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March 14, 2008
###### Abstract

We consider two-point non-self-adjoint boundary eigenvalue problems for linear matrix differential operators. The coefficient matrices in the differential expressions and the matrix boundary conditions are assumed to depend analytically on the complex spectral parameter and on the vector of real physical parameters . We study perturbations of semi-simple multiple eigenvalues as well as perturbations of non-derogatory eigenvalues under small variations of . Explicit formulae describing the bifurcation of the eigenvalues are derived. Application to the problem of excitation of unstable modes in rotating continua such as spherically symmetric MHD -dynamo and circular string demonstrates the efficiency and applicability of the theory.

Multiparameter perturbation theory for non-self-adjoint operator matrices] Perturbation of multiparameter
problems for operator matrices O.N. Kirillov]O.N. Kirillov

This work was completed with the support of the Alexander von Humboldt Foundation and the grant DFG HA 1060/43-1.\par\@mkboth\shortauthors\shorttitle \par [\par

Mathematics Subject Classification (2000). Primary 34B08; Secondary 34D10.

Keywords. operator matrix, non-self-adjoint boundary eigenvalue problem, Keldysh chain, multiple eigenvalue, diabolical point, exceptional point, perturbation, bifurcation, stability, veering, spectral mesh, rotating continua.

\par\@xsect\par

 L(λ,p)u=0,Uk(λ,p)u=0,k=1,…,m, (\theequation)

where . The differential expression of the operator is

 Lu=m∑j=0lj(x)∂m−jxu,lj(x)∈CN×N⊗C(m−j)[0,1],det[l0(x)]≠0, (\theequation)

and the boundary forms are

 Uku=m−1∑j=0Akju(j)x(x=0)+m−1∑j=0Bkju(j)x(x=1),Akj,Bkj∈CN×N. (\theequation)

Introducing the block matrix and the vector

 uT:=(uT(0),u(1)Tx(0),…,u(m−1)Tx(0),uT(1),u(1)Tx(1),…,u(m−1)Tx(1))∈C2mN (\theequation)

the boundary conditions can be compactly rewritten as [26, 27]

 Uu=[A,B]u=0, (\theequation)

where and . It is assumed that the matrices , , and are analytic functions of the complex spectral parameter and smooth functions of the real vector of physical parameters . For some fixed vector the eigenvalue , to which the eigenvector corresponds, is a root of the characteristic equation obtained after substitution of the general solution to equation into the boundary conditions (\@setrefs7) [8]. \parLet us introduce a scalar product [8]

 :=∫10v∗udx, (\theequation)

where the asterisk denotes complex-conjugate transpose (). Taking the scalar product of and a vector-function and integrating it by parts yields the Lagrange formula for the case of operator matrices (cf. [8, 22, 26, 27])

 Ω(u,v):==v∗Lu, (\theequation)

with the adjoint differential expression [8, 22]

 L†v:=m∑q=0(−1)m−q∂m−qx(l∗qv), (\theequation)

the vector

 vT:=(vT(0),v(1)Tx(0),…,v(m−1)Tx(0),vT(1),v(1)Tx(1),…,v(m−1)Tx(1))∈C2mN (\theequation)

and the block matrix

 (\theequation)

where the matrices are

 lij := m−1−j∑k=i(−1)kMkij∂k−ixlm−1−j−k, Mkij := ⎧⎪ ⎪⎨⎪ ⎪⎩k!(k−i)!i!,i+j≤m−1∩k≥i≥00,i+j>m−1∪k
\par

Extend the original matrix (cf. (\@setrefs7)) to a square matrix , which is made non-degenerate in a neighborhood of the point and the eigenvalue by an appropriate choice of the auxiliary matrices and

 U=[A,B]↪U:=(AB˜A˜B)∈C2mN×2mN,˜U:=[˜A,˜B],det(U)≠0. (\theequation)

Similar transformation for the adjoint boundary conditions yields

 V:=[C,D]↪V:=(CD˜C˜D)∈C2mN×2mN,˜V:=[˜C,˜D],det(V)≠0. (\theequation)

Then, the form in (\@setrefs13) can be represented as [8]

 Ω(u,v)=(Vv)∗˜Uu−(˜Vv)∗Uu, (\theequation)

so that without loss in generality we can assume [26, 27]

