Lemma3.4(Smooth cutoff density and limiting power law).The function g ε is C ∞ ,vanishes to all orders at y = ε ,and on every bounded interval converges in C 1 as ε ↓ 0 to

Proof. Flatness of ϑ makes the zero extension of φ ε smooth. Equation (6) is locally its convolution with u + 1 / 2 , the integral of the product after translating one factor by y . More explicitly, for every integer n ≥ 0 ,

On a bounded y -interval the integrand vanishes for all sufficiently large u , uniformly in y . Differentiation therefore acts only on the smooth factor φ ε , while u 1 / 2 is integrable at zero. This proves continuity of every displayed derivative, with no endpoint term. For y > ε the integral is over 0 ≤ u ≤ y − ε ; its absolute value is bounded by a constant times ( y − ε ) 3 / 2 times the supremum of | φ ε ( n ) | on [ ε , y ] . Every derivative thus tends to zero as y ↓ ε , agreeing with the zero extension.

Moreover 0 ≤ η + − φ ε ( η ) ≤ η 1 [ 0 , 2 ε ] ( η ) , where 1 A is one on a set A and zero outside. The defining integral and its first y -derivative give the stated C 1 convergence. Direct integration gives Equation (7). □

Let s ≥ 0 now denote a dimensionless radius, y c > 0 the central relative potential, and μ ≥ 0 the dimensionless cosmological parameter. A prime denotes differentiation of a function with respect to its scalar argument. Consider the regular radial equation

At the origin, regularity means a C 2 solution satisfying the displayed central conditions. For ( ε , μ ) = ( 0 , 0 ) , denote the solution by y 0 ; the equation then has a density proportional to ( y + ) 5 / 2 and is the Lane-Emden equation of index 5/2. Its first zero S 0 is finite and satisfies y ′ 0 ( S 0 ) < 0 [2]. A crossing y ( S ) = ε is transverse when y ′ ( S ) ≠ 0 .

Lemma3.5 (Persistence of a transverse matter boundary).For sufficiently small ε , μ > 0 ,the regular solution ofEquation (8)is strictly decreasing up to a unique first crossing y ( S ε , μ ) = ε with y ′ ( S ε , μ ) < 0 . Moreover S ε , μ → S 0 and y ′ ( S ε , μ ) → y ′ 0 ( S 0 ) < 0 as ( ε , μ ) → ( 0 , 0 ) .

Proof. The regular integral equation is

Choose d > 0 small enough that y ′ 0 < 0 on ( 0 , S 0 + d ] . Lemma 3.4 gives a common Lipschitz bound for g ε on a bounded interval containing the range of y 0 . Gronwall’s inequality for the integral equation and continuation on that interval yield y → y 0 in C 1 ( [ 0 , S 0 + d ] ) .

Since g 0 ( y c ) > 0 , sufficiently small parameters give g ε ( y c ) > μ and y ″ ( 0 ) = [ μ − g ε ( y c ) ] / 3 < 0 . By continuity there are s c > 0 and κ > 0 , independent of these parameters, with g ε ( y ( s ) ) − μ ≥ κ on [ 0 , s c ] . Hence

so y ′ < 0 near the centre. On [ s c , S 0 + d ] , y ′ 0 is bounded above by a negative constant; C 1 convergence preserves its sign. The inequalities y ( S 0 − d ) > ε and y ( S 0 + d ) < ε hold for small parameters. Strict monotonicity gives one transverse crossing between them. Convergence of its radius and derivative follows from C 1 convergence and y ′ 0 ( S 0 ) < 0 . □

Theorem3.6 (Smooth finite-mass stationary solution on an inner bounded orbit family).Let Λ > 0 . There exist a positive density scale C ρ ,a cutoff energy E c ∈ ℝ ,a smooth spherical potential Φ * ,and radii 0 < R < r 1 < r 2 < r Λ ,where r Λ = ( G M / Ω Λ 2 ) 1 / 3 and M is the resulting total mass. Let ζ be a smooth radial function equal to one for r ≤ r 1 and zero for r ≥ r 2 . The nonzero spherical distribution

togetherwith Φ * ∈ C ∞ ( ℝ 3 ) , isaclassicalself-consistentstationaryVlasov-Poisson-Λstate, meaningthatbothequationsholdpointwisewithitsowngeneratedpotential. Itsmattersupportliesintheclosedball B R ¯ = { x : | x | ≤ R } . Define H * ( x , p ) = | p | 2 / 2 + Φ * ( | x | ) and

