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Fibration of the periodical eigenfunctions manifold into hypersurfaces Ya. Dymarskii Moscow, MIPT, 2016 September 15 Ya. Dymarskii Fibration of the periodical eigenfunctions manifold into hypersurfaces

The space of of self-adjoint periodic eigenvalue and eigenfunction boundary-value problems − y ′′ + p ( x ) y = λ y , y (0) − y (2 π ) = y ′ (0) − y ′ (2 π ) = 0 , (1) � 2 π � � p ∈ C 0 (2 π ) | P := p ( x ) dx = 0 0 The spectrum consists of real eigenvalues, which have multiplicity at most 2: λ 0 ( p ) < λ − 1 ( p ) ≤ λ + 1 ( p ) < . . . < λ − k ( p ) ≤ λ + k ( p ) < . . . Eigenfunctions corresponding to eigenvalues with subscript k have precisely 2 k nondegenerate zeros on the half-open interval [0 , 2 π ) . Ya. Dymarskii Fibration of the periodical eigenfunctions manifold into hypersurfaces

The manifold of eigenfunctions with exactly 2k zeros � 2 π Y k := { y ∈ C 2 (2 π ) : y 2 dx = 1 , (1) with λ = λ ± k ( p ) , y ∼ = − y } 0 The set Y k ( k = 0 , 1 , ... ) consists of all functions y such that: 1. there exist 2k points x i ∈ [0 , 2 π ) at which y ( x i ) = y ′′ ( x i ) = 0 , y ′ ( x i ) � = 0 ; 2. the function y has no other zeros; 3. there exist derivatives y (3) ( x i ) < ∞ 4. Y k is a manifold which locally C ∞ -diffeomorphic to space P . 5. There are mappings which recover the eigenvalue and potential: � 2 π y ′′ Λ k : Y k → R , Λ k ( y ) = λ := − 1 y dx ; 2 π 0 f k : Y k → P , f k ( y ) = p := y ′′ y + Λ k ( y ) . Ya. Dymarskii Fibration of the periodical eigenfunctions manifold into hypersurfaces

The degenerate and nondegenerate eigenfunctions For k ∈ N a pair ( y , z ) ∈ Y k × Y k is said to be conjugated if these � 2 π functions are generated by the same potential p and yzdx = 0 . 0 Any eigenfunction y ∈ Y k has a unique conjugated function z = I ( y ) and I 2 ( y ) = y . If λ − ( p ) < λ + ( p ) then I ( y ± ( p )) = y ∓ ( p ) . Lacuna ( y ) is ∆Λ k ( y ) := Λ k ( y ) − Λ k ( I ( y )) , Y k (∆Λ k = C ) := { y ∈ Y k : ∆Λ k ( y ) = C } . Y k = ∪ C ∈ R Y k (∆Λ k = C ) . The set Y k (∆Λ k = 0) is called degenerate; if C � = 0 , Y k (∆Λ k = C ) is nondegenerate. 1. For any fixed C , the subset Y k (∆Λ k = C ) ⊂ Y k is a C ∞ -submanifold of codimension 1; for any C 1 � = C 2 , Y k (∆Λ k = C 1 ) ∼ = Y k (∆Λ k = C 2 ) . 2. Y k ∼ = Y k (∆Λ k = C ) × R ∼ R P 1 . Ya. Dymarskii Fibration of the periodical eigenfunctions manifold into hypersurfaces

The degenerate and nondegenerate potentials | ∆ λ k ( p ) | := λ + k ( p ) − λ − k ( p ) ≥ 0 , P ( | ∆ λ k | = C ) := { p ∈ P : | ∆ λ k ( p ) | = C ≥ 0 } . P = ∪ C ≥ 0 P ( | ∆ λ k | = C ) . 1. For any fixed C > 0 , the nondegenerate subset P ( | ∆ λ k | = C ) ⊂ P is a C ∞ -submanifold of codimension 1; P ( | ∆Λ k | = C ) × R + ∼ = P \ P ( | ∆ λ k | = 0) ∼ R P 1 . 2. The degenerate subset P ( | ∆ λ k | = 0) ⊂ P is a C ∞ -submanifold of codimension 2; P ( | ∆ λ k | = 0) ∼ ∗ . 3. For C � = 0 , f k | ± C : Y k (∆Λ k = ± C ) → P ( | ∆ λ k | = | C | ) is C ∞ -diffeomorphism. 4. For C = 0 , f k | 0 : Y k (∆Λ k = 0) → P ( | ∆ λ k | = 0) is C ∞ -bundle with R P 1 as fiber; 5. For any C , Y k (∆Λ k = C ) ∼ = P ( | ∆ λ k | = 0) × R P 1 . Ya. Dymarskii Fibration of the periodical eigenfunctions manifold into hypersurfaces

