Introducing a power of the operator in direct spectral problems

Even though the spectral theory of operators began in the nineteenth century, it is still actively developing. It includes separate directions like the problem of computing regularized trace, inverse problems, the calculation of spectral characteristics. This work, representing the school of Sadovnichiy and Dubrovsky, is devoted to the calculation of the regularized trace of the Laplace operator on a parallelepiped of arbitrary dimension. The problem of finding the regularized trace, besides its mathematical significance, has physical meaning. For example, the formula for the regularized trace of the Sturm-Liouville equation describes energy conservation in the dynamical system described by the Korteweg-de Vries equation [1]. With the aid of regularized traces calculated by Liphshic [2], a modification of the free energy of a crystal with implemented admixtures was found.

The theory of regularized traces originated with the work of Gelfand and Levitan (1953) [3], who found the asymptotics of the eigenvalues of the Sturm–Liouville operator. In the 1960s Lidskii and Sadovnichiy almost completed this theory for ordinary differential operators. They succeeded in calculating the regularized traces of boundary value problems for ordinary differential equations of arbitrary order with a complex occurrence of the parameter. Partial differential operators are much more difficult to study. Kostyuchenko and Gasimov [4], as well as other authors, obtained various results in this direction. The challenge is that for partial differential equations the resolvent has special structure, and the exact asymptotics of the eigenvalues is unknown. Sadovnichiy and Dubrovsky proposed the so-called resolvent method, which enabled them to avoid these difficulties using perturbative corrections. Subsequently, basing on this method, the followers of Dubrovsky obtained a regularized trace formula for the Laplace operator [5] or a Chebyshev-type operator [6] and investigated the existence and uniqueness [7] of the perturbing operator and the stability of the solution [8]. They proposed algorithms for reconstructing the potential for models with the Laplace operator [9]. The approach of these studies rests on the principle of lowering the power of the operator, as the Laplace operator itself is widely applied in physics rather than its powers. This paper presents a fundamentally different approach: we regard the power of an operator as an opportunity to explore physical problems not only on the interval, but also in other dimensions.

• 1.    The Regularized Trace

Take the N -dimensional parallelepiped

П = {x = (x1,x2,..., xN) : 0 < Xj < aj ,j = 1,..., N}, where aj > 0. Denote by U a separable Hilbert space with the inner product

(f, h)= [

Π

f(x)h(x)dx

and the norm ||h|| =   (h, h). In the space CAU) consider the discrete selfadjoint operator T of the Dirichlet boundary value problem

- Tv = Av, v| d_n = 0.

By a discrete operator we understand an operator whose spectrum is discrete. Assume that the eigenvalues A m E R of T , for m = (m 1 , m 2 , ..., m N ), in the ascending order, have the asymptotics A k ~ Ck N with в > 1 and C = const. For example, consider the Dirichlet boundary value problem for the Laplace operator on the rectangle. Obviously, the eigenvalues in this case satisfy the asymptotics. [10]

Next, given a bounded operator P defined everywhere on U , we can show that T + P is a discrete operator. Denote by µ _k its eigenvalues in the ascending order of their real parts.

Lemma 1. Take a discrete selfadjoint operator T E L ( U ). If the perturbing operator P satisfies ||P|| < r/2, where 0 < r < r ₀ = inf r k with r k = 1 min{A _{k +1} — A k ; A k — A _{k -1} } for k ²

k >  1 and r 1 = 2 (A 2 — A 1 ), then T + P is discrete and the following claims hold:

• (i)    if R ₀ (A) E S/q then R(A) E S/q for 1 <  q <  ж ;

• (ii)    if A k E C \ ^ r k then ц ⁸ _к E C \ ^ r k for s = 1, v _k , and v _k is the multiplicity of the eigenvalue λ _k .

Therefore, the multiplicities of the eigenvalues of T and T + P lying in the circle Y r k are equal.

Let us put the problem of finding the numbers A k so that the series

У^ (Ц к — A k — A k )

k converges and determining the sum of the series. This series is called the regularized trace of the operator T + P.

As we can see from the asymptotics, for all N > 2 the series

∞

∞

£ Xt

t =0

diverges. Therefore, we cannot apply the resolvent method when N >  2. To resolve this difficulty and treat arbitrary dimension, we introduce a new boundary value problem with the power T ^β of T :

^{(—T e} + P )v = Xv, v| d n = °.

The eigenfunctions of these problems coincide, but the eigenvalues of T ^β are equal to λ _k ^β . The asymptotics of the eigenvalues of the new operator shows that it is possible to consider the problem in arbitrary dimension when a suitable k is found.

Assume that the suitable k is found. Following[9], we obtain

Theorem 1. If β > N/2 and ||P|| <  r/2, where 0 < r < r ₀ , then for all k E N we have

ν k                      ν k

52 ^ k = ^V k ^{X e} - ^(Pv k , v k ) ^{+ a} k (P), s =1                 s =1

where

a k (p) =

21 /rk ^XTr [^R(X) И^х)) 2]

dλ.

Evaluating perturbative corrections, we see that they vanish for all n >  2. Thus, we obtain

Theorem 2. If β > N/2 and ||P|| < r/2, where 0 < r < r ₀ , then for all k E N we have the regularized trace formula

ν k                      ν k

E ^ k = v k х в - E(Pv k ,v k ).

s =1                 s =1