Theexteriorradialpotentialhasitsmaximumat r Λ . Theenergy E sep istheouterradialseparatrixlevel, separatingboundedradialmotionfromtheexteriorescapebranch. Everyoccupiedtrajectoryremainsintheinnerboundedcomponent, and

Proof. Choose the parameters from Lemma 3.5, put α = 3 Ω Λ 2 / μ , C ρ = α / ( 4 π G ) , R = S ε , μ / α , and y * ( r ) = y ( α r ) . The density ρ * = C ρ g ε ( y * ) on r < R and zero outside is C c ∞ because the crossing is transverse and g ε is flat. Define U * ( x ) = − G ∫ ℝ 3 ρ * ( y ) / | x − y | d y and Φ * = U * − Ω Λ 2 r 2 / 2 . Elliptic regularity gives smoothness [23]; outside the support, Φ * ( r ) = − G M / r − Ω Λ 2 r 2 / 2 by Newton’s theorem [21]. Integrating the radial equation at R gives R 2 y ′ * ( R ) = Ω Λ 2 R 3 − G M < 0 , hence R < r Λ . With E c = ε + Φ * ( R ) , the radial equations imply E c − Φ * = y * on [ 0 , R ] .

Choose R < r 1 < r 2 < r Λ and a smooth selector ζ equal to one on [ 0 , r 1 ] and zero on [ r 2 , ∞ ) . On supp   ∇ ζ the argument of φ ε is below ε , so the microscopic cutoff is already zero. Hence Equation (9) has density ρ * and

Thus the selector introduces neither a wall nor a force. Finally, with ξ = R / r Λ ,

Nonzero occupation requires E c − H * > ε , so continuity gives sup supp   F ∗ , ε H * ≤ E c − ε = E max . For fixed angular momentum the effective radial potential is Φ * ( r ) + L 2 / ( 2 r 2 ) . Any trajectory reaching r = r Λ must satisfy

No occupied trajectory can therefore cross that sphere. The non-negative angular-momentum term strengthens this bound; no assertion about a global maximum of the effective potential is needed. □

For a fixed ε , we henceforth abbreviate this equilibrium by F ∗ : = F ∗ , ε .

Remark3.7 (Role of the radial cut-off ζ ). The derivative of ζ is supported where the microscopic energy cutoff vanishes; it removes the disconnected exterior component without changing occupied characteristics or the self-consistent density.

Remark3.8 (Comparison with dispersing solutions). The state remains confined by its self-gravity. This differs from the dispersing solutions in [5][6], whose trajectories enter the region dominated by the outward quadratic force.

Corollary3.9 (Local persistence of the component-selected family).If one member of the radial family has a transverse first crossing and g ε ( y c ) > μ ,then the same is true in a parameterneighbourhood. There R , M , r Λ ,and Δ sep vary continuously,remain strictly ordered by R < r Λ and Δ sep > 0 ,and the same selector radii r 1 < r 2 may be used after shrinking theneighbourhood.

Proof. The integral equation above gives C 1 continuous dependence on ( y c , ε , μ ) , so the nonzero crossing derivative and strict monotonicity persist for sufficiently small parameter changes. Writing S = S ε , μ , continuity of the mass integral and G M − Ω Λ 2 R 3 = − α − 1 / 2 S 2 y ′ ( S ) > 0 gives continuity and the strict ordering; compactness of a smaller neighbourhood permits fixed selector radii. □

In dimensionless units G = 1 and 4 π G C ρ = 1 , the choice y c = 1 , ε = 0.05 , μ = 0.005 gives

Figure 1 illustrates the smooth density and the separatrix gap.

Figure 1. Representative smooth trapped equilibrium. The matter boundary is y ( R ) = ε and E max < E sep . (a) Relative potential and normalized density. (b) Total potential and outer separatrix.

For the explicit comparison state, fix its mass M > 0 and radius R > 0 . Define the gravitational frequency ω g , the net interior frequency ω * , and the exterior radius of zero radial acceleration r Λ [24] by

With X = x / R , V = p / ( ω * R ) , Q = 1 − | X | 2 − | V | 2 + | X × V | 2 , and D = { Q > 0 , | X × V | < 1 } , set

The quotient is evaluated on D and the distribution is zero elsewhere. The velocity integral is ∫ 1 D / Q d V = π 2 1 { | X | < 1 } , so

A stationary weak solution has a locally integrable distribution, satisfies Poisson’s equation after integration against smooth compactly supported test functions, and obeys

This definition does not require derivatives of F at its support boundary.

