The Asymptotic Behaviour of the Sum of Negative Eigenvalues of a Self-Adjoint Operator Given in Semi-Axis

TL;DR

This paper derives asymptotic formulas for the sum of negative eigenvalues of a specific self-adjoint differential operator on a semi-axis, providing insights into spectral properties relevant to mathematical physics.

Contribution

It introduces new asymptotic formulas for the sum of negative eigenvalues of a class of self-adjoint operators defined by differential expressions on the semi-axis.

Findings

Derived asymptotic formulas for eigenvalue sums

Analyzed spectral behavior of differential operators

Extended understanding of eigenvalue distribution in semi-infinite domains

Abstract

In this work, we find the asymptotic formulas for the sum of the negative eigenvalues smaller than $-\varepsilon$ $(\varepsilon >0)$ of a self-adjoint operator $L$ which is defined by the following differential expression $$\ell(y)=-(p(x)y'(x))'-Q(x)y(x)$$ with the boundary condition $y(0) = 0$ in the space in the space $L_{2}(0,\infty ;H)$.

Equations

$$\ell (y)=-(p(x)y^{\prime }(x){)}^{\prime }-Q(x)y(x)$$

$$y(0)=0$$

$$c_1\le p(x)\le c_2.$$

$$\underset{x\to \infty }{\lim }\Vert Q(x)\Vert =0.$$

$$(Ly)(x)=-(p(x)y^{\prime }(x){)}^{\prime }-Q(x)y(x).$$

$$\sum _{-{\lambda }_i<-\epsilon }{\lambda }_i(\epsilon >0),$$

$${\alpha }_1(x)=\underset{\Vert f\Vert =1}{\sup }(Q(x)f,f),$$

$$\Vert Q(x)\Vert =\underset{\Vert f\Vert =1}{\sup }|(Q(x)f,f)|=\underset{\Vert f\Vert =1}{\sup }(Q(x)f,f),$$

$${\psi }_j(\epsilon )=\sup \{x\in [0,\infty );{\alpha }_j(x)\ge \epsilon \}(j=1,2,\cdots )(2)$$

$$y(0)$$

$$y^{\prime }(0)$$

$$y(x_{i-1})$$

$$y^{\prime }(x_{i-1})$$

$$-p(x_i)y^{\prime \prime }(x)-Q(x_i)y(x)$$

$$-p(x_{i-1})y^{\prime \prime }(x)-Q(x_{i-1})y(x)$$

$$0=x_0<x_1<\cdots <x_M={\psi }_1(\epsilon ).$$

$$S$$

$$S^0$$

$$-{\mu }_{(1)1}\le -{\mu }_{(1)2}\le \cdots \le -{\mu }_{(1)N^0(\lambda )},-{\mu }_{(2)1}\le -{\mu }_{(2)2}\le \cdots \le -{\mu }_{(2)N^{\prime }(\lambda )}(10)$$

$$N(\lambda )<N^0(\lambda ).$$

$$\Big{(}u_{i},\>\varphi\Big{)}_{(0,\psi_{1}(\varepsilon))}=\int\limits_{0}^{\psi_{1}(\varepsilon)}\Big{(}u_{i}(x),\>\varphi(x)\Big{)}\>dx=0\quad(i=1,2,\cdots,N(\lambda))$$

$$\displaystyle\Big{(}S^{0}\varphi,\>\varphi\Big{)}_{(0,\psi_{1}(\varepsilon))}$$

$$\inf_{y\in D(S),\>\|y\|_{(0,\infty)}=1\atop{y\perp u_{i}\>(i=1,2,\cdots,N(\lambda))}}\Big{(}Sy,y\Big{)}_{(0,\infty)}\>=\>\mu_{N(\lambda)+1}$$

$$\displaystyle\bigg{(}S^{0}\Big{(}\frac{\tilde{\varphi}}{\parallel\tilde{\varphi}\parallel}\Big{)},\>\Big{(}\frac{\tilde{\varphi}}{\parallel\tilde{\varphi}\parallel}\Big{)}\bigg{)}_{(0,\psi_{1}(\varepsilon))}$$