 L=V∗˜U−˜V∗U. (\theequation)

Differentiating the equation (\@setrefs28) we find

 ∂rλL=r∑k=0(rk)[(∂r−k¯λV)∗∂kλ˜U−(∂r−k¯λ˜V)∗∂kλU]. (\theequation)

Hence, we obtain the formula for calculation of the matrix of the adjoint boundary conditions and the auxiliary matrix

 [−˜VV]∗=LU−1=(−L(0)00L(1))(AB˜A˜B)−1, (\theequation)

which exactly reproduces and extends the corresponding result of [26, 27]. \par\par\@xsect \parAssume that in the neighborhood of the point the spectrum of the boundary eigenvalue problem (\@setrefs1) is discrete. Denote and . Let us consider a smooth perturbation of parameters in the form where and is a small real number. Then, as in the case of analytic matrix functions [25], the Taylor decomposition of the differential operator matrix and the matrix of the boundary conditions are [19, 24, 26, 27]

 L(λ,p(ε))=∞∑r,s=0(λ−λ0)rr!εsLrs,U(λ,ε)=∞∑r,s=0(λ−λ0)rr!εsUrs, (\theequation)

with

 L00 = L0,Lr0=∂rλL,Lr1=n∑j=1˙pj∂rλ∂pjL, U00 = U0,Ur0=∂rλU,Ur1=n∑j=1˙pj∂rλ∂pjU, (\theequation)

where dot denotes differentiation with respect to at and partial derivatives are evaluated at , . Our aim is to derive explicit expressions for the leading terms in the expansions for multiple-semisimple and non-derogatory eigenvalues and for the corresponding eigenvectors. \par\par\@xsect \parLet at the point the spectrum contain a semi-simple -fold eigenvalue with linearly-independent eigenvectors , , , . Then, the perturbed eigenvalue and the eigenvector are represented as Taylor series in [4, 24, 28, 29, 33]

 λ=λ0+ελ1+ε2λ2+…,u=b0+εb1+ε2b2+…. (\theequation)
\par

Substituting expansions (\@setrefp2) and (\@setrefp6) into (\@setrefs1) and collecting the terms with the same powers of we derive the boundary value problems

 L0b0=0,\nobreak \ignorespacesU0b0=0, (\theequation)
 L0b1+(λ1L10+L01)b0=0,\nobreak \ignorespacesU0b1+(λ1U10+U01)b0=0, (\theequation)

Scalar product of (\@setrefp8) with the eigenvectors , of the adjoint boundary eigenvalue problem

 L†0v=0,\nobreak \ignorespacesV0v=0 (\theequation)

yields equations

 =−−λ1. (\theequation)

With the use of the Lagrange formula (\@setrefs13), (\@setrefs28) and the boundary conditions (\@setrefp8) the left hand side of (\@setrefp11) takes the form

 =(˜V0vj)∗(U01b0+λ1U10b0). (\theequation)

Together (\@setrefp11) and (\@setrefp13) result in the equations

 λ1(+(˜V0vj)∗U10b0)=−−(˜V0vj)∗U01b0. (\theequation)

Assuming in the equations (\@setrefp14) the vector as a linear combination

 b0(x)=c0u0(x)+c1u1(x)+…+cμ−1uμ−1(x), (\theequation)

and taking into account that

 b0=c0u0+c1u1+…+cμ−1uμ−1, (\theequation)

where , we arrive at the matrix eigenvalue problem (cf. [LMZ03])

 −Fc=λ1Gc. (\theequation)

The entries of the matrices and are defined by the expressions

 (\theequation)

Therefore, in the first approximation the splitting of the semi-simple eigenvalue due to variation of parameters is , where the coefficients are generically distinct roots of the -th order polynomial

 det(F+λ1G)=0. (\theequation)

For the formulas (\@setrefp18) and (\@setrefp19) describe perturbation of a simple eigenvalue

 λ=λ0−ε+v∗0˜V∗0U01u0+v∗0˜V∗0U10u0+o(ε). (\theequation)