Theorem3.10 (Finite-mass Kurth-Λequilibrium)UnderEquation (12), ( F 0 , Λ , Φ 0 Λ ) is a finite-mass stationary weak Vlasov-Poisson-Λsolution with compact support. The potential is C 1 and Poisson’s equation has no surface mass concentration at r = R .

Proof. Equation (14) follows from the velocity integral and its potential and radial derivative match at R . On the support, the scaled characteristics are harmonic, X ( t ) = X 0 cos ( ω * t ) + V 0 sin ( ω * t ) and V ( t ) = V 0 cos ( ω * t ) − X 0 sin ( ω * t ) , preserving Q and D ; testing this invariant flow against C c 1 functions gives the weak transport identity. □

Remark3.11 (Boundary regularity). The singularity F 0 , Λ ~ Q − 1 / 2 is integrable but not classically differentiable. It is used as an exact weak solution for comparison; the classical linearization in Section 5 uses the smooth state F * .

Let a ∈ C 2 ( J ) be a positive scale factor on a time interval J . A homothetic motion changes all spatial lengths by this same factor. The transformation below preserves the phase volume d z : its position and momentum determinants cancel. Define

Theorem3.12 (Homothetic Vlasov-Poisson-Λ family).The function (15) is a weak solution exactly when

Let ℰ [ F ] denotethephysicalenergy: thekineticenergyplusonehalfofthematter-potentialintegralandthefullcosmological-potentialintegral. Thus

Forthisequilibriumdefineitssecondspatialmoment I 0 = ∬ | x | 2 F 0 , Λ d z . Alongthescalingfamily,

Proof. With X = x / a and P = a p − a ˙ x , the transformed characteristics become X ˙ = P / a 2 and P ˙ = − ω * 2 X / a 2 precisely under Equation (16); the time change d τ = a − 2 d t recovers the harmonic Kurth flow. The energy formula follows by scaling. □

The homothetic family is isotropic; the tidal response below belongs to the trace-free degree-two sector.

For a solution of Equation (3), define the inertia matrix I ^ 3 , the mixed position-momentum matrix J ^ 3 , and the kinetic second-moment matrix K ^ 3 by

All unmarked phase-space integrals extend over ℝ 6 . Let Id ^ 3 be the 3 × 3 identity matrix. The quadrupole matrix and total angular momentum are, respectively,

The vector L is the total angular momentum of the distribution, distinct from the one-particle magnitude L . Repeated component indices are summed from one to three.

Proposition4.1(Exact second-moment identities).Assume the displayed moments are finite and may be differentiated,with no boundary contributions in integration by parts; these conditions hold for classical solutions of compact phase-space support. Write a F = − ∇ x Φ F Λ − T ^ 3 x for the acceleration. Then

Theexternaltidecontributes

Let ε i j k bethealternatingsymbol, equaltothesignofapermutationof ( 1 , 2 , 3 ) andzerowhenindicesrepeat. Thetidaltorqueis

Proof. Multiply Equation (3) successively by x i x j , x i p j + x j p i , and ( x × p ) i , and integrate by parts in position and momentum. The external acceleration − T ^ 3 x gives the stated matrix product and torque. □

Theorem4.2(Preservation of spherical symmetry).If the unforced problem has a unique solution in a rotation-invariant class and the initial state is spherical,then it remains spherical and Q ^ 3 ( t ) = 0 . The same holds for the homothetic family of Theorem3.12.

Proof. Rotating position and momentum produces a solution with the same initial data; uniqueness gives rotational invariance, hence I ^ 3 is proportional to the identity. □

The mass current of a distribution F is j F ( t , x ) = ∫ ℝ 3 p F ( t , x , p ) d p .

Theorem4.3(Initial quadrupole response).Let F s be a spherical stationary state with vanishing mass current and finite second spatial moment I 0 = ∬ | x | 2 F s d z . If the forced solution has twice differentiable second moments and the preceding integrations by parts are valid,then

For the uniform Kurth ball this is Q ^ ¨ 3 ( 0 ) = − ( 2 M R 2 / 5 ) T ^ 3 ( 0 ) .

Proof. Sphericity gives I ^ 3 ( 0 ) = I 0 Id ^ 3 / 3 and makes the unforced internal term isotropic. The trace-free part of Equation (18) yields the result; Equation (17) gives the uniform-ball coefficient. □

Corollary4.4(Short-time tidal expansion).If T ^ 3 is C 1 near zero and the second moments have a third time derivative that is bounded near zero,then Q ^ 3 ( t ) = − ( I 0 / 3 ) T ^ 3 ( 0 ) t 2 + O ( t 3 ) ; under twice differentiability the remainder is o ( t 2 ) .