Full text

The Asymptotic Behaviour of the Sum of Negative Eigenvalues of a Self-Adjoint Operator Given in Semi-Axis

**Özlem Bakṣi **

Department of Mathematics,

Faculty of Arts and Science, Yıldız Technical University

(34210), Davutpaṣa, İstanbul, Turkey

e-mail: baksi@yildiz.edu.tr

In this work, we find the asymptotic formulas for the sum of the negative eigenvalues smaller than $\>-\varepsilon\>\quad(\varepsilon>0)\>$ of a self-adjoint operator $\>L\>$ which is defined by the following differential expression

[TABLE]

with the boundary condition

[TABLE]

in the space $\>L_{2}(0,\infty;H)$.

AMS Subj. Classification: 34B24, 47A10

Keywords : Self-adjoint operator, Sturm-Liouville operator, spectrum, negative eigenvalues, asymptotic behaviour.

1 Introduction

Let $\>H\>$ be an infinite dimensional separable Hilbert space. Let us consider the operator $\>L\>$ in the Hilbert space $\>L_{2}(0,\infty;H)\>$ defined by the differential equation

[TABLE]

and with the boundary condition $\>y(0)=0$.

Let us assume the scalar function $\>p(x)\>$ and the operator function $\>Q(x)\>$ satisfy the following conditions:

p1) For every $\>x\in[0,\>\infty)$, there are positive constants $\>c_{1},\>c_{2}\>$ such that

[TABLE]

p2) The function $\>p(x)\>$ has continuous and bounded derivative.

p3) The function $\>p(x)\>$ is not decreasing in the interval $\>[0,\>\infty)$.

Q1) For every $\>x\in[0,\>\infty)\>$ the operator $\>Q(x):H\rightarrow H\>$ is self-adjoint, compact and positive.

Q2) The operator $\>Q(x)\>$ is monotone decreasing.

Q3) $\>Q(x)\>$ is a continuous operator function with respect to the norm in $\>B(H)\>$ and

[TABLE]

$\>D(L)\>$ denotes the set of all functions $\>y(x)\in L_{2}(0,\>\infty;H)\>$ satisfying the following conditions:

y1) $\>y(x)\>$ and $\>y^{\prime}(x)\>$ are absolute continuous with respect to the norm in the space $\>H\>$ in every finite interval $\>[0,\>a]\>$

y2) $\>l(y)=-(p(x)y^{\prime}(x))^{\prime}-Q(x)y(x)\in L_{2}(0,\>\infty;H)\>$

y3) $\>y(0)=0\>$ and,

[TABLE]

It is proved that the operator $\>L:D(L)\longrightarrow L_{2}(0,\>\infty;H)\>$ is self-adjoint, semi bounded-below and the negative part of the spectrum of the operator $\>L\>$ is discrete [1]. Let $\>-\lambda_{1}\leq-\lambda_{2}\leq\cdots\leq-\lambda_{n}\leq\cdots\>$ be negative eigenvalues of the operator $\>L\>$. In this work we find an asymptotic formula for the sum

[TABLE]

as $\>\varepsilon\rightarrow+0$.

In [2] and [3], the asymptotic formulas for the sum of the negative eigenvalues of second order differential operator with scalar coefficient are calculated. In [1], [4], [5], [6], [7] the asymptotic behaviour of the number of the negative eigenvalues are investigated.

2 Some Inequalities For the Sum of the

Eigenvalues

Let $\>\alpha_{1}(x)\geq\alpha_{2}(x)\geq\cdots\geq\alpha_{j}(x)\geq\cdots\>$ be the eigenvalues of the operator

$Q(x):H\longrightarrow H$. Since the operator function $\>Q(x)\>$ is monotone decreasing, the functions $\>\alpha_{1}(x),\alpha_{2}(x),\cdots,\alpha_{j}(x),\cdots\>$ are also monotone decreasing, [5].

Moreover, since

[TABLE]

[8] and

[TABLE]

[9] then $\>\alpha_{1}(x)=\|Q(x)\|$.

