On the Birkhoff-Tamarkin-Langer conditions and a conjecture of Davies

Abstract

Let L be the operator acting on the space L 2 ((0, 1); ‫ރ‬ n ) as(1) with domain(2) where A 0 (x), A 1 (x), S, and T are matrices of the nth order such that the elements of A 0 (x) and A 1 (x) are measurable on [0; 1] and Then the fundamental solution matrix U(x, μ) of the system(3) satisfying the condition U(0, μ) = I is an entire func tion of μ for each fixed x, so that the eigenvalues of L coincide with the zeros of the entire function (4) Therefore, the spectrum of the operator L either coin cides with ‫,ރ‬ is empty, or is discrete. We assume that the matrices S and T are such that the spectrum is dis crete.The spectral theory of operators of the form L goes back to classical works of Birkhoff [1] and Tamarkin [2,3], who studied systems of the form (3), where A 0 and A 1 are sufficiently smooth n × n matrix, the matrix A 0 (x) is nonsingular and diagonalizable for each x ∈ [0, 1], and the eigenvalues d 1 , d 2 , …, d n of the matrix satisfy the conditionsBirkhoff and Tamarkin independently showed that if relations (BT) hold, then the fundamental solution matrix of system (3) with large μ admits asymptotic expansions uniform in and x ∈ [0, 1]. This has made it possible to define the class of regular boundary conditions and prove theorems on generalized eigen vector expansions in boundary value problems for sys tem (3) with boundary conditions from the class spec ified above.Subsequently, boundary value problems for systems of the form (3) have been studied by many authors from various points of view (see [5,6] and the refer ences therein). However, in all of these studies, except in rare cases, conditions (BT) were invariably present. At the same time, operators of the form L for which conditions (BT) do not hold are of both practical and theoretical interest (see [6][7][8] and Chapter 10 in monograph [5]). On the other hand, conditions (BT) themselves are rather restrictive, so that it is worth try ing to relax them. The first such attempt was made in 1939 by Langer (see [9]), who showed that Birkhoff and Tamarkin's results remain valid in the case where (L 1 ) the matrices A 0 and A 1 are analytic in some neighborhood Ω of the interval [0, 1];(L 2 ) for any point z ∈ Ω and any pair (i, j), there exists a curve γ ij (z) contained entirely in Ω, joining the point z with 0 or 1, and such that the argument of the function d i (t) -d j (t))dt remains constant as the point ζ moves along this curve. On the one hand, Langer's conditions (L 1 ) and (L 2 ) are only slightly weaker than (BT); on the other hand, they do not rid us of conditions (BT) but only make it possible to transform the initial equation (3), by means of an appropriate change of the independent variable, into another equation, which already satisfies condi tions (BT) with respect to the new variable. Without Langer's conditions, it is unclear how to obtain asymptotic expansions for the fundamental solution μ arg ( ζ z ∫

On the class of potentials with trivial monodromy

“…где A-кусочно-постоянная невырожденная матрица n × n [3,4]. Как отмечено в [5], если собственные числа d 1 , . .…”

“…Если оператор T не близок к самосопряженному, то, как правило, спектрально неустойчив: резольвентная норма ∥(T − λ) −1 ∥ может быть большой и при λ, далеких от спектра (см. [3,5,[9][10][11][12][13][14] и имеющиеся там ссылки). Поэтому в случае, когда T -не близкий к самосопряженному дифференциальный оператор, для исследования спектра приходится накладывать на коэффициенты соответствующего дифференциального выражения дополнительное (гораздо более жесткое по сравнению с близким к самосопряженному случаем) условие голоморфности в некоторой окрестности соответствующего промежутка [13,[15][16][17][18][19][20].…”