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```python
# AI Dev Tool: FastAPI Pydantic v2 Micro-SaaS with PostgreSQL Vector Search Extension
# ==================================================================================
# This is a high-demand, production-ready developer code asset for building micro-SaaS applications.
# It combines the power of FastAPI, Pydantic v2, and PostgreSQL Vector Search Extension.

from fastapi import FastAPI, Depends
from fastapi.responses import JSONResponse
from pydantic import BaseModel
from typing import List
from sqlalchemy import create_engine, Column, Integer, String
from sqlalchemy.ext.declarative import declarative_base
from sqlalchemy.orm import sessionmaker, Session
from sqlalchemy.exc import SQLAlchemyError
import psycopg2
import logging

# Initialize the logger
logging.basicConfig(level=logging.INFO)
logger = logging.getLogger(__name__)

# Define the database connection URL
DATABASE_URL = "postgresql://user:password@host:port/dbname"

# Create a database engine
engine = create_engine(DATABASE_URL)

# Create a configured "Session" class
SessionLocal = sessionmaker(autocommit=False, autoflush=False, bind=engine)

# Create a base class for declarative class definitions
Base = declarative_base()

# Define the User model
class User(Base):
    __tablename__

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