On the other hand, since $\>lim_{x\rightarrow\infty}\alpha_{1}(x)=0$, then the function $\>\alpha_{1}\>$ has a continuous inverse function defined in the interval $\>(0,\alpha_{1}(0)]\>$. Let

[TABLE]

and $\>\psi_{1}\>$ denote the inverse function of $\>\alpha_{1}$. We consider the following operators:

1. Let $\>L^{0}\>$ and $\>L^{\prime}\>$ be operators in the space $\>L_{2}(0,\>\psi_{1}(\varepsilon);H),\>$ which are formed by expression $\>(1)\>$ and with the boundary conditions

[TABLE]

respectively. Here, $\>\varepsilon\in(0,\alpha_{1}(0)]$.

2. $\>L_{i}\>$ and $\>L_{i}^{\prime}\>$ be operators in the space $\>L_{2}(x_{i-1},\>x_{i};H)\>$ which are formed by expression $\>(1)\>$ and with the boundary conditions

[TABLE]

respectively.

3. $\>L_{i(1)}\>$ be operator in the space $\>L_{2}(x_{i-1},\>x_{i};H)\>$ which is formed by the differential equation

[TABLE]

and with the boundary conditions $\>y(x_{i-1})=y(x_{i})=0$.

4. Let $\>L^{\prime}_{i(1)}\>$ be operator in the space $\>L_{2}(x_{i-1},\>x_{i};H)\>$ which is formed by the differential equation

[TABLE]

and with boundary conditions $\>y^{\prime}(x_{i-1})=y^{\prime}(x_{i})=0$.

Let us divide the interval $\>[0,\>\psi_{1}(\varepsilon)]\>$ by the intervals at the length

[TABLE]

Here, $\>a\in(0,1)\>$ is a constant number and $\>\varepsilon\>$ is any positive number satisfying the inequality $\>\psi_{1}^{a}(\varepsilon)\geq 2$. And also $\>[|\psi_{1}^{a}(\varepsilon)|]\>$ shows exact part of $\>\psi_{1}^{a}(\varepsilon)$.

Let the partition points of the interval $\>[0,\>\psi_{1}(\varepsilon)]\>$ be

[TABLE]

Let $\>N(\lambda),\>N^{0}(\lambda),\>N^{\prime}(\lambda),\>n_{i}(\lambda)\>$ and $\>n_{i(1)}(\lambda)\>$ be numbers of eigenvalues smaller than $\>-\lambda\quad(\lambda>0)\>$ of the operators $\>L,\>L^{0},\>L^{\prime},\>L_{i}\>$ and $\>L_{i(1)}\>$, respectively. Let us write $n_{i},\>n_{i(1)}\>$ instead of $n_{i}(\varepsilon),\>n_{i(1)}(\varepsilon)\>$, respectively.

Şengül [1] proved that the inequalities

[TABLE]

are satisfied, if $\>Q(x)\>$ satisfies the conditions $\>Q1),\>Q2),\>Q3)\>$ and $\>p(x)\>$ satisfies the conditions $\>p1),\>p3)$.

We want to show that the inequalities

[TABLE]

are satisfied. Let $\>u_{1},u_{2},\cdots,u_{n},\cdots\>$ be orthonormal eigenvectors corresponding to the eigenvalues $\>-\lambda_{1},-\lambda_{2},\cdots,-\lambda_{n},\cdots$. Let us consider the following operators:

[TABLE]

Here $\>I\>$ in (6) is identity operator in the space $\>L_{2}(0,\infty;H)\>$; $\>I\>$ in (7) is identity operator in the space $\>L_{2}(0,\psi_{1}(\varepsilon);H)$. We have

[TABLE]

Since the eigenvalues smaller than $\>\lambda\>$ are $\>\mu_{i}=\lambda_{i}+\lambda\quad(i=1,2,\cdots),\>$ from (8)

[TABLE]

is obtained.By the similar way we can show that the number of negative eigenvalues of the operators $\>S^{0}\>\rm{and}\>S^{\prime}\>$ are $\>N^{0}(\lambda)\>\rm{and}\>N^{\prime}(\lambda)$, respectively. Let

[TABLE]

be negative eigenvalues of the operators $\>S^{0}\>\rm{and}\>S^{\prime}\>$ respectively. Let the orthonormal eigenvectors corresponding these eigenvalues be $\varphi_{1},\varphi_{2},\cdots,\varphi_{N^{0}{(\lambda)}}\\ \rm{and}\quad\psi_{1},\psi_{2},\cdots,\psi_{N^{\prime}{(\lambda)}}\>$ respectively.

