It is constructively proved that for class $A_{r,gamma}={qin L_{1,loc}(0,1): qleq 0, int_0^1 rq^gamma,dxleqslant 1}$, where $rin C[0,1]$ is uniformly positive weight and $gamma>1$, there exists a unique potential $hat qin A_{r,gamma}$ such that minim
al eigenvalue $lambda_0(hat q)$ of boundary problem $$-y+hat qy=lambda y, y(0)=y(1)=0 $$ is equal to $M_{r,gamma}=sup_{qin A_{r,gamma}}lambda_0(q)$. For case $gamma=1$ we obtain that there exists a unique potential $hat qinGamma_{r,gamma}$ with analogous property. Here $Gamma_{r,gamma}$ is a closure of $A_{r,gamma}$ in the space $W_{2,loc}^{-1}(0,1)$ of generalized functions.
It is proved that for class $A_gamma={qin L_1[0,1]: qgeq 0, int_0^1 q^gamma,dx=1}$, where $gammain (0,1)$, there exists a potential $q_*in A_gamma$ such that minimal eigenvalue $lambda_1(q_*)$ of boundary problem $$ -y+q_*y=lambda y, y(0)=y(1)=0 $$ i
s equal to $m_gamma=inf_{qin A_gamma}lambda_1(q)$. The equality $m_gamma=1$ for $gammaleq 1-2pi^{-2}$ and the inequality $m_gamma<1$ for $gamma>1-2pi^{-2}$ are also obtained.
We consider integrals $tau_{rho}=int_0^1rhoxi^2,dx$, where $xi$ is Wiener process and $rho$ is generalized function from some class of multipliers. In the case when multiplier $rho$ belongs to the trace-class, it is shown that $tau_{rho}$ has $chi^2$
-distribution (or analogous). An example of multiplier $rho$ not belonging to the trace-class is constructed.
Sturm-Liouville spectral problem for equation $-(y/r)+qy=lambda py$ with generalized functions $rge 0$, $q$ and $p$ is considered. It is shown that the problem may be reduced to analogous problem with $requiv 1$. The case of $q=0$ and self-similar $r$ and $p$ is considered as an example.
On the basis of the theory of Sturm--Liouville problem with distribution coefficients we get the infima and suprema of the first eigenvalue of the problem $-y + (q-lambda) y=0, y(0) -k_0^2 y(0) = y(1) + k_1^2 y(1) = 0$, where $q$ belongs to the set o
f constant-sign summable functions on $[0,1]$ such that $int_0^1 q dx=pm 1$.
The present paper deals with the spectral and the oscillation properties of a linear pencil $A-lambda B$. Here $A$ and $B$ are linear operators generated by the differential expressions $(py)$ and $-y+ cry$, respectively. In particular, it is shown t
hat the negative eigenvalues of this problem are simple and the corresponding eigenfunctions $y_{-n}$ have $n-1$ zeros in $(0,1)$.
Self-adjoint boundary problems for the equation $y^{(4)}-lambdarho y=0$ with generalized derivative $rhoin W_2^{-1}[0,1]$ of self-similar Cantor type function as a weight are considered. Using the oscillating properties of the eigenfunctions, the spe
ctral asymptotics are made more precise then in previous papers.