Proof. Use Taylor’s formula together with Q ^ 3 ( 0 ) = Q ^ ˙ 3 ( 0 ) = 0 and Theorem 4.3. □

We linearize about F * with Hamiltonian H * and use I 0 = ∬ | x | 2 F * d z for this equilibrium’s second spatial moment. Fix 0 < δ < Δ sep . Let K δ be the closed inner connected component of { H * ≤ E max + δ } containing supp   F * . The energy bound makes it compact and invariant under the stationary particle motion. Define the projections π x ( x , p ) = x and π p ( x , p ) = p .

For an integer k ≥ 1 and 0 < α < 1 , C k , α ( ℝ n ) is the normed space of functions with bounded continuous derivatives through order k whose derivatives of order k satisfy an α -Hölder bound: the difference between values at two points is bounded by a constant times the α th power of their distance. The norm includes the supremum of each derivative and these difference bounds. Define the little Hölder space c 0 k , α ( ℝ n ) as the closure of C c ∞ ( ℝ n ) in that norm. The fixed-support spaces are

Thus ℋ δ , + k , α has one additional spatial and momentum derivative. For normed spaces X , Y , let ℬ ( X , Y ) denote the bounded linear maps with their operator norm, and write ℬ ( X ) = ℬ ( X , X ) .

Let Φ t z denote the stationary Hamiltonian flow, the solution at time t issued from z of x ˙ = p , p ˙ = − ∇ x Φ * ( x ) . Define the transport and momentum-integration operators

For σ ∈ X δ k , α , its Newtonian perturbation potential is ϕ σ ( x ) = − G ∫ σ ( y ) / | x − y | d y . The sources due to the perturbed gravitational field and the imposed tide are

The family U 0 is called a strongly continuous group when U 0 ( t + s ) = U 0 ( t ) U 0 ( s ) , U 0 ( 0 ) is the identity, and ‖ U 0 ( t ) h − h ‖ → 0 as t → 0 for every h .

Lemma5.1(Invariant support and the composition group).For every T > 0 , U 0 is a strongly continuous group on ℋ δ k , α and there is a finite constant C T such that

Proof. Energy conservation makes K δ invariant and the smooth flow has uniformly bounded derivatives on [ − T , T ] × K δ . Standard composition estimates and smooth approximation give strong continuity [7][8]. Define the transport differential operator L 0 f = { f , H * } . For f ∈ ℋ δ , + k , α , ( U 0 ( t ) − U 0 ( s ) ) f = − ∫ s t U 0 ( r ) L 0 f   d r . The coefficients of L 0 are bounded on K δ , giving the Lipschitz bound. □

Lemma5.2(Momentum projection).The map P : ℋ δ k , α → X δ k , α isbounded.

Proof. All momentum supports lie in the fixed compact set π p K δ ; differentiate under the integral and integrate the corresponding Hölder seminorm. For little Hölder membership, approximate h by smooth functions multiplied by one fixed compactly supported cutoff equal to one near K δ . The same estimate on their common momentum support passes to the limit. □

Lemma5.3(Fixed-support Newtonian map and source operators).The maps S : X δ k , α → ℋ δ , + k , α and S tid : T 2 → ℋ δ , + k , α are bounded.

Proof. Put K x = π x K δ . Choose a smooth compactly supported spatial function χ equal to one near π x ( supp   ∇ p F * ) , and a bounded open neighbourhood O containing supp   χ with positive distance to its boundary. Writing ‖   ⋅   ‖ L ∞ ( O ) for the essential-supremum norm, local integrability of | x − y | − 1 gives

Interior Schauder estimates [23], which bound potential derivatives by the density’s Hölder norm, then yield

The constants depend only on G , k , α and the fixed supports, neighbourhood and cutoff, not on σ .

By the definition of c 0 k , α , choose smooth compactly supported σ n → σ in C k , α . Multiplying by a fixed cutoff equal to one near K x makes their supports lie in one compact set without changing the limit. The preceding estimate, with this common support, shows that χ ∇ ϕ σ n → χ ∇ ϕ σ in C k + 1 , α . Each approximant is smooth and compactly supported, proving little Hölder membership. Multiplication by ∇ p F * gives the bound for S , with phase-space support inside K δ . The tidal bound follows directly from the smooth fixed-support product. □

We write C 0 , 1 ( [ 0 , T ] ; X ) for Lipschitz-continuous paths with values in a normed space X .

Proposition5.4(Finite-time kernel regularity)The kernels