Lemma 2.1

*If the operator function $\>Q(x)\>$ satisfies the conditions

$\>Q1),\>Q2),\>Q3)\>$ and the function $\>p(x)\>$ satisfies the conditions $\>p1),\>p2)\>$ then*

[TABLE]

**Proof: ** To obtain a contradiction, we suppose that

[TABLE]

Then, there is a non-zero linear combination

[TABLE]

of the functions $\>\varphi_{1},\varphi_{2},\cdots,\varphi_{N^{0}{(\lambda)}}\>$ such that

[TABLE]

By using $\>(12)\>$

[TABLE]

In the similar way as proved in Glazman [10] there exists a vector function $\>\tilde{\varphi}\>$ which has the following properties:

$\tilde{\varphi}1)$ The vector function $\>\tilde{\varphi}=\tilde{\varphi}(x)\>$ has second second order continuous derivative respect to the norm in the space $\>H\>$ in the interval $\>[0,\psi_{1}(\varepsilon)]$.

$\tilde{\varphi}2)$ $\tilde{\varphi}(x)\>$ is equal to zero outside of the interval \>[a,b]\subset\big{(}0,\psi_{1}(\varepsilon)\big{)}.

$\tilde{\varphi}3)$ \Big{|}\Big{(}S^{0}\tilde{\varphi}\>,\>\tilde{\varphi}\Big{)}_{(0,\psi_{1}(\varepsilon))}-\>\Big{(}S^{0}\varphi\>,\>\varphi\Big{)}_{(0,\psi_{1}(\varepsilon))}\Big{|}\><\>-\frac{\alpha}{2}

$\tilde{\varphi}4)$ \Big{(}u_{i}\>,\>\tilde{\varphi}\Big{)}_{(0,\psi_{1}(\varepsilon))}=\>0\quad\big{(}i=1,2,\cdots,N(\lambda)\big{)}.

As it is known,

[TABLE]

Therefore

[TABLE]

By the last inequality,

[TABLE]

is obtained. By $\>(13)\quad\rm{and}\>\>(14)\>$

[TABLE]

is found. On the other hand this result in $\>(15)\>$ contradicts with the property $\tilde{\varphi}$3) . Hence

[TABLE]

Lemma 2.2

If the operator function $\>Q(x)\>$ satisfies the conditions $\>Q1),\>Q2),\>Q3)\>$ and function $\>p(x)\>$ satisfies the conditions $\>p1),\>p2)\>$ then $\>N(\lambda)\>\leq\>N^{\prime}(\lambda)\>$ for all $\>\lambda\in[\varepsilon,\>\infty)).$

Proof: Suppose for contradiction that $\>N(\lambda)\>>\>N^{\prime}(\lambda)$. Then, there is a non-zero linear combination

[TABLE]

of the vector functions $\>u_{1},\>u_{2},\cdots,\>u_{N(\lambda)}\>$ such that

[TABLE]

By using $\>(16)$

[TABLE]

is obtained. We can write the equation $\>(17)\>$ as

[TABLE]

Since

[TABLE]

then we have

[TABLE]

If we consider the equality

[TABLE]

from $\>(19)\>$

[TABLE]

is obtained. From $\>(20)$

[TABLE]

is found. By $\>(21)$

[TABLE]

is obtained. On the other hand, we have

[TABLE]

This result contradicts with (22). Therefore $\>N(\lambda)\leq N^{\prime}(\lambda)\>$. $\Box$

Let $\>-\mu_{i(1)1}\leq-\mu_{i(1)2}\leq-\mu_{i(1)3}\leq\cdots\>$ be eigenvalues of the operator $\>L_{i(1)}\>$ and let we have the following equalities

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Theorem 2.3

If the operator function $\>Q(x)\>$ and the scalar function $\>p(x)\>$ satisfy the conditions $\>Q1)-Q3)\>\rm{and}\>p1)-p3)$, then we have

[TABLE]

for small positive values of $\>\varepsilon$.