The analytic description of bundle of Y k For y ∈ Y k (∆Λ k = 0) Wronskian W ( y ) := W ( y , I ( y )) = y · ( I ( y )) ′ − y ′ · I ( y ) = const . The mapping � x � � ∆ λ 0 yI ( y ) dx exp y 2 W ( y ) F : Y k (∆Λ k = 0) × R → Y , F ( y , ∆ λ ) := || ... || L 2 is C ∞ -diffeomorphism and F ( y , ∆ λ ) ∈ Y k (∆Λ k = ∆ λ ) . The inverse mapping is F − 1 : Y → Y k (∆Λ k = 0) × R , � − 1 / 2  � x  � ∆ λ ( y ) 1 + 0 y · I ( y ) dx y W ( y (0)) F − 1 ( y ) = , ∆ λ ( y )  .   || ... || L 2  Ya. Dymarskii Fibration of the periodical eigenfunctions manifold into hypersurfaces

Levels of functional Λ 1. For any fixed C , the subset Y k (Λ k = C ) ⊂ Y k is a C ∞ -submanifold of codimension 1; for any C 1 � = C 2 , Y k (Λ k = C 1 ) ∼ = Y k (Λ k = C 2 ) . 2. Y k ∼ = Y k (Λ k = C ) × R ∼ R P 1 . On Y k consider the vector field � 2 π y 4 dx − y 2 y ⇒ ˙ 0 y = v ( y ) := ˙ λ ( v ( y )) = 1 ⇒ � 2 π 0 ( y ′ ) 2 dx 4 there exists the vector flow F t : Y k → Y k ( −∞ < t < ∞ ) F t ( Y k (Λ k = C )) = Y k (Λ k = C + t ) . Ya. Dymarskii Fibration of the periodical eigenfunctions manifold into hypersurfaces

The parametrization of manifolds Y k and P H k ⊂ C 2 (2 π ) is the set of functions η that satisfy the conditions 1. η ( x ) ∈ C 2 (2 π ) , 2. η ( x ) > 0 , � 2 π 3. η ( x ) dx = 2 π k , 0 4. � x � x � 2 π � 2 π sin 2 0 η ( t ) dt cos 2 0 η ( t ) dt dx = 0 , dx = 0 . η ( x ) η ( x ) 0 0 The set H k is homotopy trivial C ∞ -manifold. � x 0 η ( t ) dt , where ϕ ∈ R P 1 By definition θ ( x ; ϕ, η ) := ϕ + Ya. Dymarskii Fibration of the periodical eigenfunctions manifold into hypersurfaces

The parametrization of manifold Y k Υ : H k × R × R P 1 → Y k , Υ( η, ∆ λ, ϕ ) := � x � ∆ λ � y sign (∆ λ ) = const sin 2 θ ( t ; ϕ, η ) η 1 / 2 ( x ) exp · cos( θ ( x ; ϕ, η )) . dt 4 η ( t ) 0 Υ is C ∞ -diffeomorphism. Ya. Dymarskii Fibration of the periodical eigenfunctions manifold into hypersurfaces

The parametrization of manifold P 2 η + 3( η ′ ) 2 r ( x ) := ( y k ) ′′ = − η ′′ − η 2 + 4 η 2 y k ∆ λη ′ sin 2 θ ( x ; ϕ, η ) + ∆ λ 2 sin 2 2 θ ( ... ) − ∆ λ cos 2 θ ( ... ) + ∆ λ 2 , 2 η 2 16 η 2 � 2 π λ k = − 1 r ( x ) dx , 2 π 0 Φ : H k × R + × R P 1 → P , Φ( η, ∆ λ, ϕ ) = p ( x ) := r ( x ) + λ k . Φ is C ∞ -diffeomorphism. Ya. Dymarskii Fibration of the periodical eigenfunctions manifold into hypersurfaces

Literature 1 Ya.M. Dymarskii Мanifold Method in the Eigenvector Theory of Nonlinear Operators // Jornal of Mathematical Sciences – 2008. 2 Ya. M. Dymarskii, Yu. A. Evtushenko Foliation of the space of periodic boundary-value problems by hypersurfaces to fixed lengths of the nth spectral lacuna // Sbornik: Mathematics 207:5, 2016, P. 678–701 Ya. Dymarskii Fibration of the periodical eigenfunctions manifold into hypersurfaces