By Manfred Möller, Vyacheslav Pivovarchik

The theoretical a part of this monograph examines the distribution of the spectrum of operator polynomials, concentrating on quadratic operator polynomials with discrete spectra. the second one half is dedicated to purposes. usual spectral difficulties in Hilbert areas are of the shape A-λI for an operator A, and self-adjoint operators are of specific curiosity and significance, either theoretically and when it comes to functions. A attribute function of self-adjoint operators is that their spectra are actual, and plenty of spectral difficulties in theoretical physics and engineering might be defined by utilizing them. in spite of the fact that, a wide classification of difficulties, specifically vibration issues of boundary stipulations counting 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 in most cases not more genuine, yet nonetheless show sure styles. The distribution of those spectra is the focus of the current quantity. For a few periods of quadratic operator polynomials, inverse difficulties also are thought of. the relationship among the spectra of such quadratic operator polynomials and generalized Hermite-Biehler capabilities is mentioned in detail.

Many functions are completely investigated, reminiscent of the Regge challenge and damped vibrations of delicate strings, Stieltjes strings, beams, superstar graphs of strings and quantum graphs. a few chapters summarize complex history fabric, that's supplemented with specified proofs. with reference to the reader’s history wisdom, in basic terms the fundamental homes of operators in Hilbert areas and famous effects from advanced research are assumed.

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**Sample text**

Hence this multiplicity is ﬁnite, and since the spectrum of type I is symmetric with respect to the origin, there are at most ﬁnitely many eigenvalues of type I on the positive imaginary axis, and therefore at most ﬁnitely many of the eigenvalues of type I are on the imaginary axis. 5. 1. For all λ, η ∈ C, N (L(λ, η))∩N (K) = N (L(λ))∩N (K), that is, N (L(λ, η))∩ N (K) is independent of η. In particular, m0 (λ) ≤ mI (λ). 2. Let λ = 0, η ∈ (0, 1] and assume that N (L(λ)) ∩ N (K) = {0}. Then no eigenvector y0 ∈ N (L(λ)) ∩ N (K) of L(·, η) at λ has an associated vector.

By Theorem yk on [0, ε), i. 5. 6, part 3, we may assume that Kyj (η) = 0 for at least one η and therefore for all η ∈ (0, ε) and suﬃciently small ε. 7) is also valid for negative indices ˆ (η) > 0 j under the assumptions made in this section, which shows that Im λ −j ˆ (η) = 0 for for all η ∈ (0, ε). 6) shows that λ j η ∈ (0, ε). For all indices j = 1, . . , κ2 , we therefore have ˆ j (η) ˆ j (η) (M yj (η), yj (η)) λ 2λ − η. 4) If (M yj (0), yj (0)) = 0, then (Kyj (0), yj (0)) > 0. 4) tends to 0 as η → 0.

Hence, consider an eigenvalue λ = −iτ of L(·, and a corresponding normed eigenvector y. Then (−τ 2 [(1 − η)I + ηM ] − τ K − A)y = 0, which leads to τ 2 ([(1 − η)I + ηM ]y, y) + τ (Ky, y) + (Ay, y) = 0. 8) Since M + K 0, there is ε > 0 such that M + K ≥ εI. 8) gives τ= −(Ay, y) β ≤ . 9) 16 Chapter 1. 8) is diﬀerent from 0 and hence positive, then τ > 0 implies that (Ay, y) ≤ 0, and (Ky, y)2 − 4([(1 − η)I + ηM ]y, y)(Ay, y) 2([(1 − ηI)I + ηM ]y, y) −2(Ay, y) = . 2 (Ky, y) + (Ky, y) − 4([(1 − η)I + ηM ]y, y)(Ay, y) √ √ Using that (Ay, y) ≤ 0 and that r + s ≥ 12 s for r, s ≥ 0, we obtain τ= −(Ky, y) + τ≤ (Ky, y) + −2(Ay, y) .