Proof : Let us consider the operator $\>L_{i(1)}\>$ which is formed by the differential expression

[TABLE]

with the boundary conditions $\>y(x_{i-1})=y(x_{i})=0$.

We wish to obtain the eigenvalues of the operator $\>L_{i(1)}$. In order to find the eigenvalues, we will solve the eigenvalues problem

[TABLE]

in the space $\>L_{2}(a,b)$. Here, $\>a=x_{i-1},\>b=x_{i}\>$ and $\>d=p(x_{i})$. Moreover, $\>\gamma\>$ is an eigenvalue of the operator $\>Q(b):H\longrightarrow H$. The eigenvalues of boundary-value problem (28) are in the form

[TABLE]

So, the eigenvalues of the operator $\>L_{i(1)}\>$ are of the form

[TABLE]

Since the eigenvalues of the operator $\>Q(x):H\longrightarrow H\>$ are $\>\alpha_{1}(x)\geq\alpha_{2}(x)\geq\cdots\geq\alpha_{j}(x)\geq\cdots\>$ then the eigenvalues of the operator $\>L_{i(1)}\>$ are

[TABLE]

therefore $\>n_{i(1)}\>$ is the number of pairs $\>(m,j)\qquad(m,j\geq 1)\>$ satisfying the inequality

[TABLE]

By using (24), (25) and (29), we obtain

[TABLE]

For the sum $\>\sum\limits_{m=1}^{[|b_{j}(\varepsilon,x_{i})|]-1}a_{j}(x_{i},m)\>$ in (30)

[TABLE]

is obtained. If we consider that the functions $\>\beta_{j}(\varepsilon,x)\quad(j=1,2,\cdots)\>$ are decreasing, from(27), (30) and (31)

[TABLE]

is obtained. $\Box$

Theorem 2.4

If the operator function $\>Q(x)\>$ and the scalar function $\>p(x)\>$ satisfy the conditions $\>Q1)-Q3),p1)-p3)$, then we have

[TABLE]

for small positive values of $\>\varepsilon$.

Here, $\>l_{\varepsilon}=\sum\limits_{\alpha_{j}(0)\geq\varepsilon}1\>$.

Proof : We can easily show that $\>L_{i}<L_{i(1)}\>$. In the case, it is known that

[TABLE]

[11]. On the other hand, from variation principles of R. Courant [12], we have

[TABLE]

From (32) and (33)

[TABLE]

is obtained. From (5) and (34)

[TABLE]

is found. By using (35), we can show that the inequality

[TABLE]

is satisfied. By the Theorem 2.1 and (36)

[TABLE]

is obtained. Since the functions $\>\alpha_{j}(x)\quad(j=1,2,\cdots)\>$ are decreasing, then we have

[TABLE]

From (37) and (38)

[TABLE]

is obtained. By using (27) on the rigth-hand side of inequality (39)

[TABLE]

is found. Here, $\>i_{0}\>$ is a natural number satisfying the following condition:

[TABLE]

By using (27) and (40)

[TABLE]

is obtained. From (24), (25) and (26)

[TABLE]

is found for the expression $\>{1\over\delta}\>\beta_{j}(\varepsilon,x)$. From (27) and (42),

[TABLE]

is obtained. From (3), (41) and (43)

[TABLE]

is found.$\Box$

Let $\>-\mu_{i(1)1}^{\prime}\leq-\mu_{i(1)2}^{\prime}\leq-\mu_{i(1)3}^{\prime}\leq\cdots\>$ be eigenvalues of the operator $\>L_{i(1)}^{\prime}\>$ and $\>n_{i(1)}^{\prime}(\lambda)\>$ be number of the eigenvalues smaller than $\>-\lambda\>\>(\lambda>0)\>$ of the operator $\>L_{i(1)}^{\prime}$. Moreover, we will simply write $\>n_{i(1)}^{\prime}\>$ instead of $\>n_{i(1)}^{\prime}(\varepsilon)$.