The formulas (\@setrefp18),(\@setrefp19), and (\@setrefp20) generalize the corresponding results of the works [24] to the case of the multiparameter non-self-adjoint boundary eigenvalue problems for operator matrices. \par\par\@xsect \parLet at the point the spectrum contain a -fold eigenvalue with the Keldysh chain of length , consisting of the eigenvector and the associated vectors . The vectors of the Keldysh chain solve the following boundary value problems [8, 22]

 L0u0=0,U0u0=0, (\theequation)
 L0uj=−j∑r=11r!∂rλLuj−r,U0uj=−j∑r=11r!∂rλUuj−r. (\theequation)
\par

Consider vector-functions , , , . Let us take scalar product of the differential equation (\@setrefp21) and the vector-function . For each we take the scalar product of the equation (\@setrefp22) and the vector-function . Summation of the results yields the expression

 μ−1∑j=0j∑r=01r!<∂rλLuj−r,vμ−1−j>=0 (\theequation)

Applying the Lagrange identity (\@setrefs13), (\@setrefs28) and taking into account relation (\@setrefs29), we transform (\@setrefp23) to the form

 μ−1∑j=0 + μ−1∑k=0μ−1−k∑j=0[j∑r=0(1r!∂r¯λVvj−r)∗]∂kλ˜Uk!uμ−1−j−k = 0. (\theequation)

Equation (\@setrefp24) is satisfied in case when the vector-functions , , , originate the Keldysh chain of the adjoint boundary value problem, corresponding to the -fold eigenvalue [26, 27]

 L†0v0=0,\nobreak \ignorespacesV0v0=0, (\theequation)
 L†0vj=−j∑r=11r!∂r¯λL†vj−r,\nobreak \ignorespacesV0vj=−j∑r=11r!∂r¯λVvj−r. (\theequation)
\par

Taking the scalar product of equation (\@setrefp22) and the vector and employing the expressions (\@setrefs13), (\@setrefs28) we arrive at the orthogonality conditions

 j∑r=11r![<∂rλLuj−r,v0>+v∗0˜V∗0∂rλUuj−r]=0,j=1,…,μ−1. (\theequation)
\par

Substituting into equations (\@setrefs1) the Newton-Puiseux series for the perturbed eigenvalue and eigenvector [7, 13, 26, 27]

 λ=λ0+λ1ε1/μ+…,u=w0+w1ε1/μ+…, (\theequation)

where , taking into account expansions (\@setrefp2) and (\@setrefp26) and collecting terms with the same powers of , yields boundary value problems serving for determining the functions ,

 L0wr=−r−1∑j=0⎛⎝r−j∑σ=11σ!Lσ0∑|α|σ=r−jλα1…λασ⎞⎠wj, (\theequation)
 U0wr=−r−1∑j=0r−j∑σ=1⎛⎝∑|α|σ=r−jλα1…λασ⎞⎠1σ!Uσ0wj, (\theequation)

where and , , are positive integers. The vector-function is a solution of the following boundary value problem

 L0wμ=−L01w0−μ−1∑j=0⎛⎝μ−j∑σ=11σ!Lσ0∑|α|σ=μ−jλα1…λασ⎞⎠wj, (\theequation)
 U0wμ=−U01w0−μ−1∑j=0μ−j∑σ=1⎛⎝∑|α|σ=μ−jλα1…λασ⎞⎠1σ!Uσ0wj. (\theequation)

Comparing equations (\@setrefp31) and (\@setrefp32) with the expressions (\@setrefp22) we find the first functions in the expansions (\@setrefp26)

 wr=r∑j=1uj∑|α|j=rλα1…λαj. (\theequation)

With the vectors (\@setrefp33) we transform the equations (\@setrefp31) and (\@setrefp32) into

 L0wμ=−L01u0−λμ1μ∑r=11r!∂rλLuμ−r+μ−1∑j=1L0uj∑|α|j=μλα1…λαj, (\theequation)
 U0wμ=−U1u0−λμ1μ∑r=11r!∂rλUuμ−r+μ−1∑j=1U0uj∑|α|j=μλα1…λαj. (\theequation)

