By Manfred Möller, Vyacheslav Pivovarchik

The theoretical a part of this monograph examines the distribution of the spectrum of operator polynomials, targeting quadratic operator polynomials with discrete spectra. the second one half is dedicated to purposes. normal spectral difficulties in Hilbert areas are of the shape A-λI for an operator A, and self-adjoint operators are of specific curiosity and value, either theoretically and when it comes to functions. A attribute function of self-adjoint operators is that their spectra are genuine, and lots of spectral difficulties in theoretical physics and engineering may be defined through the use of them. even if, a wide type of difficulties, specifically vibration issues of boundary stipulations reckoning on the spectral parameter, are represented through operator polynomials which are quadratic within the eigenvalue parameter and whose coefficients are self-adjoint operators. The spectra of such operator polynomials are commonly not more actual, yet nonetheless convey sure styles. The distribution of those spectra is the main target of the current quantity. For a few sessions of quadratic operator polynomials, inverse difficulties also are thought of. the relationship among the spectra of such quadratic operator polynomials and generalized Hermite-Biehler services is mentioned in detail.

Many functions are completely investigated, corresponding to the Regge challenge and damped vibrations of soft strings, Stieltjes strings, beams, megastar graphs of strings and quantum graphs. a few chapters summarize complex heritage fabric, that is supplemented with precise proofs. with reference to the reader’s history wisdom, basically the fundamental homes of operators in Hilbert areas and recognized effects from advanced research are assumed.

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In this case, σj = 0 for all j = 1, . . , p and m = 0. 42 Chapter 2. 4) we obtain ∂2 vj (λ, sj ) − iλσj (sj )vj (λ, sj ) + λ2 ρj (sj )vj (λ, sj ) = 0, ∂s2j j = 1, . . 5) vj (λ, 0) = 0, j = 1, . . 7) ∂ vj (λ, lj ) − λ2 mv1 (λ, l1 ) + iλνv1 (λ, l1 ) = 0. 8) is described by the operator pencil L(λ) = λ2 M − iλK − A, p L2 (0, lj ) ⊕ C. A is deﬁned by where A, K and M act in the Hilbert space j=1 ⎛ ⎛ ⎞ v1 ⎜ ⎜ .. ⎟ ⎜ A⎜ ⎟ = ⎜ ⎝vp ⎠ ⎜ ⎝ c −v1 .. ⎞ ⎟ ⎟ ⎟ , −vp ⎟ ⎟ p ⎠ vj (lj ) j=1 ⎧⎛ ⎞ v1 ⎪ ⎪ ⎪ ⎨⎜ ..

1. 3. If κ > 0, then the numbers i − i|λ−j − i |, j = 1, . . , κ, are not eigenvalues. 4. If κ ≥ 2, then in each of the intervals (i − i|λ−(j+1) − i |, i − i|λ−j − i |), j = 1, . . , κ − 1, the number of eigenvalues, counted with multiplicity, is odd. 5. If κ > 0, then the interval (i − i|λ−1 − i |, i ] contains no or an even number ˆ = {0}, they are all diﬀerent of eigenvalues, counted with multiplicity. If N (A) from i , otherwise one of them is the simple eigenvalue i . 4 shows a priori in case ν < 2m that the spectrum lies in a horizontal strip of the complex plane.

D(A) = ⎜ . ⎟ : vj ∈ W22 (0, lj ), vj (0) = 0, vj (lj ) = c, j ⎪⎝vp ⎠ ⎪ ⎪ ⎩ c whereas ⎛ ⎛ ⎞ σ1 . . 0 ρ1 . . 0 0 ⎜ .. ⎜ .. . .. ⎟ .. ⎜ ⎜ . . ⎟ . M =⎜. ⎟, K = ⎜ . ⎝ 0 . . σp ⎝ 0 . . ρp 0 ⎠ 0 ... 0 m 0 ... 0 ⎫ ⎪ ⎪ ⎪ ⎬ = 1, . . , p ⎪ ⎪ ⎪ ⎭ , ⎞ 0 .. ⎟ ⎟. 1. The operators A, K and M are self-adjoint, M ≥ 0 and K ≥ 0 are bounded, M + K 0, and A 0 has a compact resolvent. If m > 0, then M 0, and if σ1 ≥ ε, . . , σp ≥ ε for some ε > 0, then K 0. Proof. The statements about M and K are obvious, so that we turn our attention [1] to A.