Theorem 2.5

If the operator function $\>Q(x)\>$ and the scalar function $\>p(x)\>$ satisfy the conditions $\>Q1)-Q3),\>p1)-p3)\>$ then the inequality

[TABLE]

is satisfied for the small positive values of $\>\varepsilon$.

Proof: The eigenvalues of the operator $\>L_{i(1)}^{\prime}\>$ are in the form

[TABLE]

Therefore $\>n_{i(1)}^{\prime}\>$ is the number of the pairs $\>(m,j)\quad(m,j\geq 1)\>$ satisfying the inequality

[TABLE]

From (24), (25), and (44)

[TABLE]

is found. It is easy to see that

[TABLE]

We consider that the functions $\>\beta_{j}(\varepsilon,x)\quad(j=1,2,\cdots)\>$ are monotone decreasing, by (45) and (46),

[TABLE]

is obtained.$\>\Box$

Let $\>n_{i}^{\prime}(\lambda)\>$ be number of the eigenvalues smaller than $\>-\lambda\quad(\lambda>0)\>$ of the operator $\>L_{i}^{\prime}\>$, $\>-\mu_{1}^{\prime}\leq-\mu_{2}^{\prime}\leq-\mu_{3}^{\prime}\leq\cdots\>$ be eigenvalues of the operator $\>L_{1}^{\prime}\>$ and $\>n_{i}^{\prime}(\varepsilon)=n_{i}^{\prime}$.

Theorem 2.6

If the operator function $\>Q(x)\>$ and the scalar function $\>p(x)\>$ satisfy the conditions $\>Q1)-Q3),\>\rm{and}\>p1)-p3)$, then we have

[TABLE]

for the small values of $\>\varepsilon\>$.

Proof : We can easily show that $\>L_{i}^{\prime}>L_{i(1)}^{\prime}$. In this case we have

[TABLE]

[11]. On the other hand, from variation principles of R. Courant [12], we have

[TABLE]

From (47) and (48),

[TABLE]

is obtained. From (5) and (49)

[TABLE]

is found. By using (50),we have

[TABLE]

By using Theorem 2.3 and (51)

[TABLE]

is found. Since the functions $\>\alpha_{j}(x)\quad(j=1,2,\cdots)\>$ are monotone decreasing, then we have

[TABLE]

From (52) and (53)

[TABLE]

is obtained. Here, $\>i_{0}\>$ is a natural number satisfying the conditions

[TABLE]

From (2)

[TABLE]

From (54) and (55)

[TABLE]

is found.$\Box$

Let

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Let $\>L_{(i)}\>$ be operator in the space $\>L_{2}(0,\delta_{i};H)\>$ which is formed by the expression (1) and with the boundary condition

[TABLE]

Moreover, let $\>L_{(i)}^{(0)}\>$ be operator which is formed by the expression

[TABLE]

and with the boundary condition (58).

Let $\>-\mu_{(i)1}\leq-\mu_{(i)2}\leq\cdots\>$ and $\>-\mu_{(i)1}^{(0)}\leq-\mu_{(i)2}^{(0)}\leq\cdots\>$ be eigenvalues smaller than $\>-\lambda,\quad(\lambda>0)\>$ of the operators $\>L_{(i)}\>$ and $\>L_{(i)}^{(0)}\>$, respectively.

Moreover, let $\>n_{(i)}(\lambda)\>$ and $\>n_{(i)}^{(0)}(\lambda)\>$ be numbers of the eigenvalues smaller than $\>-\lambda,\>(\lambda>0)\>$ of the operators $\>L_{(i)}\>$ and $\>L_{(i)}^{(0)}\>$, respectively.