Applying the expression following from the Lagrange formula

 = v∗0˜V∗0U01u0+λμ1μ∑r=11r!v∗0˜V∗0Ur0uμ−r (\theequation) − μ−1∑j=1v∗0˜V∗0U0uj∑|α|j=μλα1…λαj,

and taking into account the equations for the adjoint Keldysh chain (\@setrefp24a) and (\@setrefp25a) yields the coefficient in (\@setrefp26). Hence, the splitting of the -fold non-derogatory eigenvalue due to perturbation of the parameters is described by the following expression, generalizing the results of the works [19, 24, 26, 27]

 λ=λ0+μ  ⎷−ε+v∗0˜V∗0U01u0∑μr=11r!(+v∗0˜V∗0Ur0uμ−r)+o(ε1μ). (\theequation)

For equation (\@setrefp37) is reduced to the equation (\@setrefp20) for a simple eigenvalue. \par\par\@xsect \parConsider a circular string of displacement , radius , and mass per unit length that rotates with the speed and passes at through a massless eyelet generating a constant frictional follower force on the string, as shown in Fig. \@setreffig1. The circumferential tension in the string is assumed to be constant; the stiffness of the spring supporting the eyelet is and the damping coefficient of the viscous damper is . Introducing the non-dimensional variables and parameters

 t=τr√Pρ,w=Wr,Ω=γr√ρP,k=KrP,μ=FP,d=D√ρP, (\theequation)

and assuming we arrive at the non-self-adjoint boundary eigenvalue problem for a scalar differential operator [15]

 Lu=λ2u+2Ωλu′−(1−Ω2)u′′=0, (\theequation)
 u(0)−u(2π)=0,u′(0)−u′(2π)=λd+k1−Ω2u(0)+μ1−Ω2u′(0), (\theequation)

where prime denotes differentiation with respect to . Parameters , , , and express the speed of rotation, and damping, stiffness, and friction coefficients. \par

\par

For the unconstrained rotating string with , , and the eigenfunctions and of the adjoint problems, corresponding to purely imaginary eigenvalue and , coincide. With assumed as a solution of (\@setrefst4), the characteristic equation follows from (\@setrefst5)

 8λsinπλi(1−Ω)sinπλi(1+Ω)e−2πλΩΩ2−1Ω2−1=0. (\theequation)
\par\par

The eigenvalues with the eigenfunctions , , found from equation (\@setrefst8) form the spectral mesh in the plane , Fig. \@setreffig1. The lines and , where , intersect each other at the node with

 Ωεδmn=n−mmδ−nε,λεδmn=inm(δ−ε)mδ−nε, (\theequation)

where the double eigenvalue has two linearly independent eigenfunctions

 uεn=cos(nφ)−εisin(nφ),uδm=cos(mφ)−δisin(mφ). (\theequation)
\par\par\par

Using the perturbation formulas for semi-simple eigenvalues (\@setrefp18) and (\@setrefp19) with the eigenelements (\@setrefst11) and (\@setrefst13) we find an asymptotic expression for the eigenvalues originated after the splitting of the double eigenvalues at the nodes of the spectral mesh due to interaction of the rotating string with the external loading system

 λ=λεδnm+iεn+δm2ΔΩ+in+m8πnm(dλεδnm+k)+ε+δ8πμ±√c, (\theequation)

with and

 c = (iεn−δm2ΔΩ+im−n8πmn(dλεδnm+k)+ε−δ8πμ)2 (\theequation) − (dλεδnm+k−iεnμ)(dλεδnm+k−iδmμ)16π2nm.
\par
\par

Due to action of gyroscopic forces and an external spring double eigenvalues split in the subcritical region (, and ) as

 λ=λεδnm+im−n2ΔΩ+in+m8πnmk±i√k216π2nm+(m−n8πmnk−m+n2ΔΩ)2, (\theequation)

while in the supercritical region (, and , )

 λ=λεδnm+im+|n|2ΔΩ+i|n|−m8π|n|mk± ⎷k216π2|n|m−(|n|−m2ΔΩ−m+|n|8πm|n|k)2. (\theequation)

