, (?u)). définit l'ensemble résolvant de A : ?(A) = {? ? C | A ? ? I est inversible

, On définit le spectre de A : sp(A) = C \ ?(A)

, On définit le spectre essentiel de A : sp ess (A) = {? ? C | A ? ? I n'est pas Fredholm}

, A) un opérateur autoadjoint. On a ? ? sp ess (A) si, et seulement si, il existe une suite (u m )

B. Proposition, Pour (H 2 (R d, vol.10, p.1

. Sy, Function ( ' tau ' , real = True , positive = True ) ( s )

. Sy, Function ( ' kappa ' , real = True ) ( s )

N. Sy, Function ( 'N ' , real = True , positive = True ) ( s )

. Sy, Function ( 'b ' , real = True , positive = True ) ( s ) On définit les séries formelles

, n(s,?h 2 ) notée nsi et ? ? n(s,?h 2 )

, n(s,?h 2 ) noté nxi

. Sy, Function ( ' n0 ' , real = True , positive = True )

. Sy,

, eta = sy . symbols ( ' eta0 : '+ str ( nbTerm +3) , cls = sy . Function , real = True ) 26 in range

, ) ** q * n [0]*( sy . factorial ( q ) * k ** q -eta

. Dxn-+=-n-;-q-+1]*-z-**-q-*-h-**, * q ) /( n [0]* sy . factorial ( q ) ) in range (1 , nbTerm +3) : 41 ina += (( -1) ** q *

, La matrice ME et les fonctions FtoVExp et VtoFExp permettent de résoudre l'équation différentielle en ? à s fixé ?? ?? ? + 4b(s)? ? ? = S dans R ne

, La solution est ? = VtoFExp(ME * FtoVExp(S))

. Ne-=-ne-+1,

;. Mte-=-sy, . Ne, . Ne-)-in, and . Range, 57 vec [j ,0] = fct . coeff (r , j ) in range ( NE ) : 63 fct += vec [j ,0]* r **( j +1) /( j +1) na +1) : 69 MTA [2* j -4 ,2* j ] = -j *( j -1) na +1) : 75 MTA

, MTA [2* j -2 ,2* j +1] = -4* j * a * t

. Ma-=-mta,

F. Def, vec = sy . zeros ( NA -1, vol.1, p.82

, * z * t ) ) in range ( na +1) : 85 vec [2* j ,0] = fctAi . coeff (z , j ) 86 for j in range ( na ) : 87 vec [2* j +1 ,0] = fctAp . coeff (z , j ) in range ( na ) : 93 fct += vec [2* j +1 ,0]* z **( j +1) * sy . airyai ( -a -2* z * t ) 94 for j in range ( na +1) : 95 fct += vec

, ph : h **3* expr . diff ( s ) +2* sy . I * ph * expr 98 dzPhi = lambda expr : h * expr . diff ( z )

, ph : h **3* expr . diff ( s ) +2* sy . I * ph * expr -h **3*2* r * b . diff ( s ) * expr 101 drPsi = lambda expr : expr . diff ( r ) -2* b *

D. , Cas général 2D

. Sy, , vol." ----------------------------------------------------------

, nbTerm +3) ) ) . expand (

, nbTerm +3) ) ) . expand (

, nbTerm +1) ) ) . expand (

, Equation differentielle sur psi , O ( h^" + str ( nbTerm +1) + " ) " )

, edPsi += ( -ir1 * dsPsi ( ir1 * psis , dTh ) )

, Equation differentielle sur phi , O ( h^" + str ( nbTerm +3) + " ) " )

, edPhi += ( -iz1 * dsPhi ( iz1 * phis , dTh ) )

, Equation saut fonctions , O ( h^" + str ( nbTerm +1) + " ) " )

, saut0 += ( psi . subs (r ,0) -phi . subs (z ,0) . subs

, Equation saut derivees , O ( h^" + str ( nbTerm +1) + " ) " )

, psir . subs (r ,0) -N * phiz . subs (z ,0) . subs, p.1

, diff ( x ) , sy . sqrt ( sy . Rational (i ,2) ) * GH [i -1]( x ) 15 -sy . sqrt ( sy . Rational ( i +1 ,2) ) * GH [ i +1]( x ) ) for i in range

, Rational ( nb -1 ,2) ) * GH

, ( x ) , sy . sqrt ( sy . Rational (i ,2) ) * GH

, Rational ( i +1 ,2) ) * GH [ i +1]( x ) ) for i in range

. Sy,

, ) ** q * n0 *( sy . factorial ( q ) -eta [q -3]) ) in range (1 , nbTerm +3) : 47 n10 += n [ q ]*( x * h / t ) ** q /( n0 * sy . factorial ( q ) ) in range, nbTerm +3) : 55 ina +=

=. Sq and . Multx,

. Lq-=--sq,

. =-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=,

, diff ( x ) , sy . sqrt ( sy . Rational (i ,2) ) * GH [i -1]( x ) 98 -sy . sqrt ( sy . Rational ( i +1 ,2) ) * GH [ i +1]( x ) ) for

, diff ( x ) , sy . sqrt ( sy . Rational ( nb ,2) ) * GH

, ( x ) , sy . sqrt ( sy . Rational (i ,2) ) * GH

, Rational ( i +1 ,2) ) * GH [ i +1]( x ) ) for i in range

, Rational ( nb ,2) ) * GH, vol.x ) ) ) ( " ----------------------------------------------------------

, Equation differentielle sur phi , O ( h^" + str ( nbTerm +3) + " ) " )

, ( i ) ) /( sy . pi **( sy . in ( l for l in range ( nb ) if l %2) : 75 proj += c

. Sy,

, ) ** q * n0 *( sy . factorial ( q ) -eta [q -3]) ) in range (1 , nbTerm +3) : 96 n10 += n [ q ]*( x * h / t ) ** q /( n0 * sy . factorial ( q ) ) in range (1 , nbTerm +3) : 104 ina += (( -1) ** q * n10 ** q ), p.167

, diff ( x ) , sy . sqrt ( sy . Rational (i ,2) ) * GH [i -1]( x ) 169 -sy . sqrt ( sy . Rational ( i +1 ,2) ) * GH [ i +1]( x ) ) for

, Rational ( nb ,2) ) * GH

, ( x ) , sy . sqrt ( sy . Rational (i ,2) ) * GH

, Rational ( i +1 ,2) ) * GH [ i +1]( x ) ) for i in range

*. Gh, , vol.x ) ) ) ( " ----------------------------------------------------------

, Equation differentielle sur psi , O ( h^" + str ( nbTerm +1) + " ) " )

, Equation differentielle sur phi , O ( h^" + str ( nbTerm +3) + " ) " )

, Equation saut fonctions , O ( h^" + str ( nbTerm +1) + " ) " )

, Equation saut derivees , O ( h^" + str ( nbTerm +1) + " ) " )

, * psi ) . subs (r ,0) -h * N * phi . subs ( trace1 ) ), vol.expand () ( " ----------------------------------------------------------, pp.1-1

.. .. Puits-d'ordre-Élevé,

R. E. , Pour une étude plus générale des puits plats

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