Since $\>L_{(i)}\geq L_{(i)}^{(0)}\>$, then we have

[TABLE]

[11]. By using (59), we can show that

[TABLE]

Here, $\>n_{(i)}=n_{(i)}(\varepsilon),\quad n_{(i)}^{(0)}=n_{(i)}^{(0)}(\varepsilon)$. $\>\delta_{-1}=\psi_{1}(\varepsilon)\>$ and from the formula (56)

[TABLE]

is obtained. From the last relation, we find

[TABLE]

for the values of $\>\varepsilon\>$ satisfying the inequality $\>\psi_{1}^{a}(\varepsilon)>2$.

Theorem 2.7

If the operator function $\>Q(x)\>$ and the scalar function $\>p(x)\>$ satisfy the conditions $\>Q1)-Q3),\>\rm{and}\>p1)-p3)$, then we have

[TABLE]

for small positive values of $\>\varepsilon$.

Proof : By the similar way to the proof of Theorem 2.6, the following inequality

[TABLE]

can be proved. If we replace the equation (57) in (62), then we have

[TABLE]

If we apply the inequality (63) for the eigenvalues of the operator $\>L_{(i)}\>$, then

[TABLE]

is obtained. From (61) and (64)

[TABLE]

is found. By using (45) and (46)

[TABLE]

is obtained. Moreover, if we use the equation (42), then we get

[TABLE]

From (60), (66) and (67),

[TABLE]

is obtained. By using inequality (56), we find

[TABLE]

Here, $\>i_{0}\in N\>$ is a constant satisfying the condition

[TABLE]

From (68) and (69), we get

[TABLE]

From (61), (63), (65) and (70),

[TABLE]

is found. From (57), (67) and (71),

[TABLE]

is obtained. By the Theorem 2.6 and (72), we have

[TABLE]

is obtained. $\Box$

3 Asymptotic Formulas For The Sum Of

Negative Eigenvalues

In this section, we find asymptotic formulas for the sum $\>\sum\limits_{-\lambda_{i}<-\varepsilon}\lambda_{i}\>$ as $\>\varepsilon\rightarrow+0\>$.

Let us denote the functions of the form $\>\ln_{0}x=x,\quad\ln_{n}x=\ln(\ln_{n-1}x)\>$ by $\>\ln_{n}x\>\quad(n=0,1,2,\cdots)\>$ and we suppose that the function $\>\alpha_{1}(x)=\|Q(x)\|\>$ satisfies the following condition:

$\alpha 1)$ There are a number $\>\xi>0\>$ and a natural number $\>n\geq 1\>$ such that the function $\>\alpha_{1}(x)-(\ln_{n}x)^{-\xi}\>$ is neither negative nor monotone increasing in the interval $\>[b,\infty)\quad(b>0).\>$

Theorem 3.1

If the conditions $\>Q1)-Q3),\>p1)-p3)\>\rm{and}\>\alpha 1)\>$ are satisfied and the series $\>\sum\limits_{j=1}^{\infty}[\alpha_{j}(0)]^{m}\>$ is convergent for a constant $\>m\in(0,\infty)\>$ , then the asymptotic formula

[TABLE]

is satisfied as $\>\varepsilon\rightarrow+0\>$. Here, $\>\beta\>$ is a positive constant.

Proof: By using Theorem 2.4 and Theorem 2.5, we have

[TABLE]

for the small positive values of $\>\varepsilon$. If we take $\>a={1\over 2}\>$ and consider (3)

[TABLE]

is found. Let us take $\>f(\varepsilon)=\psi_{1}(\varepsilon)[\ln\psi_{1}(\varepsilon)]^{-1}$. By using the function $\>p(x)\>$ which satisfies the condition (p1) and the inequality (42)

[TABLE]

is obtained. Şengül showed

[TABLE]

for the small values of $\>\varepsilon>0$, [1]. From (74) and (75)

[TABLE]

is found. From (73) and (76)

[TABLE]

is obtained. Since the series $\>\sum\limits_{m=1}^{\infty}[\alpha_{j}(0)]^{m}\>$ is convergent then we have

[TABLE]

From last inequality

[TABLE]

is found. Since the function $\>\alpha_{1}(x)\>$ satisfy the condition $\alpha 1)$, we have

[TABLE]

for the small values of $\>\varepsilon>0$. From the last inequality above,

[TABLE]

is obtained. From (77), (78) and (79)

[TABLE]

is found. We can rewrite inequality (80)

[TABLE]

as $\>\varepsilon\rightarrow 0$. From (2), (42) and (81)

[TABLE]

as $\>\varepsilon\rightarrow 0$, is obtained. $\Box$

Let us assume that the function $\>\alpha_{1}(x)\>$ satisfies the following condition:

$\alpha 2)$ For every $\>\eta>0\>$

[TABLE]

Here, $\>a_{0}\>$ is a constant in the interval $\>(0,\frac{2}{3})$.