Therefore, for the spectral mesh collapses into separated curves demonstrating avoided crossings; for the eigenvalue branches overlap forming the bubbles of instability with eigenvalues having positive real parts, see Fig. \@setreffig2. From (\@setrefst22) a linear approximation follows to the boundary of the domains of supercritical flutter instability in the plane (gray resonance tongues in Fig. \@setreffig2)

 k=4π|n|m(|n|−m)(√|n|±√m)2(Ω−|n|+m|n|−m). (\theequation)
\par
\par

At the nodes with the external damper creates a circle of complex eigenvalues being a latent source of subcritical flutter instability responsible for the emission of sound in the squealing brake and the singing wine glass [38]

 (Reλ+d4π)2+n2Ω2=d216π2,Imλ=n, (\theequation)
 n2Ω2−(Imλ−n)2=d216π2,Reλ=−d4π, (\theequation)

as shown in Fig. \@setreffig3. Non-conservative perturbation yields eigenvalues with

 Imλ=n±12π√2π2n2Ω2±πnΩ√4π2n2Ω2+μ2, (\theequation)
 Reλ=±12π√−2π2n2Ω2±πnΩ√4π2n2Ω2+μ2, (\theequation)

so that both the real and imaginary parts of the eigenvalue branches show a degenerate crossing, touching at the node , Fig. \@setreffig3. Deformation patterns of the spectral mesh and first-order approximations of the instability tongues obtained by the perturbation theory and shown in Fig. \@setreffig2 and Fig. \@setreffig3 are in a good qualitative and quantitative agreement with the results of numerical calculations of [15]. \par\par\@xsect \parConsider a non-self-adjoint boundary eigenvalue problem appearing in the theory of MHD -dynamo [14, 32, 33, 37] \par

 Lu:=l0∂2xu+l1∂xu+l2u=0,Uu:=[A,B]u=0, (\theequation)

with the matrices of the differential expression

 l0=(10−α(x)1),l1=∂xl0,l2=⎛⎜⎝−l(l+1)x2−λα(x)α(x)l(l+1)x2−l(l+1)x2−λ⎞⎟⎠, (\theequation)

and of the boundary conditions

 A=⎛⎜ ⎜ ⎜⎝1000010000000000⎞⎟ ⎟ ⎟⎠,B=⎛⎜ ⎜ ⎜⎝00000000βl+1−β0β00100⎞⎟ ⎟ ⎟⎠, (\theequation)

where it is assumed that with . For the fixed the differential expression depends on the parameters and , while interpolates between the idealistic () and physically realistic () boundary conditions [14, 32, 33, 36]. \parThe matrix of the boundary conditions and auxiliary matrix for the adjoint differential expression follow from the formula (\@setrefs31) where the matrices , and are chosen as

 ˜A=(0I00),˜B=(000I). (\theequation)
\par

In our subsequent consideration we assume that and interpret and as perturbing parameters. It is known [33] that for and the spectrum of the unperturbed eigenvalue problem (\@setrefdm1) forms the spectral mesh in the plane , as shown in Fig. \@setreffig4. The eigenelements of the spectral mesh are

 λεn=−(πn)2+εα0πn,λδm=−(πm)2+δα0πm,ε,δ=±1, (\theequation)
 (\theequation)

The branches (\@setrefdm6) intersect and originate a double semi-simple eigenvalue with two linearly independent eigenvectors (\@setrefdm7) at the node , where [33]

 λν0=εδπ2nm,αν0=επn+δπm. (\theequation)

Taking into account that the components of the eigenfunctions of the adjoint problems are related as and , we find from equations (\@setrefp18) and (\@setrefp19) the asymptotic formula for the perturbed eigenvalues, originating after the splitting of the double semi-simple eigenvalues at the nodes of the spectral mesh

 λ = λν0−εδπ2mnβ+π2(δm+εn)Δα0 ± π2√((δm−εn)Δα0)2+4mn(εγΔα−(−1)n+mπnβ)(δγΔα−(−1)n+mπmβ),

where

 Δα0:=αν0−α0,Δα:=∫10Δα(x)cos((εn−δm)πx)dx. (\theequation)

When and , one of the two simple eigenvalues (\@setrefdm10) remains unshifted in all orders of the perturbation theory with respect to the parameter : . The sign of the first-order increment to another eigenvalue depends on the sign of , which is directly determined by the Krein signature of the modes involved in the crossing [33]. This is in the qualitative and quantitative agreement with the results of numerical calculations of [36] shown in Fig. \@setreffig4. \par