Theorem 3.2

We suppose that the operator function $\>Q(x)\>$, the scalar function $\>p(x)\>$ satisfy the condition $\>Q1)-Q3),\>p1)-p3)\>$ and $\>\alpha_{1}(x)\>$ also satisfies the condition $\alpha 2)$ . In addition the series $\>\sum\limits_{j=1}^{\infty}[\alpha_{j}(0)]^{m}\>$ is convergent for a constant $\>m\>$ satisfying the condition

[TABLE]

then the asymptotic formula

[TABLE]

is satisfied as $\>\varepsilon\rightarrow 0$. Where $\>t_{0}\>$ is a positive constant.

Proof : By Theorem 2.4 and Theorem 2.5, we have

[TABLE]

for the small values of $\>\varepsilon>0$. Since the function $\>\alpha_{1}(x)\>$ is decreasing,

[TABLE]

in the interval $\>[0,\psi_{1}(2\varepsilon)]$. Since the function $\>p(x)\>$ satisfies the condition p1) and (42), (84) then we find

[TABLE]

If we consider that the function $\>\alpha_{1}(x)\>$ satisfies the condition $\alpha 2)$ and

$\>\lim_{\varepsilon\rightarrow 0}\psi_{1}(\varepsilon)=\infty$, then we have

[TABLE]

From the last equality above, we obtain

[TABLE]

for the small value of $\>\varepsilon>0$. From (85) and (86)

[TABLE]

is found. We limit the integral $\>\int\limits_{0}^{\delta}\alpha_{1}^{\frac{3}{2}}(x)\mathrm{d}x\>$ at the right hand side of the inequality (83). Since the function $\>\alpha_{1}(x)\>$ satisfies the condition $\alpha 2)$, then we have

[TABLE]

Therefore we have

[TABLE]

On the other hand, from (3)

[TABLE]

is obtained. If we take $\>x=\psi_{1}(\varepsilon)\>$ in the inequality (88), then we find

[TABLE]

or

[TABLE]

From (89), (90) and (91), we have

[TABLE]

From (78), (91) and (92)

[TABLE]

[TABLE]

are found. From (87), (93) and (94) we obtain

[TABLE]

and

[TABLE]

Here,

[TABLE]

[TABLE]

There is a number $\>\omega=\omega(t)>0\qquad(0<\eta<\omega)\>$ such that

[TABLE]

[TABLE]

for every $\>t>0\>$. If we take

[TABLE]

in the inequalities (97) and (98), then we have

[TABLE]

Since the number $\>m\>$ satisfies the condition (82), we have $\>a\in(0,1)\>$ and $\>t_{0}>0$.

From (83), (95),(96) and (99) we obtain

[TABLE]

By (42), (97) and (100) we have the asymptotic formula

[TABLE]

as $\>\varepsilon\rightarrow 0$. $\Box$

Example 3.3

*Let $\>H=L_{2}[0,\pi]\>$ be a separable Hilbert space and

$\>e_{i}=\sqrt{\frac{2}{\pi}}\>\sin ix\quad(i=1,2,\cdots)\>$ be a standard basis in $\>H\>$. Let $\>Q(x):H\rightarrow H$*

[TABLE]

for all $\>x\in[0,\infty)$. $\>Q(x)\>$ is a self adjoint, completely continuous and positive operator function. The eigenvalues of $\>Q(x)\>$ are in the form

[TABLE]

Here $\>b>e^{3}\>$ is a constant such that $\>\ln x>(\ln\ln x)^{2}$.